From 42254921757645c16b3c05a1de52d424b385c09c Mon Sep 17 00:00:00 2001 From: Carlos H Villa Pinto Date: Sun, 12 Nov 2017 22:31:24 -0200 Subject: [PATCH] Initial commit --- .gitignore | 11 + .hgignore | 11 + HACKING.txt | 19 + LICENSE | 674 +++++++++++++++++++++++ MANIFEST.in | 2 + NEWS.txt | 9 + README.md | 61 +++ bootstrap.py | 113 ++++ buildout.cfg | 13 + setup.py | 46 ++ setuptools-33.1.1.zip | Bin 0 -> 702770 bytes src/psifa/__init__.py | 7 + src/psifa/api.py | 440 +++++++++++++++ src/psifa/example.py | 45 ++ src/psifa/patternsearch.py | 291 ++++++++++ src/psifa/quasinewton.py | 339 ++++++++++++ src/psifa/utils.py | 270 ++++++++++ src/tests/__init__.py | 3 + src/tests/hstestcase.py | 1033 ++++++++++++++++++++++++++++++++++++ 19 files changed, 3387 insertions(+) create mode 100644 .gitignore create mode 100644 .hgignore create mode 100644 HACKING.txt create mode 100644 LICENSE create mode 100644 MANIFEST.in create mode 100644 NEWS.txt create mode 100644 README.md create mode 100644 bootstrap.py create mode 100644 buildout.cfg create mode 100644 setup.py create mode 100644 setuptools-33.1.1.zip create mode 100644 src/psifa/__init__.py create mode 100644 src/psifa/api.py create mode 100644 src/psifa/example.py create mode 100644 src/psifa/patternsearch.py create mode 100644 src/psifa/quasinewton.py create mode 100644 src/psifa/utils.py create mode 100644 src/tests/__init__.py create mode 100644 src/tests/hstestcase.py diff --git a/.gitignore b/.gitignore new file mode 100644 index 0000000..dfb5e01 --- /dev/null +++ b/.gitignore @@ -0,0 +1,11 @@ +*.pyc + +.installed.cfg +bin +develop-eggs + +*.egg-info + +tmp +build +dist diff --git a/.hgignore b/.hgignore new file mode 100644 index 0000000..c5f1c19 --- /dev/null +++ b/.hgignore @@ -0,0 +1,11 @@ +syntax: glob + +.installed.cfg +bin +develop-eggs + +*.egg-info + +tmp +build +dist diff --git a/HACKING.txt b/HACKING.txt new file mode 100644 index 0000000..5a16441 --- /dev/null +++ b/HACKING.txt @@ -0,0 +1,19 @@ +Development setup +================= + +To create a buildout, + + $ python bootstrap.py + $ bin/buildout + +Release HOWTO +============= + +To make a release, + + 1) Update release date/version in NEWS.txt and setup.py + 2) Run 'python setup.py sdist' + 3) Test the generated source distribution in dist/ + 4) Upload to PyPI: 'python setup.py sdist register upload' + 5) Increase version in setup.py (for next release) + diff --git a/LICENSE b/LICENSE new file mode 100644 index 0000000..94a9ed0 --- /dev/null +++ b/LICENSE @@ -0,0 +1,674 @@ + GNU GENERAL PUBLIC LICENSE + Version 3, 29 June 2007 + + Copyright (C) 2007 Free Software Foundation, Inc. + Everyone is permitted to copy and distribute verbatim copies + of this license document, but changing it is not allowed. + + Preamble + + The GNU General Public License is a free, copyleft license for +software and other kinds of works. + + The licenses for most software and other practical works are designed +to take away your freedom to share and change the works. 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Interpretation of Sections 15 and 16. + + If the disclaimer of warranty and limitation of liability provided +above cannot be given local legal effect according to their terms, +reviewing courts shall apply local law that most closely approximates +an absolute waiver of all civil liability in connection with the +Program, unless a warranty or assumption of liability accompanies a +copy of the Program in return for a fee. + + END OF TERMS AND CONDITIONS + + How to Apply These Terms to Your New Programs + + If you develop a new program, and you want it to be of the greatest +possible use to the public, the best way to achieve this is to make it +free software which everyone can redistribute and change under these terms. + + To do so, attach the following notices to the program. It is safest +to attach them to the start of each source file to most effectively +state the exclusion of warranty; and each file should have at least +the "copyright" line and a pointer to where the full notice is found. + + + Copyright (C) + + This program is free software: you can redistribute it and/or modify + it under the terms of the GNU General Public License as published by + the Free Software Foundation, either version 3 of the License, or + (at your option) any later version. + + This program is distributed in the hope that it will be useful, + but WITHOUT ANY WARRANTY; without even the implied warranty of + MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the + GNU General Public License for more details. + + You should have received a copy of the GNU General Public License + along with this program. If not, see . + +Also add information on how to contact you by electronic and paper mail. + + If the program does terminal interaction, make it output a short +notice like this when it starts in an interactive mode: + + Copyright (C) + This program comes with ABSOLUTELY NO WARRANTY; for details type `show w'. + This is free software, and you are welcome to redistribute it + under certain conditions; type `show c' for details. + +The hypothetical commands `show w' and `show c' should show the appropriate +parts of the General Public License. Of course, your program's commands +might be different; for a GUI interface, you would use an "about box". + + You should also get your employer (if you work as a programmer) or school, +if any, to sign a "copyright disclaimer" for the program, if necessary. +For more information on this, and how to apply and follow the GNU GPL, see +. + + The GNU General Public License does not permit incorporating your program +into proprietary programs. If your program is a subroutine library, you +may consider it more useful to permit linking proprietary applications with +the library. If this is what you want to do, use the GNU Lesser General +Public License instead of this License. But first, please read +. diff --git a/MANIFEST.in b/MANIFEST.in new file mode 100644 index 0000000..6f11de8 --- /dev/null +++ b/MANIFEST.in @@ -0,0 +1,2 @@ +include README.md +include NEWS.txt diff --git a/NEWS.txt b/NEWS.txt new file mode 100644 index 0000000..34fd111 --- /dev/null +++ b/NEWS.txt @@ -0,0 +1,9 @@ +News +==== + +0.1 +--- + +*Release date: 05-Out-2017* + +* Alpha 0.1 released. diff --git a/README.md b/README.md new file mode 100644 index 0000000..d7a6ffb --- /dev/null +++ b/README.md @@ -0,0 +1,61 @@ +# PSIFA + +**PSIFA** (**P**attern **S**earch and **I**mplicit **F**iltering **A**lgorithm) is a derivative-free optimization algorithm developed by Ferreira, Ehrhardt and Santos (2017) that has been designed for linearly constrained problems with noise in the objective function. It combines some elements of the pattern search approach of Lewis and Torczon (2000) with ideas from the implicit filtering method of Kelley (2011). + + + +## Requirements + +- numpy (>= 1.13) +- scipy (>= 0.19) + +## Example of use + +```python +from psifa.api import psifa +import numpy as np +import pprint + +# Objective function. +def f(x): + y = (1. / (27. * np.sqrt(3.))) * ((x[0] - 3.)**2 - 9.) * x[1]**3 + eval_success = True + return y, eval_success + +# Problem constraints. +A = np.array([ + [1. / np.sqrt(3.), -1.], + [1., np.sqrt(3.)], + [-1., -np.sqrt(3.)] +]) +l = np.array([0., 0., -6.]) +u = np.array([np.inf, np.inf, np.inf]) +bl = np.array([0., 0.]) +bu = np.array([np.inf, np.inf]) +n = 2 + +# Initial guess for the solution. +x0 = np.array([1., 0.5]) + +# Find a global optimum using PSIFA. +x, y, iterations, alpha, evaluations, history, exit_cause = \ + psifa(f, x0, A, l, u, n, bl, bu) + +# Show the results. +print('Solution: x = {}'.format(x)) +print('Function value: f(x) = {}'.format(y)) +print('Step length: alpha = {}'.format(alpha)) +print('Number of iterations: {}'.format(iterations)) +print('Function evaluations: {}'.format(evaluations)) +print('Exit cause: {}'.format(exit_cause)) +print('History:') +pprint.PrettyPrinter(indent=2).pprint(history) +``` + +## License + +[GNU GPLv3](LICENSE) diff --git a/bootstrap.py b/bootstrap.py new file mode 100644 index 0000000..63aebb9 --- /dev/null +++ b/bootstrap.py @@ -0,0 +1,113 @@ +############################################################################## +# +# Copyright (c) 2006 Zope Corporation and Contributors. +# All Rights Reserved. +# +# This software is subject to the provisions of the Zope Public License, +# Version 2.1 (ZPL). A copy of the ZPL should accompany this distribution. +# THIS SOFTWARE IS PROVIDED "AS IS" AND ANY AND ALL EXPRESS OR IMPLIED +# WARRANTIES ARE DISCLAIMED, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED +# WARRANTIES OF TITLE, MERCHANTABILITY, AGAINST INFRINGEMENT, AND FITNESS +# FOR A PARTICULAR PURPOSE. +# +############################################################################## +"""Bootstrap a buildout-based project + +Simply run this script in a directory containing a buildout.cfg. +The script accepts buildout command-line options, so you can +use the -c option to specify an alternate configuration file. + +$Id: bootstrap.py 102545 2009-08-06 14:49:47Z chrisw $ +""" + +import os, shutil, sys, tempfile, urllib2 +from optparse import OptionParser + +tmpeggs = tempfile.mkdtemp() + +is_jython = sys.platform.startswith('java') + +# parsing arguments +parser = OptionParser() +parser.add_option("-v", "--version", dest="version", + help="use a specific zc.buildout version") +parser.add_option("-d", "--distribute", + action="store_true", dest="distribute", default=True, + help="Use Disribute rather than Setuptools.") + +options, args = parser.parse_args() + +if options.version is not None: + VERSION = '==%s' % options.version +else: + VERSION = '' + +USE_DISTRIBUTE = options.distribute +args = args + ['bootstrap'] + +to_reload = False +try: + import pkg_resources + if not hasattr(pkg_resources, '_distribute'): + to_reload = True + raise ImportError +except ImportError: + ez = {} + if USE_DISTRIBUTE: + exec urllib2.urlopen('http://python-distribute.org/distribute_setup.py' + ).read() in ez + ez['use_setuptools'](to_dir=tmpeggs, download_delay=0, no_fake=True) + else: + exec urllib2.urlopen('http://peak.telecommunity.com/dist/ez_setup.py' + ).read() in ez + ez['use_setuptools'](to_dir=tmpeggs, download_delay=0) + + if to_reload: + reload(pkg_resources) + else: + import pkg_resources + +if sys.platform == 'win32': + def quote(c): + if ' ' in c: + return '"%s"' % c # work around spawn lamosity on windows + else: + return c +else: + def quote (c): + return c + +cmd = 'from setuptools.command.easy_install import main; main()' +ws = pkg_resources.working_set + +if USE_DISTRIBUTE: + requirement = 'distribute' +else: + requirement = 'setuptools' + +if is_jython: + import subprocess + + assert subprocess.Popen([sys.executable] + ['-c', quote(cmd), '-mqNxd', + quote(tmpeggs), 'zc.buildout' + VERSION], + env=dict(os.environ, + PYTHONPATH= + ws.find(pkg_resources.Requirement.parse(requirement)).location + ), + ).wait() == 0 + +else: + assert os.spawnle( + os.P_WAIT, sys.executable, quote (sys.executable), + '-c', quote (cmd), '-mqNxd', quote (tmpeggs), 'zc.buildout' + VERSION, + dict(os.environ, + PYTHONPATH= + ws.find(pkg_resources.Requirement.parse(requirement)).location + ), + ) == 0 + +ws.add_entry(tmpeggs) +ws.require('zc.buildout' + VERSION) +import zc.buildout.buildout +zc.buildout.buildout.main(args) +shutil.rmtree(tmpeggs) diff --git a/buildout.cfg b/buildout.cfg new file mode 100644 index 0000000..06a9eec --- /dev/null +++ b/buildout.cfg @@ -0,0 +1,13 @@ +[buildout] +parts = python scripts +develop = . +eggs = PSIFA + +[python] +recipe = zc.recipe.egg +interpreter = python +eggs = ${buildout:eggs} + +[scripts] +recipe = zc.recipe.egg:scripts +eggs = ${buildout:eggs} diff --git a/setup.py b/setup.py new file mode 100644 index 0000000..6dd4d1d --- /dev/null +++ b/setup.py @@ -0,0 +1,46 @@ +from setuptools import setup, find_packages +import sys, os + +here = os.path.abspath(os.path.dirname(__file__)) +README = open(os.path.join(here, 'README.md')).read() +NEWS = open(os.path.join(here, 'NEWS.txt')).read() + + +version = '0.1' + +install_requires = [ + # List your project dependencies here. + # For more details, see: + # http://packages.python.org/distribute/setuptools.html#declaring-dependencies + 'numpy', + 'scipy', +] + + +setup(name='PSIFA', + version=version, + description="Python implementation of the Pattern Search Implicit Filtering Algorithm (PSIFA)", + long_description=README + '\n\n' + NEWS, + classifiers=[ + # Get strings from http://pypi.python.org/pypi?%3Aaction=list_classifiers + 'Development Status :: 3 - Alpha', + 'Intended Audience :: Science/Research', + 'Programming Language :: Python :: 2.7', + 'Programming Language :: Python :: 3.6', + 'Topic :: Scientific/Engineering :: Mathematics', + ], + keywords='derivative-free-optimization linearly-constrained-minimization noisy-optimization global-convergence degenerate-constraints', + author='Carlos H. Villa Pinto', + author_email='chvillap@gmail.com', + url='https://github.com/chvillap', + license='GNU GPLv3', + packages=find_packages('src'), + package_dir = {'': 'src'}, + include_package_data=True, + zip_safe=False, + install_requires=install_requires, + entry_points={ + 'console_scripts': + ['PSIFA=psifa:main'] + } +) diff --git a/setuptools-33.1.1.zip b/setuptools-33.1.1.zip new file mode 100644 index 0000000000000000000000000000000000000000..9f4eb152ca3bf6556b118edc2a286cc90e89e2a8 GIT binary patch literal 702770 zcmV(rK<>X#O9KQH000080O6i6N|XZ)C~SrR0A`2*0384T0CQz@b#QcVZ)|fdGcztR zE-^1dNI_0VMN=+vb9A)5+jbjCmL>SkukfO}hh!E25;uZWR#Y#AL`5)57q%!xWLB

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z14z$W{Z7^5+i*@CiFUdGW<#77*VngTBc96dxXWten4&VCoK+XNSH)mqV9uYcK4{cI zj(?%!1$9J%(^qhJJet+Xv~(>GjN{E=+$kB{O+1bNB9=e5{Y?xv%)-FZI0DYdV!K)c z;{QMZe~KUXTheP1x-UyS_+PhA+%MOzk+^+;coI0@vBmXAaX*Q+Mx~Q9`^WTTRb`>1o?T6(X z?Ko|G)&>Z3Cl-5MQy*N`aa(h&!OtOyg~!*Z1D6hNQ;an_Bc8<4!EKR&O9Qth!5WPQ zA7W{&Z%}}XFM-osU=82$6tVbNtp?T;z}G}zjX*ci1Xf!LtjEUJQD6<5&lex_E8ZJ+ re*s)f+&ju^m;!+tV=jAyzi%w-9i;${1~@ 0: + N = null(Aeq, rational=True) + # N = eqnull(Aeq, n) + else: + N = np.array([]) + + if most_general_kind == 'equality': + # In this case the pattern Pk will be the set of columns of N and -N + # (after some normalization). + Pk = np.hstack((N, -N)) + Pk = np.apply_along_axis( + lambda z: z / np.linalg.norm(z, np.inf), 0, Pk) + elif most_general_kind == 'box': + # In this case the pattern Pk will be the set of columns of eye(n) and + # -eye(n). Additional unit vectors are also included for the sake of + # heuristics. + V = np.eye(n) + Pk = np.hstack((V, -V)) + if n > 1: + ones = np.ones((n, 1)) + ones = np.apply_along_axis( + lambda z: z / np.linalg.norm(z, np.inf), 0, ones) + Pk = np.hstack((Pk, ones, -ones)) + elif most_general_kind == 'inequality': + # In this case we find the epsilon-active inequality constraints and + # then build the pattern using a special procedure. + Vin, Veq, Vb, epsilon_active = find_epsilon_active_constraints( + A, x0, l, u, N, alpha, prec3) + Pk, Tb = build_pattern(Vin, Veq, Vb, Aeq, N, epsilon_active) + + # The history of the optimization process is represented as a dictionary of + # lists. Each iteration will add a new element to (some of) these arrays. + history = { + 'xps': [x0], # x provided by the patern search iteration. + 'xqn': [np.nan], # x provided by the quasi-Newton iteration. + 'x': [x0], # Best x among xps and xqn. + 'yps': [y0], # Value of the objective function at xps. + 'yqn': [np.nan], # Value of the objective function at xqn. + 'y': [y0], # Minimum y among yps and yqn. + 'alpha': [alpha0], # Step length. + 'hessian': [hessian], # Hessian approximation at x. + 'grad': [grad], # Gradient approximation at x. + 'x_old': [0.], # Old value for x. + } + + # The set of constraints is also represented as a single dictionary so + # we don't need to send a lot of parameters to some functions. + constrained = { + 'A': A, + 'Aeq': Aeq, + 'eq': eq, + 'l': l, + 'u': u, + 'bl': bl, + 'bu': bu, + 'n': n, + 'N': N, + } + + # ------------------------------------------------------------------------- + # Main iteration. + + # In the end, the final solution and its function value will be stored in + # these two variables. + xk = x0 + yk = y0 + + # The loop continues until we reach the maximum number of evaluations + # of the objective function, or the step length goes below the tolerance, + # or the value of the objective function varies too little from some + # iteration to the next one. + while evaluations < max_evals and alpha > prec1 and y_var > prec2: + iterations += 1 + + y_old = yk + + # This flag decides whether the quasi-Newton iteration needs to be + # computed. Its value can be True only if the pattern search step + # was successful in finding feasible directions at the current x. + quasinewton_needed = False + + # W_stencil is a 2D array whose columns are the directions found at the + # pattern search step. + # y_stencil is a 1D array of function values obtained at the x points + # reached using the directions found by the pattern search step. + # These arrays will be needed in the quasi-Newton iteration. From them + # we'll be able to compute the stencil gradient that approximates the + # derivative of the objective function at x. + W_stencil = np.empty((Pk.shape[0], 0)) + y_stencil = [] + + # --------------------------------------------------------------------- + # Inner iteration (searches for the best direction in Pk). + + # Inner iterations counter. + inner_iterations = 0 + + # Flag that tells whether the search for a candidate x that decreases + # the value of the objective function was successful in this iteration. + search_success = False + + # The loop continues until the search is successful, or the same stop + # conditions of the main loop are satisfied. + while evaluations < max_evals and alpha > prec1 and not search_success: + inner_iterations += 1 + + # Change applied to the objective function's value. + # (Not necessarily a decrease, though) + decrease = -min(alpha**p, 1.) * gamma + eta + + # Iterate over all directions in Pk. + for i in range(Pk.shape[1]): + dk = Pk[:, i] + + # Use the i-th direction and the step length to calculate the + # next candidate point, and then check its feasibility. + x = xk + alpha * dk + is_feasible = check_feasibility( + x, A, Aeq, eq, l, u, bl, bu, prec4) + + # Check the history of previously evaluated x points to avoid + # evaluating the same point twice. + if is_feasible: + found_x, found_dk, position = search_history( + history['x'], W_stencil, x, dk, prec5) + + # If necessary, evaluate the objective function at the new + # trial point x. + if found_x and not found_dk: + y = history['y'][position] + y_stencil.append(y) + W_stencil = np.hstack((W_stencil, dk.reshape(-1, 1))) + eval_success = False + if not found_x: + y, eval_success = f(x) + evaluations += 1 + + # Update the history. + W_stencil = np.hstack((W_stencil, dk.reshape(-1, 1))) + y_stencil.append(y) + history['x'].append(x) + history['y'].append(y) + + # Update x and y if the iteration was successful in + # finding a better trial point. + if y < (y_old + decrease) and \ + eval_success and not search_success: + xp = x + yk = y + search_success = True + if y < yk and eval_success and search_success: + xp = x + yk = y + + # Update the best x and y values if needed, and activate + # the quasi-Newton flag as well. + if y < y_min: + x_min = xk + y_min = yk + quasinewton_needed = True + + # If the search fails, we decrease the step length and reset the + # successes counter (beta) to zero. + if not search_success: + if alpha > alpha_old: + alpha = alpha_old + else: + alpha *= 0.5 + beta = 0 + W_stencil = np.empty((Pk.shape[0], 0)) + y_stencil = [] + + # Update eta for the next iteration. + eta = gamma / (iterations**3) + + # Update the history of pattern search solutions. + history['xps'].append(xp if search_success else xk) + history['yps'].append(yk) + + # print('xps = {}'.format(history['xps'][-1])) + + # --------------------------------------------------------------------- + # Quasi-Newton iteration. + + # Compute the quasi-Newton iteration and save new points/values in + # the histories. + if quasinewton_needed: + if alpha > np.sqrt(prec1): + xq, yq, history, evals, x_new_status = quasinewton( + f, W_stencil, y_stencil, xk, y_old, yk, history, + alpha, constrained, Tb, max_red, prec5) + + evaluations += evals + history['xqn'].append(xq) + history['yqn'].append(yq) + else: + yq = yk + 1. + history['xqn'].append(np.nan) + history['yqn'].append(np.nan) + else: + yq = yk + 1. + history['xqn'].append(np.nan) + history['yqn'].append(np.nan) + + # print('xqn = {}'.format(history['xqn'][-1])) + + # Compare the objective function values at xps and xqn. Then select the + # point whose function value is the lowest. + if yq < yk and x_new_status == 1: + yk = yq + xk = xq + else: + xk = history['xps'][-1] + + # Update the minimums. + if yk <= y_min: + x_min = xk + y_min = yk + + # If the step length needs to be reduced, we set beta to zero in order + # to stop increasing alpha. + if beta > 0 and yk > y_old: + beta = 0 + + # Update the variation in the objective function value. + y_var = abs(yk - y_old) + if alpha > 1e-2: + y_var = 1. + + alpha_old = alpha + if beta == 0 or alpha < 1.: + beta = 1 + else: + # Increase alpha so we may take larger steps in the next + # iterations. + alpha = min(rho**beta * alpha, 10.) + beta += 1 + + # print('xk = {}'.format(xk)) + + # Update the pattern and working set. + if most_general_kind == 'inequality': + Vin, Veq, Vb, epsilon_active = find_epsilon_active_constraints( + A, xk, l, u, N, min(alpha, 1.), prec3) + + Pk, Tb = build_pattern(Vin, Veq, Vb, Aeq, N, epsilon_active) + + # Look for any of the reasons that may cause the optimization process to + # stop. These are: (1) reaching the maximum number of evaluations of the + # objective function; (2) Reaching a very low value for the step length; + # or (3) Having no significant variation in the objective function value + # from one iteration to another. + exit_cause = [] + if alpha < prec1: + exit_cause.append('TP') + if y_var < prec2: + exit_cause.append('VF') + if evaluations >= max_evals: + exit_cause.append('AF') + + # Update values for the next iteration. + x = x_min + y = y_min + + return x, y, iterations, alpha, evaluations, history, exit_cause diff --git a/src/psifa/example.py b/src/psifa/example.py new file mode 100644 index 0000000..8585da5 --- /dev/null +++ b/src/psifa/example.py @@ -0,0 +1,45 @@ +from __future__ import print_function +from psifa.api import psifa +import numpy as np +import pprint + + +# Objective function. +def f(x): + y = (1. / (27. * np.sqrt(3.))) * ((x[0] - 3.)**2 - 9.) * x[1]**3 + eval_success = True + return y, eval_success + + +# Problem constraints. +A = np.array([ + [1. / np.sqrt(3.), -1.], + [1., np.sqrt(3.)], + [-1., -np.sqrt(3.)] +]) +l = np.array([0., 0., -6.]) +u = np.array([np.inf, np.inf, np.inf]) +bl = np.array([0., 0.]) +bu = np.array([np.inf, np.inf]) +n = 2 + +# Initial guess for the solution. +x0 = np.array([1., 0.5]) + +# Find a global optimum using PSIFA. +x, y, iterations, alpha, evaluations, history, exit_cause = \ + psifa(f, x0, A, l, u, n, bl, bu, max_evals=2000, prec1=1e-8) + +# Show the results. +print('------------------------------------------------------------') +print(' Results ') +print('------------------------------------------------------------') +print('Solution: x = {}'.format(x)) +print('Function value: f(x) = {}'.format(y)) +print('Step length: alpha = {}'.format(alpha)) +print('Number of iterations: {}'.format(iterations)) +print('Function evaluations: {}'.format(evaluations)) +print('Exit cause: {}'.format(exit_cause)) +print('History:') +pp = pprint.PrettyPrinter(indent=2) +pp.pprint(history) diff --git a/src/psifa/patternsearch.py b/src/psifa/patternsearch.py new file mode 100644 index 0000000..a3321f9 --- /dev/null +++ b/src/psifa/patternsearch.py @@ -0,0 +1,291 @@ +import numpy as np +from psifa.utils import null + + +def classify_constraints(A, l, u): + """Identifies the different kinds of constraints in A and return them in + separate arrays. + + Parameters + ---------- + A : ndarray + 2D array representing the constraints. + l : ndarray + 1D array of constraint lower bounds. + u : ndarray + 1D array of constraint upper bounds. + + Return values + ------------- + Aeq : ndarray + 2D array of equality constraints. + A : ndarray + 2D array of inequality constraints. + l : ndarray + 1D array of inequality constraint lower bounds. + u : ndarray + 1D array of inequality constraint upper bounds. + eq : ndarray + 1D array of equality constraint values. + most_general_kind : str + Name the most general kind of constraint originally contained in `A`. + Values can be: + `'box'`: Box constraints; + `'equality'`: Equality constraints; + `'inequality'` Inequality constraints. + """ + + Aeq = np.array([]) + eq = np.array([]) + + # Only box constraints exist if A is empty. + if A.size == 0: + most_general_kind = 'box' + else: + # Locate equality and inequality constraints. + ind_eq = l == u + ind_in = l != u + + # Leave only equality constraints for Aeq and eq. + if np.any(ind_eq): + Aeq = A[ind_eq, :] + eq = u[ind_eq] + + # Leave only inequality constraints for A, l and u. + A = A[ind_in, :] + l = l[ind_in] + u = u[ind_in] + + # If A is empty at this point, we have no inequality constraints. + if A.size == 0: + most_general_kind = 'equality' + else: + most_general_kind = 'inequality' + + return Aeq, A, l, u, eq, most_general_kind + +# ----------------------------------------------------------------------------- + + +def check_feasibility(x, A, Aeq, eq, l, u, bl, bu, prec4): + """Checks whether a trial point `x` is feasible or not according to the + problem's constraints. + + Parameters + ---------- + x : ndarray + Current trial point as a 1D array. + A : ndarray + 2D array of inequality constraints. + Aeq : ndarray + 2D array of equality constraints. + eq : ndarray + 1D array of equality constraint values. + l : ndarray + 1D array of inequality constraint lower bounds. + u : ndarray + 1D array of inequality constraint upper bounds. + bl : ndarray + 1D array of box constraint lower bounds. + bu : ndarray + 1D array of box constraint upper bounds. +. prec4 : float + Tolerance for violating constraints. + + Return value + ------------ + is_feasible : bool + `True` if `x` is a feasible point; `False` otherwise. + """ + + # Check whether any inequality constraint is being violated at x. + # Some tolerance is allowed to compensate eventual numerical + # roundoff errors. + if A.size > 0: + b = np.dot(A, x) + l_ok = l <= b + prec4 + u_ok = u >= b + prec4 + else: + l_ok = np.array([True]) + u_ok = np.array([True]) + + # Analogously, check the equality constraints too. + eq_ok = True + if Aeq.size > 0: + b = np.dot(Aeq, x) + eq_ok = np.linalg.norm(b - eq, np.inf) < prec4 + + # Finally, check the box constraints. + bl_ok = x >= bl - prec4 + bu_ok = x <= bu + prec4 + + # The trial point is feasible if it passes all tests. + is_feasible = np.all(l_ok) and np.all(u_ok) and \ + np.all(bl_ok) and np.all(bu_ok) and eq_ok + + return is_feasible + +# ----------------------------------------------------------------------------- + + +def find_epsilon_active_constraints(A, x, l, u, Z, tol, prec3): + """Identifies all epsilon-active constraints at the current trial point `x` + return them in separate arrays according to their kinds. + + Parameters + ---------- + A : ndarray + 2D array of inequality constraints. + x : ndarray + Current trial point as a 1D array. + l : ndarray + 1D array of inequality constraint lower bounds. + u : ndarray + 1D array of inequality constraint upper bounds. + Z : ndarray + Null space of the array of equality constraints. + tol : float + Step length tolerance for the epsilon-active inequality constraints. + prec3 : float + Numerical tolerance that decides whether a constraint can be + temporarily taken as an equality constraint. + + Return values + ------------- + Vin : ndarray + 2D array of epsilon-active inequality constraints. + Veq : ndarray + 2D array of epsilon-active equality constraints. + Vb : ndarray + 2D array of epsilon-active inequality constraints that were temporarily + taken as equality constraints. + epsilon_active: bool + `True` if any constraint is epsilon-active at `x`; `False` otherwise. + """ + + Vin = np.array([]) + Veq = np.array([]) + Vb = np.array([]) + + # Inner products. + b = np.dot(A, x) + c = np.dot(Z.T, A) if Z.size > 0 else 1. + + # Find the epsilon-active inequality constraints. + # OBS: np.where() returns a 1-sized tuple with the result inside it. + il = np.where(np.abs((b - l) / c) <= tol)[0] + iu = np.where(np.abs((b - u) / c) <= tol)[0] + + # Find the inequality constraints that are active both at the lower bound + # and at the upper bound (thus being temporarily taken as equality + # constraints in order to avoid making the working set degenerate). + eq = np.intersect1d(il, iu) + + # Remove the constraints temporarily taken as equality constraints from + # the lists of inequality constraint indices. + il = np.setdiff1d(il, eq, assume_unique=True) + iu = np.setdiff1d(iu, eq, assume_unique=True) + # for j in range(eq.size): + # il = il[il != eq[j]] + # iu = iu[iu != eq[j]] + + # If at least one inequality constraint has reached one of its bounds, we + # add to the pattern the directions that are in the null space of such + # constraints. This way we allow the trial point to move over the bound's + # face. + le = np.where(np.abs((b - l) / c) <= prec3)[0] + ue = np.where(np.abs((b - u) / c) <= prec3)[0] + + # Now that the epsilon-active constraints are known, use this information + # to compute the matrices of generators that later we use to calculate the + # feasible directions. + if eq.size == 0: + Vin = np.hstack((-A[il, :].T, A[iu, :].T)) + if eq.size > 0 or le.size > 0 or ue.size > 0: + Vin = np.hstack((-A[il, :].T, A[iu, :].T)) + if np.any(eq): + Veq = A[eq, :].T + Vb = np.hstack((A[le, :].T, A[ue, :].T)) + + # If all matrices are empty (no epsilon-active constraint at x), set Vin + # as the identity matrix. The others remain empty. + if Vin.size == 0 and Veq.size == 0 and Vb.size == 0: + Vin = np.eye(A.shape[1]) + epsilon_active = False + else: + epsilon_active = True + + return Vin, Veq, Vb, epsilon_active + +# ----------------------------------------------------------------------------- + + +def build_pattern(Vin, Veq, Vb, Aeq, Z, epsilon_active): + """Builds a pattern of feasible directions from the current trial point, + given the current working set. + + Parameters + ---------- + Vin : ndarray + 2D array of epsilon-active inequality constraints. + Veq : ndarray + 2D array of epsilon-active equality constraints. + Vb : ndarray + 2D array of epsilon-active inequality constraints that were temporarily + taken as equality constraints. + Aeq : ndarray + 2D array of equality constraints. + Z : ndarray + Null space of the array of equality constraints. + epsilon_active : bool + `True` if any constraint is epsilon-active at `x`; `False` otherwise. + + Return values + ------------- + Pk : ndarray + 2D array whose columns represent the feasible directions from `x`. + Tb : ndarray + Null space of `Vb`. + """ + + m = Vin.shape[0] + Tb = np.array([]) + + # If no epsilon-active constraint exists, build a pattern that contains + # only eye(n) and -eye(n), since we can move to any direction from the + # current point. + # OBS: notice that Vin == eye(n) when epsilon_active == False. + if not epsilon_active: + ones = np.ones((m, 1)) + Pk = np.hstack((Vin, -Vin, ones, -ones)) + else: + # Build a basis for the null space of each matrix of epsilon-active + # constraints. + if Veq.size > 0: + Aeq = np.vstack((Aeq, Veq.T)) + Teq = null(Aeq) + else: + Teq = Z + Tb = null(Vb.T) + Tin = null(Vin.T) + F = np.dot(Vin, np.linalg.pinv(np.dot(Vin.T, Vin))) + + # Build a pattern composed by the generated bases. + # OBS: empty matrices can't be stacked as in MATLAB, so these + # conditionals below are necessary. + Pk = np.empty((m, 0)) + if Teq.size > 0: + Pk = np.hstack((Pk, -Teq, Teq)) + if Tin.size > 0: + Pk = np.hstack((Pk, Tin, -Tin)) + if F.size > 0: + Pk = np.hstack((Pk, F, -F)) + if Tb.size > 0: + Pk = np.hstack((Pk, Tb, -Tb)) + + if Pk.size > 0: + # Normalize the feasible directions (columns of Pk). + Pk = np.apply_along_axis( + lambda z: z / np.linalg.norm(z), 0, Pk) + + return Pk, Tb diff --git a/src/psifa/quasinewton.py b/src/psifa/quasinewton.py new file mode 100644 index 0000000..c652330 --- /dev/null +++ b/src/psifa/quasinewton.py @@ -0,0 +1,339 @@ +import numpy as np +from psifa.utils import search_history + + +def quasinewton(f, W_stencil, y_stencil, x, y, yps, history, alpha, + constrained, Tb, max_red, prec5): + """Performs the quasi-Newton iteration of PSIFA, which obtains a new + trial point `x`. + + Parameters + ---------- + f : function + Objective function. + W_stencil : ndarray + 2D array of directions found in the pattern search step. + y_stencil : ndarray + 1D array of function values obtained at the points reached using + the directions found in the pattern search step. + x : ndarray + Current trial point as a 1D array. + y : float + Objective function value at `x`. + yps : float + Objective function value at `xps` (pattern search solution). + history : dict + Historical information of the optimization process. + alpha : float + Step length. + constrained : dict + Constraints and other problem info. + Tb : ndarray + Null space of the epsilon-active inequality constraints temporarily + taken as equality constraints. + max_red : float + Maximum number of reductions in the line search step. + prec5 : float + Maximum distance to consider when checking whether a trial point was + previously evaluated before. + + Return values + ------------- + x_new : ndarray + New trial point resulting from the quasi-Newton step. + y_new : float + Objective function value at `x_new`. + history : dict + Updated history. + evals : int + Number of internal evaluations of the objective function. + x_new_status : int + Flag that tells how the quasi-Newton solution compares to the current + trial point and the pattern search solution (`xps`). Values can be: + `0` : `x_new` is not better than any of them. + `1` : `x_new` is a better solution than `x`. + `2` : `x_new` is not better than `x`, but it is better than `xps`. + """ + + x_new_status = 0 + evals = 0 + x_new = x + x_history = history['x'] + y_history = history['y'] + + # List of points the objective function was evaluated at, and the values at + # these points. + evaluated_x = [] + evaluated_y = [] + + # Compute the stencil gradient at x. + # Depending on the gradient value, we either may or may not update our + # definition of the previous value for x. + Df, grad, zero_grad = stencil_gradient( + W_stencil, y_stencil, y, alpha, history['grad']) + x_old = x if zero_grad else history['x_old'] + + # Compute the BFGS update to obtain a (possibly) feasible direction + # from the stencil gradient. + hessian = history['hessian'][-1] + dk, hessian = bfgs_direction(hessian, x, x_old, Df, grad) + + # As the BFGS direction still may be unfeasible because of the constraints, + # we perform the ratio test to obtain both a feasible (projected) direction + # and the step length we may take along it. + keep_trying = True + attempt = 1 + while keep_trying: + if attempt == 2: + keep_trying = False + + dk, alpha = ratio_test(constrained, x, dk, Tb) + + # --------------------------------------------------------------------- + # Line search + + # We have no guarantee that the direction provided is going downwards, + # so in the line search step we check the opposite direction too. + # The number of reductions in the step length is limited because we + # don't want to evaluate the objective function too many times. + step_reductions = 0 # j + while alpha > 0 and step_reductions < max_red: + # Get a new trial point and the corresponding function value at it. + x_new = x + alpha * dk + + # Look for the new point in the history and evaluate the objective + # function at such point if needed. + unused = np.array([]) + found_x, _, position = search_history( + x_history, unused, x_new, unused, prec5) + if found_x: + y_new = y_history[position] + else: + y_new, eval_success = f(x_new) + evaluated_x.append(x_new) + evaluated_y.append(y_new) + evals += 1 + + # If we get a lower value for the objective function at this new + # point, mark the search as successful. + if y_new < y and (found_x or eval_success): + y = y_new + x_old = x + x = x_new + alpha = 0 + keep_trying = False + x_new_status = 1 + elif y_new < yps: + xt = x_new + yps = y_new + x_new_status = 2 + + # Reduce alpha and try again. + alpha *= 0.5 + step_reductions += 1 + + # Now try the opposite direction, if needed. + dk = -dk + attempt += 1 + + # Set the outputs. + y_new = y + if x_new_status == 2: + y_new = yps + x_old = x + x = xt + x_new_status = 1 + x_new = x + + # Update the history. + history['x'] += evaluated_x + history['y'] += evaluated_y + history['x_old'] = x_old + + return x_new, y_new, history, evals, x_new_status + +# ----------------------------------------------------------------------------- + + +def stencil_gradient(W_stencil, y_stencil, yps, alpha, grad): + """Computes the stencil gradient at `x` from a set of feasible directions + and a particular step length. + + Parameters + ---------- + W_stencil : ndarray + 2D array of directions found in the pattern search step. + y_stencil : ndarray + 1D array of function values obtained at the points reached using + the directions found in the pattern search step. + yps : float + Objective function value at `xps` (pattern search solution). + alpha : float + Step length. + grad : float + Previously known gradient approximation at `x`. + + Return values + ------------- + Df : ndarray + Stencil gradient for the given `x`, `alpha` and `W_stencil`. + grad : ndarray + Previous stencil gradient or the same as Df if `grad_computed == True`. + grad_computed : bool + `True` if `grad` had not been computed before and now it has; `False` + otherwise. + """ + + grad_computed = False + + # The stencil gradient is computed using the pseudoinverse of W_stencil + # and differences of previously computed function values. + W_pinv = np.linalg.pinv(W_stencil) + delta = y_stencil - yps + Df = np.dot(delta, W_pinv) / alpha + + # If grad is zero, it means it hasn't been computed before. + # So grad is actually computed only once (set as Df). + if grad == 0.: + grad = Df + grad_computed = True + + return Df, grad, grad_computed + +# ----------------------------------------------------------------------------- + + +def bfgs_direction(hessian, x, x_old, Df, grad): + """Obtains a (possibly) feasible quasi-Newton direction at `x` using the + BFGS update. + + Parameters + ---------- + hessian : ndarray + Approximation for the Hessian matrix at `x`. + x : ndarray + Current trial point. + x_old : ndarray + Previous trial point. + Df : ndarray + Stencil gradient at `x`. + grad : ndarray + Gradient approximation at `x`. + + Return values + ------------- + dk : ndarray + Direction obtained by the BFGS algorithm. + hessian : ndarray + New approximation for the Hessian matrix. + """ + + # Update the Hessian matrix for the current trial point x. + # OBS: do not update if any denominator is zero. + s = np.atleast_2d(x - x_old).T + gdiff = np.atleast_2d(Df - grad).T + HS = np.dot(hessian, s) + + if 0 < np.linalg.norm(gdiff) < np.inf and \ + 0 < np.linalg.norm(s) < np.inf and \ + 0 < np.linalg.norm(HS) < np.inf and \ + np.dot(gdiff.T, s) != 0 and \ + np.dot(s.T, HS) != 0: + hessian += np.dot(gdiff, gdiff.T) / \ + np.dot(gdiff.T, s) - np.dot(HS, HS.T) / np.dot(s.T, HS) + + # Check the condition number of the Hessian matrix in order to find out + # whether it is ill-conditioned (the computation of its inverse, or + # solution of a linear system of equations, is prone to large numerical + # errors). + if np.linalg.cond(hessian) > 1e6: + hessian = np.eye(hessian.shape[0]) + + # The BFGS direction is finally obtained from the solution of the + # following linear system. + dk = np.linalg.solve(hessian, -Df) + + return dk, hessian + +# ----------------------------------------------------------------------------- + + +def ratio_test(constrained, x, dk, Tb): + """Performs the ratio test and provides both a feasible (projected) + direction and the step length we may take along it. + + Parameters + ---------- + constrained : dict + Constraints and other problem info. + x : ndarray + Current trial point as a 1D array. + dk : ndarray + BFGS direction. + Tb : ndarray + Null space of the epsilon-active inequality constraints temporarily + taken as equality constraints. + + Return values + ------------- + dk : ndarray + Projected direction. + step : float + Step length we may take along `dk`. + """ + + A = constrained['A'] + Aeq = constrained['Aeq'] + l = constrained['l'] + u = constrained['u'] + bl = constrained['bl'] + bu = constrained['bu'] + N = constrained['N'] + + # Numerical tolerance. + tolr = 2. * np.finfo(float).eps + + # Ratio test parameters. + t1 = 1. + t2 = 1. + + # Project the quasi-Newton direction onto the nullspace of the constraint + # array, in order to obtain a feasible direction. + if Aeq.size > 0: + if Tb.size > 0: + TbN = np.hstack((Tb, N)) + dk = np.dot(np.dot(TbN, TbN.T), dk) + else: + dk = np.dot(np.dot(N, N.T), dk) + elif Tb.size > 0: + dk = np.dot(np.dot(Tb, Tb.T), dk) + + # Perform the ratio test for inequality constraints. + if A.size > 0: + Ad = np.dot(A, dk) + Ax = np.dot(A, x) + + t1_ind = Ad > 0 + t2_ind = Ad < 0 + t1 = np.inf + t2 = np.inf + if np.any(t1_ind): + t1 = ((u[t1_ind] - tolr) - Ax[t1_ind]) / Ad[t1_ind] + if np.any(t2_ind): + t2 = ((l[t2_ind] + tolr) - Ax[t2_ind]) / Ad[t2_ind] + + # Perform the ratio test for box constraints. + t3_ind = dk < 0 + t4_ind = dk > 0 + t3 = np.inf + t4 = np.inf + if np.any(t3_ind): + t3 = np.min((x[t3_ind] - (bl[t3_ind] + tolr)) / -dk[t3_ind]) + if np.any(t4_ind): + t4 = np.min((x[t4_ind] - (bu[t4_ind] - tolr)) / -dk[t4_ind]) + + # Get the greatest step size that we can take along the quasi-Newton + # direction (or its projection). + step = np.min(np.hstack((t1, t2, t3, t4, 1.))) + + return dk, step diff --git a/src/psifa/utils.py b/src/psifa/utils.py new file mode 100644 index 0000000..b19652b --- /dev/null +++ b/src/psifa/utils.py @@ -0,0 +1,270 @@ +import numpy as np +from scipy.linalg import qr +from fractions import Fraction + + +def preprocess_constraints(A, l, u): + """Normalizes constraints and their lower/upper bounds using a common factor. + + Parameters + ---------- + A : ndarray + 2D array representing the constraints. + l : ndarray + 1D array of constraint lower bounds. + u : ndarray + 1D array of constraint upper bounds. + + Return values + ------------- + A : ndarray + Normalized constraints. + l : ndarray + Normalized constraint lower bounds. + u : ndarray + Normalized constraint upper bounds. + """ + + # Normalizes each row independently. + for i in range(A.shape[0]): + norm = np.linalg.norm(A[i, :], np.inf) + A[i, :] /= norm + l[i] /= norm + u[i] /= norm + + return A, l, u + +# ----------------------------------------------------------------------------- + + +def search_history(x_history, W_stencil, x, dk, prec5): + """Searches the history in order to find out whether the objective function + was already evaluated at a given trial point `x`. + + Parameters + ---------- + x_history : list + List of arrays representing the `x` points the objective function was + already evaluated at. + W_stencil: ndarray + 2D array of directions already evaluated in the current iteration of + the main loop. + x : ndarray + Current trial point as a 1D array. + dk : ndarray + Current candidate direction as a 1D array. + prec5 : float + Maximum distance used to assess equality between `x` and the points in + the history. + + Return values + ------------- + found_x : bool + `True` if `x` was found in the history; `False` otherwise. + found_dk : bool + `True` if `dk` was already evaluated in the current iteration of the + main loop; `False` otherwise. + position : int + Index of `x` in the history, or -1 if it was never evaluated before. + """ + + found_x = False + found_dk = False + position = -1 + + # First check if dk was already evaluated in the current iteration of the + # main loop. If it was, there is no need to search x_history. + if W_stencil.size > 0: + for i in range(W_stencil.shape[1]): + diff = W_stencil[:, i] - dk + if np.linalg.norm(diff, np.inf) < prec5: + found_x = True + found_dk = True + break + + # If dk was not found, we look for x in the history. + if not found_dk: + for i in range(len(x_history)): + diff = x_history[i] - x + if np.linalg.norm(diff, np.inf) < prec5: + found_x = True + position = i + break + + return found_x, found_dk, position + +# ----------------------------------------------------------------------------- + + +def null(A, rational=False): + """Computes an orthonormal basis for the null space of `A`. + + Parameters + ---------- + A : ndarray + Input array. + r : bool + If `True`, returns a "rational" basis obtained from the reduced + row echelon form. + + Return value + ------------ + Z : ndarray + Orthonormal basis for the null space of A. + """ + + Z = np.array([]) + + if A.size > 0: + A = np.atleast_2d(A) + m, n = A.shape + + if rational: + # Rational basis. + R, pivcol = rref(A) + r = pivcol.size + nopiv = np.delete(np.arange(n), pivcol) + Z = np.zeros((n, n - r), dtype=A.dtype) + if n > r: + Z[nopiv, :] = np.eye(n - r, dtype=A.dtype) + if r > 0: + Z[pivcol, :] = -R[:r, nopiv] + else: + # Orthonormal basis. + _, S, V = np.linalg.svd(A, full_matrices=m <= n) + if m > 1: + s = np.diag(S) + elif m == 1: + s = S[0] + else: + s = 0. + tol = max(m, n) * np.max(s) * np.finfo(A.dtype).eps + r = np.sum(s > tol) + Z = V[:, r:n] + + return Z + +# ----------------------------------------------------------------------------- + + +def rref(A, tol=None): + """Produces the reduced row echelon form of `A`. + + Parameters + ---------- + A : ndarray + Input array. + tol: float + Tolerance to be used in the rank tests. + + Return value + ------------ + R : ndarray + Reduced row echelon form of `A`. + jb : ndarray + A vector so that: + r = length(jb) is this algorithm's idea of the rank of A; + x[jb] are the bound variables in a linear system, Ax = b; + A[:,jb] is a basis for the range of A; + R[1:r,jb] is the r-by-r identity matrix. + """ + + R = A.copy() + m, n = R.shape + + # Vectorized version of the rat() function. + vec_rat = np.vectorize(rat) + + # Does it appear that elements of A are ratios of small integers? + num, den = vec_rat(R) + rats = np.allclose(R, num / den) + + # Compute the default tolerance if none was provided. + if tol is None: + tol = max(m, n) * np.finfo(R.dtype).eps * np.linalg.norm(R, np.inf) + + # Loop over the entire matrix. + i = j = 0 + jb = [] + while i < m and j < n: + # Find value and index of largest element in the remainder of column j. + p = np.max(np.abs(R[i:m, j])) + k = np.argmax(np.abs(R[i:m, j])) + i + + if p <= tol: + # The column is negligible, zero it out. + R[i:m, j] = np.zeros((m - i,)) + j += 1 + else: + # Remember the column index. + jb.append(j) + # Swap i-th and k-th rows. + R[[i, k], j:n] = R[[k, i], j:n] + # Divide the pivot row by the pivot element. + R[i, j:n] = R[i, j:n] / R[i, j] + # Subtract multiples of the pivot row from all the other rows. + for k in np.append(np.arange(i), np.arange(i + 1, m)): + R[k, j:n] -= np.dot(R[k, j], R[i, j:n]) + i += 1 + j += 1 + + # Return "rational" numbers if appropriate. + if rats: + num, den = vec_rat(R) + R = num / den + + return R, np.array(jb) + +# ----------------------------------------------------------------------------- + + +def rat(x): + """Produces a rational fraction approximation for `x`. + + Parameters + ---------- + x : float + Input floating point number. + + Return value + ------------ + num : int + Fraction numerator. + den : int + Fraction denominator. + """ + + f = Fraction(x).limit_denominator() + num, den = f.numerator, f.denominator + + return num, den + +# ----------------------------------------------------------------------------- + + +def eqnull(Aeq, ndims): + """Computes the null space of a given 2D array of equality constraints. + + Parameters + ---------- + Aeq : ndarray + 2D array of equality constraints. + ndims : int + Dimensionality of the optimization space. + + Return value + ------------ + Anull : ndarray + Null space of `Aeq`. + """ + + Anull = np.eye(ndims) + + if Aeq.size > 0: + Q, _ = qr(Aeq.T, mode='economic') + if Aeq.shape[0] <= ndims: + Anull = np.eye(ndims) - np.dot(Q, Q.T) + else: + raise ValueError('Infeasible') + + return Anull diff --git a/src/tests/__init__.py b/src/tests/__init__.py new file mode 100644 index 0000000..b62be50 --- /dev/null +++ b/src/tests/__init__.py @@ -0,0 +1,3 @@ +__doc__ = """ +Unit tests for PSIFA. +""" diff --git a/src/tests/hstestcase.py b/src/tests/hstestcase.py new file mode 100644 index 0000000..5c6a006 --- /dev/null +++ b/src/tests/hstestcase.py @@ -0,0 +1,1033 @@ +from __future__ import print_function +import unittest +import numpy as np +from psifa.api import psifa + + +class HSTestCase(unittest.TestCase): + """Test case with a collection of optimization problems from + Hock and Schittkowski (1981). + """ + + def test_problem_HS01(self): + def f(x): + y = 100. * (x[1] - x[0]**2)**2 + (1. - x[0])**2 + eval_success = True + return y, eval_success + + A = np.array([]) + l = np.array([]) + u = np.array([]) + bl = np.array([-np.inf, -1.5]) + bu = np.array([np.inf, np.inf]) + n = 2 + x0 = np.array([-2., 1.]) + xc = np.array([1., 1.]) + yc = 0. + + x1, y1, iterations, alpha, evaluations, history, exit_cause = \ + psifa(f, x0, A, l, u, n, bl, bu, max_evals=2000, prec1=1e-8) + + self.assertTrue(np.allclose(x1, xc) or np.isclose(y1, yc) or + np.allclose(xc, x1) or np.isclose(yc, y1) or + y1 < yc) + self.assertTrue(iterations <= 130) + self.assertTrue(evaluations <= 498) + self.assertTrue('VF' in exit_cause) + + # ------------------------------------------------------------------------- + + def test_problem_HS03(self): + def f(x): + y = x[1] + 10.**(-5) * (x[1] - x[0])**2 + eval_success = True + return y, eval_success + + A = np.array([]) + l = np.array([]) + u = np.array([]) + bl = np.array([-np.inf, 0.]) + bu = np.array([np.inf, np.inf]) + n = 2 + x0 = np.array([10., 1.]) + xc = np.array([0., 0.]) + yc = 0. + + x1, y1, iterations, alpha, evaluations, history, exit_cause = \ + psifa(f, x0, A, l, u, n, bl, bu, max_evals=2000, prec1=1e-8) + + self.assertTrue(np.allclose(x1, xc) or np.isclose(y1, yc) or + np.allclose(xc, x1) or np.isclose(yc, y1) or + y1 < yc) + self.assertTrue(iterations <= 13) + self.assertTrue(evaluations <= 102) + self.assertTrue('VF' in exit_cause) + + # ------------------------------------------------------------------------- + + def test_problem_HS04(self): + def f(x): + y = 1. / 3. * (x[0] + 1.)**3 + x[1] + eval_success = True + return y, eval_success + + A = np.array([]) + l = np.array([]) + u = np.array([]) + bl = np.array([1., 0.]) + bu = np.array([np.inf, np.inf]) + n = 2 + x0 = np.array([1.125, 0.125]) + xc = np.array([1., 0.]) + yc = 8. / 3. + + x1, y1, iterations, alpha, evaluations, history, exit_cause = \ + psifa(f, x0, A, l, u, n, bl, bu, max_evals=2000, prec1=1e-8) + + self.assertTrue(np.allclose(x1, xc) or np.isclose(y1, yc) or + np.allclose(xc, x1) or np.isclose(yc, y1) or + y1 < yc) + self.assertTrue(iterations <= 12) + self.assertTrue(evaluations <= 105) + self.assertTrue('TP' in exit_cause) + + # ------------------------------------------------------------------------- + + def test_problem_HS05(self): + def f(x): + y = np.sin(x[0] + x[1]) + (x[0] - x[1])**2 - 1.5 * x[0] + \ + 2.5 * x[1] + 1. + eval_success = True + return y, eval_success + + A = np.array([]) + l = np.array([]) + u = np.array([]) + bl = np.array([-1.5, -3.]) + bu = np.array([4., 3.]) + n = 2 + x0 = np.array([0., 0.]) + xc = np.array([-np.pi / 3. + 0.5, -np.pi / 3. - 0.5]) + yc = -0.5 * np.sqrt(3.) - np.pi / 3. + + x1, y1, iterations, alpha, evaluations, history, exit_cause = \ + psifa(f, x0, A, l, u, n, bl, bu, max_evals=2000, prec1=1e-8) + + self.assertTrue(np.allclose(x1, xc) or np.isclose(y1, yc) or + np.allclose(xc, x1) or np.isclose(yc, y1) or + y1 < yc) + self.assertTrue(iterations <= 59) + self.assertTrue(evaluations <= 247) + self.assertTrue('VF' in exit_cause) + + # ------------------------------------------------------------------------- + + def test_problem_HS09(self): + def f(x): + y = np.sin(np.pi * x[0] / 12.) * np.cos(np.pi * x[1] / 16.) + eval_success = True + return y, eval_success + + A = np.array([[4., -3.]]) + l = np.array([0.]) + u = np.array([0.]) + bl = np.array([-np.inf, -np.inf]) + bu = np.array([np.inf, np.inf]) + n = 2 + x0 = np.array([0., 0.]) + xc = np.array([-3., -4.]) + yc = -0.5 + + x1, y1, iterations, alpha, evaluations, history, exit_cause = \ + psifa(f, x0, A, l, u, n, bl, bu, max_evals=2000, prec1=1e-8) + + self.assertTrue(np.allclose(x1, xc) or np.isclose(y1, yc) or + np.allclose(xc, x1) or np.isclose(yc, y1) or + y1 < yc) + self.assertTrue(iterations <= 14) + self.assertTrue(evaluations <= 50) + self.assertTrue('VF' in exit_cause) + + # ------------------------------------------------------------------------- + + def test_problem_HS21(self): + def f(x): + y = 0.01 * x[0]**2 + x[1]**2 - 100. + eval_success = True + return y, eval_success + + A = np.array([[10., -1.]]) + l = np.array([10.]) + u = np.array([np.inf]) + bl = np.array([2., -50.]) + bu = np.array([50., 50.]) + n = 2 + x0 = np.array([2., 2.]) + xc = np.array([2., 0.]) + yc = -99.96 + + x1, y1, iterations, alpha, evaluations, history, exit_cause = \ + psifa(f, x0, A, l, u, n, bl, bu, max_evals=2000, prec1=1e-8) + + self.assertTrue(np.allclose(x1, xc) or np.isclose(y1, yc) or + np.allclose(xc, x1) or np.isclose(yc, y1) or + y1 < yc) + self.assertTrue(iterations <= 18) + self.assertTrue(evaluations <= 179) + self.assertTrue('VF' in exit_cause) + + # ------------------------------------------------------------------------- + + def test_problem_HS24(self): + def f(x): + y = (1. / (27. * np.sqrt(3.))) * ((x[0] - 3.)**2 - 9.) * x[1]**3 + eval_success = True + return y, eval_success + + A = np.array([[1. / np.sqrt(3.), -1.], + [1., np.sqrt(3.)], + [-1., -np.sqrt(3.)]]) + l = np.array([0., 0., -6.]) + u = np.array([np.inf, np.inf, np.inf]) + bl = np.array([0., 0.]) + bu = np.array([np.inf, np.inf]) + n = 2 + x0 = np.array([1., 0.5]) + xc = np.array([3., np.sqrt(3.)]) + yc = -1. + + x1, y1, iterations, alpha, evaluations, history, exit_cause = \ + psifa(f, x0, A, l, u, n, bl, bu, max_evals=2000, prec1=1e-8) + + self.assertTrue(np.allclose(x1, xc) or np.isclose(y1, yc) or + np.allclose(xc, x1) or np.isclose(yc, y1) or + y1 < yc) + self.assertTrue(iterations <= 17) + self.assertTrue(evaluations <= 112) + self.assertTrue('TP' in exit_cause) + + # ------------------------------------------------------------------------- + + def test_problem_HS25(self): + def f(x): + y = 0. + for i in range(1, 100): + a = 25. + (-50. * np.log(0.01 * i))**(2. / 3.) + b = -0.01 * i + np.exp((-1. / x[0]) * (a - x[1])**x[2]) + y += b**2 + eval_success = True + return y, eval_success + + A = np.array([]) + l = np.array([]) + u = np.array([]) + bl = np.array([0.1, 0., 0.]) + bu = np.array([100., 25.6, 5.]) + n = 3 + x0 = np.array([100., 12.5, 3.]) + # xc = np.array([50., 25., 1.5]) + # yc = 0. + xc = np.array([92.7664, 23.9537, 1.6871]) + yc = 0.0048065 + + x1, y1, iterations, alpha, evaluations, history, exit_cause = \ + psifa(f, x0, A, l, u, n, bl, bu, max_evals=2000, prec1=1e-8) + + self.assertTrue(np.allclose(x1, xc) or np.isclose(y1, yc) or + np.allclose(xc, x1) or np.isclose(yc, y1) or + y1 < yc) # fail + self.assertTrue(iterations <= 157) + self.assertTrue(evaluations <= 2008) + self.assertTrue('AF' in exit_cause) + + # ------------------------------------------------------------------------- + + def test_problem_HS28(self): + def f(x): + y = (x[0] + x[1])**2 + (x[1] + x[2])**2 + eval_success = True + return y, eval_success + + A = np.array([[1., 2., 3.]]) + l = np.array([1.]) + u = np.array([1.]) + bl = np.array([-np.inf, -np.inf, -np.inf]) + bu = np.array([np.inf, np.inf, np.inf]) + n = 3 + x0 = np.array([-4., 1., 1.]) + xc = np.array([0.5, -0.5, 0.5]) + yc = 0. + + x1, y1, iterations, alpha, evaluations, history, exit_cause = \ + psifa(f, x0, A, l, u, n, bl, bu, max_evals=2000, prec1=1e-8) + + self.assertTrue(np.allclose(x1, xc) or np.isclose(y1, yc) or + np.allclose(xc, x1) or np.isclose(yc, y1) or + y1 < yc) # fail + # self.assertTrue(iterations <= 94) # fail + self.assertTrue(evaluations <= 672) + self.assertTrue('VF' in exit_cause) + + # ------------------------------------------------------------------------- + + def test_problem_HS35(self): + def f(x): + y = 9. - 8. * x[0] - 6. * x[1] - 4. * x[2] + 2. * x[0]**2 + \ + 2. * x[1]**2 + x[2]**2 + 2. * x[0] * x[1] + 2. * x[0] * x[2] + eval_success = True + return y, eval_success + + A = np.array([[-1., -1., -2.]]) + l = np.array([-3.]) + u = np.array([np.inf]) + bl = np.array([0., 0., 0.]) + bu = np.array([np.inf, np.inf, np.inf]) + n = 3 + x0 = np.array([0.5, 0.5, 0.5]) + xc = np.array([4. / 3., 7. / 9., 4. / 9.]) + yc = 1. / 9. + + x1, y1, iterations, alpha, evaluations, history, exit_cause = \ + psifa(f, x0, A, l, u, n, bl, bu, max_evals=2000, prec1=1e-8) + + self.assertTrue(np.allclose(x1, xc) or np.isclose(y1, yc) or + np.allclose(xc, x1) or np.isclose(yc, y1) or + y1 < yc) + self.assertTrue(iterations <= 67) + self.assertTrue(evaluations <= 336) + self.assertTrue('VF' in exit_cause) + + # ------------------------------------------------------------------------- + + def test_problem_HS36(self): + def f(x): + y = -x[0] * x[1] * x[2] + eval_success = True + return y, eval_success + + A = np.array([[-1., -2., -2.]]) + l = np.array([-72.]) + u = np.array([np.inf]) + bl = np.array([0., 0., 0.]) + bu = np.array([20., 11., 42.]) + n = 3 + x0 = np.array([10., 10., 10.]) + # xc = np.array([20., 11., 15.]) + # yc = -3300. + xc = np.array([17.28, 11., 16.36]) + yc = -3109.71 + + x1, y1, iterations, alpha, evaluations, history, exit_cause = \ + psifa(f, x0, A, l, u, n, bl, bu, max_evals=2000, prec1=1e-8) + + self.assertTrue(np.allclose(x1, xc) or np.isclose(y1, yc) or + np.allclose(xc, x1) or np.isclose(yc, y1) or + y1 < yc) + self.assertTrue(iterations <= 21) + # self.assertTrue(evaluations <= 201) # fail + self.assertTrue('TP' in exit_cause) + + # ------------------------------------------------------------------------- + + def test_problem_HS37(self): + def f(x): + y = -x[0] * x[1] * x[2] + eval_success = True + return y, eval_success + + A = np.array([[-1., -2., -2.], + [1., 2., 2.]]) + l = np.array([-72., 0.]) + u = np.array([np.inf, np.inf]) + bl = np.array([0., 0., 0.]) + bu = np.array([42., 42., 42.]) + n = 3 + x0 = np.array([10., 10., 10.]) + xc = np.array([24.000130885714569, + 11.999993260028832, + 11.999941297113882]) + yc = -3456. + + x1, y1, iterations, alpha, evaluations, history, exit_cause = \ + psifa(f, x0, A, l, u, n, bl, bu, max_evals=2000, prec1=1e-8) + + self.assertTrue(np.allclose(x1, xc) or np.isclose(y1, yc) or + np.allclose(xc, x1) or np.isclose(yc, y1) or + y1 < yc) + self.assertTrue(iterations <= 47) + self.assertTrue(evaluations <= 289) + self.assertTrue('VF' in exit_cause) + + # ------------------------------------------------------------------------- + + def test_problem_HS38(self): + def f(x): + y = 100 * (x[1] - x[0]**2)**2 + (1. - x[0])**2 + \ + 90. * (x[3] - x[2]**2)**2 + (1. - x[2])**2 + \ + 10.1 * ((x[1] - 1.)**2 + (x[3] - 1.)**2) + \ + 19.8 * (x[1] - 1.) * (x[3] - 1.) + eval_success = True + return y, eval_success + + A = np.array([]) + l = np.array([]) + u = np.array([]) + bl = np.array([-10., -10., -10., -10.]) + bu = np.array([10., 10., 10., 10.]) + n = 4 + x0 = np.array([-3., -1., -3., -1.]) + # xc = np.array([1., 1., 1., 1.]) + # yc = 0. + xc = np.array([1.28472396, 1.65645037, -0.49984457, 0.26141631]) + yc = 2.60824579786 + + x1, y1, iterations, alpha, evaluations, history, exit_cause = \ + psifa(f, x0, A, l, u, n, bl, bu, max_evals=2000, prec1=1e-8) + + self.assertTrue(np.allclose(x1, xc) or np.isclose(y1, yc) or + np.allclose(xc, x1) or np.isclose(yc, y1) or + y1 < yc) + self.assertTrue(iterations <= 135) + self.assertTrue(evaluations <= 2011) + self.assertTrue('AF' in exit_cause) + + # ------------------------------------------------------------------------- + + def test_problem_HS41(self): + def f(x): + y = 2. - x[0] * x[1] * x[2] + eval_success = True + return y, eval_success + + A = np.array([[1., 2., 2., -1.]]) + l = np.array([0.]) + u = np.array([0.]) + bl = np.array([0., 0., 0., 0.]) + bu = np.array([1., 1., 1., 2.]) + n = 4 + x0 = np.array([0., 0.5, 0.5, 2.]) + # xc = np.array([2. / 3., 1. / 3., 1. / 3., 2.]) + # yc = 52. / 27. + xc = np.array([0.67188, 0.33203, 0.33203, 2.]) + yc = 1.9259 + + x1, y1, iterations, alpha, evaluations, history, exit_cause = \ + psifa(f, x0, A, l, u, n, bl, bu, max_evals=2000, prec1=1e-8) + + self.assertTrue(np.allclose(x1, xc) or np.isclose(y1, yc) or + np.allclose(xc, x1) or np.isclose(yc, y1) or + y1 < yc) + self.assertTrue(iterations <= 10) + self.assertTrue(evaluations <= 75) + self.assertTrue('VF' in exit_cause) + + # ------------------------------------------------------------------------- + + def test_problem_HS44(self): + def f(x): + y = x[0] - x[1] - x[2] - x[0] * x[2] + \ + x[0] * x[3] + x[1] * x[2] - x[1] * x[3] + eval_success = True + return y, eval_success + + A = np.array([[-1., -2., 0., 0.], + [-4., -1., 0., 0.], + [-3., -4., 0., 0.], + [0., 0., -2., -1.], + [0., 0., -1., -2.], + [0., 0., -1., -1.]]) + l = np.array([-8., -12., -12., -8., -8., -5.]) + u = np.array([np.inf, np.inf, np.inf, np.inf, np.inf, np.inf]) + bl = np.array([0., 0., 0., 0.]) + bu = np.array([np.inf, np.inf, np.inf, np.inf]) + n = 4 + x0 = np.array([0., 0., 0., 0.]) + xc = np.array([0., 3., 0., 4.]) + yc = -15. + + x1, y1, iterations, alpha, evaluations, history, exit_cause = \ + psifa(f, x0, A, l, u, n, bl, bu, max_evals=2000, prec1=1e-8) + + self.assertTrue(np.allclose(x1, xc) or np.isclose(y1, yc) or + np.allclose(xc, x1) or np.isclose(yc, y1) or + y1 < yc) + self.assertTrue(iterations <= 12) + # self.assertTrue(evaluations <= 154) # fail + self.assertTrue('TP' in exit_cause) + + # ------------------------------------------------------------------------- + + def test_problem_HS45(self): + def f(x): + y = 2. - (1. / 120.) * x[0] * x[1] * x[2] * x[3] * x[4] + eval_success = True + return y, eval_success + + A = np.array([]) + l = np.array([]) + u = np.array([]) + bl = np.array([0., 0., 0., 0., 0.]) + bu = np.array([1., 2., 3., 4., 5.]) + n = 5 + x0 = np.array([0., 1., 2., 3., 4.]) + xc = np.array([1., 2., 3., 4., 5.]) + yc = 1. + + x1, y1, iterations, alpha, evaluations, history, exit_cause = \ + psifa(f, x0, A, l, u, n, bl, bu, max_evals=2000, prec1=1e-8) + + self.assertTrue(np.allclose(x1, xc) or np.isclose(y1, yc) or + np.allclose(xc, x1) or np.isclose(yc, y1) or + y1 < yc) + self.assertTrue(iterations <= 16) + self.assertTrue(evaluations <= 259) + self.assertTrue('TP' in exit_cause) + + # ------------------------------------------------------------------------- + + def test_problem_HS48(self): + def f(x): + y = (x[0] - 1.)**2 + (x[1] - x[2])**2 + (x[3] - x[4])**2 + eval_success = True + return y, eval_success + + A = np.array([[1., 1., 1., 1., 1.], + [0., 0., 1., -2., -2.]]) + l = np.array([5., -3.]) + u = np.array([5., -3.]) + bl = np.array([-np.inf, -np.inf, -np.inf, -np.inf, -np.inf]) + bu = np.array([np.inf, np.inf, np.inf, np.inf, np.inf]) + n = 5 + x0 = np.array([3., 5., -3., 2., -2.]) + xc = np.array([1., 1., 1., 1., 1.]) + yc = 0. + + x1, y1, iterations, alpha, evaluations, history, exit_cause = \ + psifa(f, x0, A, l, u, n, bl, bu, max_evals=2000, prec1=1e-8) + + self.assertTrue(np.allclose(x1, xc) or np.isclose(y1, yc) or + np.allclose(xc, x1) or np.isclose(yc, y1) or + y1 < yc) + self.assertTrue(iterations <= 19) + self.assertTrue(evaluations <= 245) + self.assertTrue('VF' in exit_cause) + + # ------------------------------------------------------------------------- + + def test_problem_HS49(self): + def f(x): + y = (x[0] - x[1])**2 + (x[2] - 1.)**2 + \ + (x[3] - 1.)**4 + (x[4] - 1.)**6 + eval_success = True + return y, eval_success + + A = np.array([[1., 1., 1., 4., 0.], + [0., 0., 1., 0., 5.]]) + l = np.array([7., 6.]) + u = np.array([7., 6.]) + bl = np.array([-np.inf, -np.inf, -np.inf, -np.inf, -np.inf]) + bu = np.array([np.inf, np.inf, np.inf, np.inf, np.inf]) + n = 5 + x0 = np.array([10., 7., 2., -3., 0.8]) + # xc = np.array([1., 1., 1., 1., 1.]) + # yc = 0. + xc = np.array([1.55468750, 1.55859375, 0.98828125, + 0.72460938, 1.00234375]) + yc = 0.00590429 + + x1, y1, iterations, alpha, evaluations, history, exit_cause = \ + psifa(f, x0, A, l, u, n, bl, bu, max_evals=2000, prec1=1e-8) + + self.assertTrue(np.allclose(x1, xc) or np.isclose(y1, yc) or + np.allclose(xc, x1) or np.isclose(yc, y1) or + y1 < yc) + self.assertTrue(iterations <= 187) + self.assertTrue(evaluations <= 2003) + self.assertTrue('AF' in exit_cause) + + # ------------------------------------------------------------------------- + + def test_problem_HS50(self): + def f(x): + y = (x[0] - x[1])**2 + (x[1] - x[2])**2 + \ + (x[2] - x[3])**4 + (x[3] - x[4])**2 + eval_success = True + return y, eval_success + + A = np.array([[1., 2., 3., 0., 0.], + [0., 1., 2., 3., 0.], + [0., 0., 1., 2., 3.]]) + l = np.array([6., 6., 6.]) + u = np.array([6., 6., 6.]) + bl = np.array([-np.inf, -np.inf, -np.inf, -np.inf, -np.inf]) + bu = np.array([np.inf, np.inf, np.inf, np.inf, np.inf]) + n = 5 + x0 = np.array([35., -31., 11., 5., -5.]) + xc = np.array([1., 1., 1., 1., 1.]) + yc = 0. + + x1, y1, iterations, alpha, evaluations, history, exit_cause = \ + psifa(f, x0, A, l, u, n, bl, bu, max_evals=2000, prec1=1e-8) + + self.assertTrue(np.allclose(x1, xc) or np.isclose(y1, yc) or + np.allclose(xc, x1) or np.isclose(yc, y1) or + y1 < yc) + self.assertTrue(iterations <= 43) + self.assertTrue(evaluations <= 272) + self.assertTrue('VF' in exit_cause) + + # ------------------------------------------------------------------------- + + def test_problem_HS51(self): + def f(x): + y = (x[0] - x[1])**2 + (x[0] + x[2] - 2.)**2 + \ + (x[3] - 1.)**2 + (x[4] - 1.)**2 + eval_success = True + return y, eval_success + + A = np.array([[1., 3., 0., 0., 0.], + [0., 0., 1., 1., -2.], + [0., 1., 0., 0., -1.]]) + l = np.array([4., 0., 0.]) + u = np.array([4., 0., 0.]) + bl = np.array([-np.inf, -np.inf, -np.inf, -np.inf, -np.inf]) + bu = np.array([np.inf, np.inf, np.inf, np.inf, np.inf]) + n = 5 + x0 = np.array([2.5, 0.5, 2., -1., 0.5]) + xc = np.array([1., 1., 1., 1., 1.]) + yc = 0. + + x1, y1, iterations, alpha, evaluations, history, exit_cause = \ + psifa(f, x0, A, l, u, n, bl, bu, max_evals=2000, prec1=1e-8) + + self.assertTrue(np.allclose(x1, xc) or np.isclose(y1, yc) or + np.allclose(xc, x1) or np.isclose(yc, y1) or + y1 < yc) + self.assertTrue(iterations <= 71) + self.assertTrue(evaluations <= 302) + self.assertTrue('VF' in exit_cause) + + # ------------------------------------------------------------------------- + + def test_problem_HS52(self): + def f(x): + y = (4. * x[0] - x[1])**2 + (x[1] + x[2] - 2.)**2 + \ + (x[3] - 1.)**2 + (x[4] - 1.)**2 + eval_success = True + return y, eval_success + + A = np.array([[1., 3., 0., 0., 0.], + [0., 0., 1., 1., -2.], + [0., 1., 0., 0., -1.]]) + l = np.array([0., 0., 0.]) + u = np.array([0., 0., 0.]) + bl = np.array([-np.inf, -np.inf, -np.inf, -np.inf, -np.inf]) + bu = np.array([np.inf, np.inf, np.inf, np.inf, np.inf]) + n = 5 + x0 = np.array([0, 0, 0., 0., 0.]) + # xc = np.array([-33., 11., 180., -158., 11.]) / 349. + # yc = 1859. / 349. + xc = np.array([-0.09472656, 0.03157552, 0.51529948, + -0.45214844, 0.03157552]) + yc = 5.32664861 + + x1, y1, iterations, alpha, evaluations, history, exit_cause = \ + psifa(f, x0, A, l, u, n, bl, bu, max_evals=2000, prec1=1e-8) + + self.assertTrue(np.allclose(x1, xc) or np.isclose(y1, yc) or + np.allclose(xc, x1) or np.isclose(yc, y1) or + y1 < yc) + self.assertTrue(iterations <= 9) + self.assertTrue(evaluations <= 91) + self.assertTrue('VF' in exit_cause) + + # ------------------------------------------------------------------------- + + def test_problem_HS53(self): + def f(x): + y = (x[0] - x[1])**2 + (x[1] + x[2] - 2.)**2 + \ + (x[3] - 1.)**2 + (x[4] - 1.)**2 + eval_success = True + return y, eval_success + + A = np.array([[1., 3., 0., 0., 0.], + [0., 0., 1., 1., -2.], + [0., 1., 0., 0., -1.]]) + l = np.array([0., 0., 0.]) + u = np.array([0., 0., 0.]) + bl = np.array([-10., -10., -10., -10., -10.]) + bu = np.array([10., 10., 10., 10., 10.]) + n = 5 + x0 = np.array([0, 0, 0., 0., 0.]) + # xc = np.array([-33., 11., 27., -5., 11.]) / 43. + # yc = 176. / 43. + xc = np.array([-0.76757812, 0.25585938, 0.62695312, + -0.11523438, 0.25585938]) + yc = 4.09302521 + + x1, y1, iterations, alpha, evaluations, history, exit_cause = \ + psifa(f, x0, A, l, u, n, bl, bu, max_evals=2000, prec1=1e-8) + + self.assertTrue(np.allclose(x1, xc) or np.isclose(y1, yc) or + np.allclose(xc, x1) or np.isclose(yc, y1) or + y1 < yc) + self.assertTrue(iterations <= 8) + self.assertTrue(evaluations <= 95) + self.assertTrue('VF' in exit_cause) + + # ------------------------------------------------------------------------- + + def test_problem_HS54(self): + def f(x): + h = (((x[0] - 1e6)**2) / 6.4e7 + + (x[0] - 1e4) * (x[1] - 1.) / 2e4 + + (x[1] - 1.)**2) * ((x[2] - 2e6)**2) / (0.96 * 4.9e13) + \ + ((x[3] - 10.)**2) / 2.5e3 + (x[4] - 1e-3)**2 / 2.5e-3 + \ + (x[5] - 1e8)**2 / 2.5e17 + y = -np.exp(-h / 2.) + eval_success = True + return y, eval_success + + A = np.array([[1., 4e3, 0., 0., 0., 0.]]) + l = np.array([1.76e4]) + u = np.array([1.76e4]) + bl = np.array([0., -10., 0., 0., -1., 0.]) + bu = np.array([2e4, 10., 1e7, 20., 1., 2e8]) + n = 6 + x0 = np.array([1.76e4, 0, 0., 0., 0., 0.]) + # xc = np.array([91600. / 7., 79. / 70., 2e6, 10., 1e-3, 1e8]) + # yc = -np.exp(-27. / 280.) + xc = np.array([1.7600e4, -1.9531e-5, 0., 3.38281250, 0., 0.]) + yc = 0. + + x1, y1, iterations, alpha, evaluations, history, exit_cause = \ + psifa(f, x0, A, l, u, n, bl, bu, max_evals=2000, prec1=1e-8) + + self.assertTrue(np.allclose(x1, xc) or np.isclose(y1, yc) or + np.allclose(xc, x1) or np.isclose(yc, y1) or + y1 < yc) + self.assertTrue(iterations <= 10) + self.assertTrue(evaluations <= 129) + self.assertTrue('VF' in exit_cause) + + # ------------------------------------------------------------------------- + + def test_problem_HS55(self): + def f(x): + y = x[0] + 2. * x[1] + 4. * x[4] + np.exp(x[0] * x[3]) + eval_success = True + return y, eval_success + + A = np.array([[1., 2., 0., 0., 5., 0.], + [1., 1., 1., 0., 0., 0.], + [0., 0., 0., 1., 1., 1.], + [1., 0., 0., 1., 0., 0.], + [0., 1., 0., 0., 1., 0.], + [0., 0., 1., 0., 0., 1.]]) + l = np.array([6., 3., 2., 1., 2., 2.]) + u = np.array([6., 3., 2., 1., 2., 2.]) + bl = np.array([0., 0., 0., 0., 0., 0.]) + bu = np.array([1., np.inf, np.inf, 1., np.inf, np.inf]) + n = 6 + x0 = np.array([0.447794849394555, 1.482598283131551, + 1.069606867473851, 0.552205150605459, + 0.517401716868434, 0.930393132526135]) + xc = np.array([0., 4. / 3., 5. / 3., 1., 2. / 3., 1. / 3.]) + yc = 19. / 3. + + x1, y1, iterations, alpha, evaluations, history, exit_cause = \ + psifa(f, x0, A, l, u, n, bl, bu, max_evals=2000, prec1=1e-8) + + self.assertTrue(np.allclose(x1, xc) or np.isclose(y1, yc) or + np.allclose(xc, x1) or np.isclose(yc, y1) or + y1 < yc) + self.assertTrue(iterations <= 10) + self.assertTrue(evaluations <= 41) + self.assertTrue('TP' in exit_cause) + + # ------------------------------------------------------------------------- + + def test_problem_HS62(self): + def f(x): + y = -32.174 * \ + (255. * np.log((x[0] + x[1] + x[2] + 0.03) / + (0.09 * x[0] + x[1] + x[2] + 0.03)) + + 280. * np.log((x[1] + x[2] + 0.03) / + (0.07 * x[1] + x[2] + 0.03)) + + 290. * np.log((x[2] + 0.03) / (0.13 * x[2] + 0.03))) + eval_success = True + return y, eval_success + + A = np.array([[1., 1., 1.]]) + l = np.array([1.]) + u = np.array([1.]) + bl = np.array([0., 0., 0.]) + bu = np.array([1., 1., 1.]) + n = 3 + x0 = np.array([0.7, 0.2, 0.1]) + xc = np.array([0.617812690138817, + 0.328202229738235, + 0.053985080122948]) + yc = -2.627251448e4 + + x1, y1, iterations, alpha, evaluations, history, exit_cause = \ + psifa(f, x0, A, l, u, n, bl, bu, max_evals=2000, prec1=1e-8) + + self.assertTrue(np.allclose(x1, xc) or np.isclose(y1, yc) or + np.allclose(xc, x1) or np.isclose(yc, y1) or + y1 < yc) # fail + # self.assertTrue(iterations <= 67) # fail + # self.assertTrue(evaluations <= 300) # fail + # self.assertTrue('TP' in exit_cause) + self.assertTrue('VF' in exit_cause) + + # ------------------------------------------------------------------------- + + def test_problem_HS76(self): + def f(x): + y = x[0]**2 + 0.5 * x[1]**2 + x[2]**2 + 0.5 * x[3]**2 - \ + x[0] * x[2] - x[2] * x[3] - x[0] - 3. * x[1] + x[2] - x[3] + eval_success = True + return y, eval_success + + A = np.array([[-1., -2., -1., -1.], + [-3., -1., -2., 1.], + [0., 1., 4., 0.]]) + l = np.array([-5., -4., 1.5]) + u = np.array([np.inf, np.inf, np.inf]) + bl = np.array([0., 0., 0., 0.]) + bu = np.array([np.inf, np.inf, np.inf, np.inf]) + n = 4 + x0 = np.array([0.5, 0.5, 0.5, 0.5]) + # xc = np.array([0.2727273, 2.090909, 0.26e-10, 0.5454545]) + # yc = -4.681818090904359 + xc = np.array([0.47114044, 2.08220996, 0., 0.36443964]) + yc = -4.62602936 + + x1, y1, iterations, alpha, evaluations, history, exit_cause = \ + psifa(f, x0, A, l, u, n, bl, bu, max_evals=2000, prec1=1e-8) + + self.assertTrue(np.allclose(x1, xc) or np.isclose(y1, yc) or + np.allclose(xc, x1) or np.isclose(yc, y1) or + y1 < yc) # fail + # self.assertTrue(iterations <= 16) # fail + # self.assertTrue(evaluations <= 165) # fail + self.assertTrue('TP' in exit_cause) + + # ------------------------------------------------------------------------- + + def test_problem_HS86(self): + def f(x): + e = np.array([-15., -27., -36., -18., -12.]) + c = np.array([[30., -20., -10., 32., -10.], + [-20., 39., -6., -31., 32.], + [-10., -6., 10., -6., -10.], + [32., -31., -6., 39., -20.], + [-10., 32., -10., -20., 30.]]) + d = np.array([4., 8., 10., 6., 2.]) + s1 = 0. + s2 = 0. + s3 = 0. + for j in range(5): + s1 += e[j] * x[j] + for i in range(5): + s2 += c[i, j] * x[i] * x[j] + s3 += d[j] * x[j]**3 + y = s1 + s2 + s3 + eval_success = True + return y, eval_success + + A = np.array([[-16., 2., 0., 1., 0.], + [0., -2., 0., 4., 2.], + [-3.5, 0., 2., 0., 0.], + [0., -2., 0., -4., -1.], + [0., -9., -2., 1., -2.8], + [2., 0., -4., 0., 0.], + [-1., -1., -1., -1., -1.], + [-1., -2., -3., -2., -1.], + [1., 2., 3., 4., 5.], + [1., 1., 1., 1., 1.]]) + l = np.array([-40., -2., -0.25, -4., -4., -1., -40., -60., 5., 1.]) + u = np.array([np.inf] * 10) + bl = np.array([0., 0., 0., 0., 0.]) + bu = np.array([np.inf, np.inf, np.inf, np.inf, np.inf]) + n = 5 + x0 = np.array([0., 0., 0., 0., 1.]) + xc = np.array([0.299999998116435, + 0.333467595429803, + 0.399999998635825, + 0.428310114302271, + 0.223964911296049]) + yc = -32.34867897 + + x1, y1, iterations, alpha, evaluations, history, exit_cause = \ + psifa(f, x0, A, l, u, n, bl, bu, max_evals=2000, prec1=1e-8) + + self.assertTrue(np.allclose(x1, xc) or np.isclose(y1, yc) or + np.allclose(xc, x1) or np.isclose(yc, y1) or + y1 < yc) # fail + self.assertTrue(iterations <= 86) # fail + # self.assertTrue(evaluations <= 920) # fail + self.assertTrue('VF' in exit_cause) + + # ------------------------------------------------------------------------- + + def test_problem_HS105(self): + # Requires the "y105.mat" file. + pass + + # ------------------------------------------------------------------------- + + def test_problem_HS110(self): + def f(x): + s = 0. + p = 1. + for i in range(10): + s += (np.log(x[i] - 2.))**2 + (np.log(10. - x[i]))**2 + p *= x[i] + y = s - p**0.2 + eval_success = True + return y, eval_success + + A = np.array([]) + l = np.array([]) + u = np.array([]) + bl = np.array([2.001] * 10) + bu = np.array([9.999] * 10) + n = 10 + x0 = np.array([9.] * 10) + xc = np.array([9.35025655] * 10) + yc = -45.77846971 + + x1, y1, iterations, alpha, evaluations, history, exit_cause = \ + psifa(f, x0, A, l, u, n, bl, bu, max_evals=2000, prec1=1e-8) + + self.assertTrue(np.allclose(x1, xc) or np.isclose(y1, yc) or + np.allclose(xc, x1) or np.isclose(yc, y1) or + y1 < yc) + self.assertTrue(iterations <= 51) + # self.assertTrue(evaluations <= 1411) # fail + # self.assertTrue('AF' in exit_cause) + self.assertTrue('VF' in exit_cause) + + # ------------------------------------------------------------------------- + + def test_problem_HS112(self): + def f(x): + c = np.array([-6.089, -17.164, -34.054, -5.914, -24.721, + -14.986, -24.1, -10.708, -26.662, -22.179]) + y = 0. + for i in range(10): + y += x[i] * (c[i] + np.log(x[0] / (x[1] + np.sum(x)))) + eval_success = True + return y, eval_success + + A = np.array([[1., 2., 2., 0., 0., 1., 0., 0., 0., 1.], + [0., 0., 0., 1., 2., 1., 1., 0., 0., 0.], + [0., 0., 1., 0., 0., 0., 1., 1., 2., 1.]]) + l = np.array([2., 1., 1.]) + u = np.array([2., 1., 1.]) + bl = np.array([1e-6] * 10) + bu = np.array([np.inf] * 10) + n = 10 + x0 = np.array([0.8, 0.5, 0.01, 0.01, 0.1, + 0.01, 0.78, 0.01, 0.015, 0.17]) + xc = np.array([0.000001000013433, + 0.999987491743295, + 0.000004802332244, + 0.999960037649762, + 0.000014076754094, + 0.000006088443443, + 0.000005720398607, + 0.999971681334066, + 0.000004736271518, + 0.000008323392047]) + # yc = -79.3911 + yc = -71.5236117652 + + x1, y1, iterations, alpha, evaluations, history, exit_cause = \ + psifa(f, x0, A, l, u, n, bl, bu, max_evals=2000, prec1=1e-8) + + self.assertTrue(np.allclose(x1, xc) or np.isclose(y1, yc) or + np.allclose(xc, x1) or np.isclose(yc, y1) or + y1 < yc) + self.assertTrue(iterations <= 20) + # self.assertTrue(evaluations <= 299) # fail + self.assertTrue('TP' in exit_cause) + + # ------------------------------------------------------------------------- + + def test_problem_HS118(self): + def f(x): + y = 0. + for i in range(5): + j = 3 * i + y += 2.3 * x[j + 0] + 1e-4 * x[j + 0]**2 + \ + 1.7 * x[j + 1] + 1e-4 * x[j + 1] + \ + 2.2 * x[j + 2] + 1.5e-4 * x[j + 2]**2 + eval_success = True + return y, eval_success + + A = np.array([[-1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], + [0, 0, -1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0], + [0, -1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, -1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, -1, 0, 0, 1, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, -1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, -1, 0, 0, 1, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 1, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 1, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 1, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 1], + [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 1, 0], + [1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1], + [0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0], + [0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0]]) \ + .astype(float) + l = np.array([-7., -7., -7., -7., -7., -7., -7., -7., -7., + -7., -7., -7., 60., 70., 100., 50., 85.]) + u = np.array([6., 6., 7., 6., 6., 7., 6., 6., 7., 6., 6., + 7., np.inf, np.inf, np.inf, np.inf, np.inf]) + bl = np.array([8., 43., 3., 0., 0., 0., 0., 0., + 0., 0., 0., 0., 0., 0., 0.]) + bu = np.array([21., 57., 16., 90., 120., 60., 90., 120., + 60., 90., 120., 60., 90., 120., 60.]) + n = 15 + x0 = np.array([20., 55., 15., 20., 60., 20., 20., 60., + 20., 20., 60., 20., 20., 60., 20.]) + # xc = np.array([8., 49., 3., 1., 56., 0., 1., 63., + # 6., 3., 70., 12., 5., 77., 18]) + # yc = 664.8204500 + # xc = np.array([8.2923, 4.8709e1, 3.0093, 1.3630, + # 5.0331e1, 3.7989e-4, 7.3630, 5.7277e1, + # 5.3619, 1.3302e1, 6.3991e1, 7.7069, + # 1.6259e1, 7.0872e1, 1.2869e1]) + # yc = 667.65 + xc = np.array([8.2923146073, 48.7085129072, 3.0093215000, + 1.3629991880, 50.3307789916, 0.0003798947, + 7.3629991880, 57.2765915265, 5.3619399377, + 13.3020232957, 63.9910701197, 7.7069065846, + 16.2591149366, 70.8721322093, 12.8687528540]) + yc = 667.6476831493 + + x1, y1, iterations, alpha, evaluations, history, exit_cause = \ + psifa(f, x0, A, l, u, n, bl, bu, max_evals=2000, prec1=1e-8) + + self.assertTrue(np.allclose(x1, xc) or np.isclose(y1, yc) or + np.allclose(xc, x1) or np.isclose(yc, y1) or + y1 < yc) # fail + self.assertTrue(iterations <= 89) + self.assertTrue(evaluations <= 2019) + self.assertTrue('AF' in exit_cause) + + # ------------------------------------------------------------------------- + + def test_problem_HS119(self): + # Requires the "p119.mat" file. + pass + + +if __name__ == '__main__': + unittest.main()