Mercurial > octave-nkf
view scripts/optimization/sqp.m @ 19627:446c46af4b42 stable
strip trailing whitespace from most source files
* Makefile.am, NEWS, build-aux/common.mk, configure.ac,
doc/Makefile.am, doc/doxyhtml/Makefile.am,
doc/interpreter/Makefile.am, doc/interpreter/arith.txi,
doc/interpreter/audio.txi, doc/interpreter/basics.txi,
doc/interpreter/bugs.txi, doc/interpreter/container.txi,
doc/interpreter/cp-idx.txi, doc/interpreter/data.txi,
doc/interpreter/debug.txi, doc/interpreter/diagperm.txi,
doc/interpreter/diffeq.txi, doc/interpreter/doccheck/README,
doc/interpreter/doccheck/spellcheck, doc/interpreter/emacs.txi,
doc/interpreter/errors.txi, doc/interpreter/eval.txi,
doc/interpreter/expr.txi, doc/interpreter/external.txi,
doc/interpreter/fn-idx.txi, doc/interpreter/func.txi,
doc/interpreter/geometry.txi, doc/interpreter/geometryimages.m,
doc/interpreter/gpl.txi, doc/interpreter/grammar.txi,
doc/interpreter/gui.txi, doc/interpreter/image.txi,
doc/interpreter/install.txi, doc/interpreter/interp.txi,
doc/interpreter/interpimages.m, doc/interpreter/intro.txi,
doc/interpreter/io.txi, doc/interpreter/java.txi,
doc/interpreter/linalg.txi, doc/interpreter/macros.texi,
doc/interpreter/matrix.txi, doc/interpreter/munge-texi.pl,
doc/interpreter/nonlin.txi, doc/interpreter/numbers.txi,
doc/interpreter/obsolete.txi, doc/interpreter/octave-config.1,
doc/interpreter/octave.texi, doc/interpreter/oop.txi,
doc/interpreter/op-idx.txi, doc/interpreter/optim.txi,
doc/interpreter/package.txi, doc/interpreter/plot.txi,
doc/interpreter/poly.txi, doc/interpreter/preface.txi,
doc/interpreter/quad.txi, doc/interpreter/set.txi,
doc/interpreter/signal.txi, doc/interpreter/sparse.txi,
doc/interpreter/sparseimages.m, doc/interpreter/splineimages.m,
doc/interpreter/stats.txi, doc/interpreter/stmt.txi,
doc/interpreter/strings.txi, doc/interpreter/system.txi,
doc/interpreter/testfun.txi, doc/interpreter/tips.txi,
doc/interpreter/var.txi, doc/interpreter/vectorize.txi,
doc/liboctave/Makefile.am, doc/liboctave/array.texi,
doc/liboctave/bugs.texi, doc/liboctave/cp-idx.texi,
doc/liboctave/dae.texi, doc/liboctave/diffeq.texi,
doc/liboctave/error.texi, doc/liboctave/factor.texi,
doc/liboctave/fn-idx.texi, doc/liboctave/gpl.texi,
doc/liboctave/install.texi, doc/liboctave/intro.texi,
doc/liboctave/liboctave.texi, doc/liboctave/matvec.texi,
doc/liboctave/nleqn.texi, doc/liboctave/nlfunc.texi,
doc/liboctave/ode.texi, doc/liboctave/optim.texi,
doc/liboctave/preface.texi, doc/liboctave/quad.texi,
doc/liboctave/range.texi, doc/refcard/Makefile.am,
doc/refcard/refcard.tex, etc/HACKING, etc/NEWS.1, etc/NEWS.2,
etc/NEWS.3, etc/OLD-ChangeLogs/ChangeLog,
etc/OLD-ChangeLogs/doc-ChangeLog,
etc/OLD-ChangeLogs/scripts-ChangeLog,
etc/OLD-ChangeLogs/src-ChangeLog, etc/OLD-ChangeLogs/test-ChangeLog,
etc/PROJECTS, etc/README.Cygwin, etc/README.MacOS, etc/README.MinGW,
etc/README.gnuplot, etc/gdbinit, etc/icons/Makefile.am,
examples/@polynomial/end.m, examples/@polynomial/subsasgn.m,
examples/Makefile.am, examples/standalonebuiltin.cc,
libgui/Makefile.am, libgui/qterminal/libqterminal/README,
libgui/qterminal/libqterminal/unix/BlockArray.cpp,
libgui/qterminal/libqterminal/unix/BlockArray.h,
libgui/qterminal/libqterminal/unix/Character.h,
libgui/qterminal/libqterminal/unix/CharacterColor.h,
libgui/qterminal/libqterminal/unix/Emulation.cpp,
libgui/qterminal/libqterminal/unix/Emulation.h,
libgui/qterminal/libqterminal/unix/Filter.cpp,
libgui/qterminal/libqterminal/unix/Filter.h,
libgui/qterminal/libqterminal/unix/History.cpp,
libgui/qterminal/libqterminal/unix/History.h,
libgui/qterminal/libqterminal/unix/KeyboardTranslator.cpp,
libgui/qterminal/libqterminal/unix/KeyboardTranslator.h,
libgui/qterminal/libqterminal/unix/LineFont.h,
libgui/qterminal/libqterminal/unix/QUnixTerminalImpl.cpp,
libgui/qterminal/libqterminal/unix/QUnixTerminalImpl.h,
libgui/qterminal/libqterminal/unix/Screen.cpp,
libgui/qterminal/libqterminal/unix/Screen.h,
libgui/qterminal/libqterminal/unix/ScreenWindow.cpp,
libgui/qterminal/libqterminal/unix/ScreenWindow.h,
libgui/qterminal/libqterminal/unix/TerminalCharacterDecoder.cpp,
libgui/qterminal/libqterminal/unix/TerminalCharacterDecoder.h,
libgui/qterminal/libqterminal/unix/Vt102Emulation.h,
libgui/qterminal/libqterminal/win32/QWinTerminalImpl.cpp,
libgui/qterminal/qterminal/main.cpp,
libgui/src/m-editor/file-editor-tab.cc, libgui/src/octave-gui.cc,
libgui/src/octave-qt-link.cc, libinterp/corefcn/data.cc,
libinterp/corefcn/defun-int.h, libinterp/corefcn/det.cc,
libinterp/corefcn/gl2ps-renderer.cc, libinterp/corefcn/graphics.cc,
libinterp/corefcn/graphics.in.h, libinterp/corefcn/ls-mat5.cc,
libinterp/corefcn/lu.cc, libinterp/corefcn/oct-tex-parser.yy,
libinterp/corefcn/oct-tex-symbols.in, libinterp/corefcn/quadcc.cc,
libinterp/corefcn/zfstream.cc, libinterp/dldfcn/__eigs__.cc,
libinterp/dldfcn/__voronoi__.cc, libinterp/gendoc.pl,
libinterp/genprops.awk, libinterp/mk-errno-list, libinterp/mk-pkg-add,
libinterp/mkbuiltins, libinterp/mkdefs, libinterp/mkdocs,
libinterp/mkops, libinterp/octave-value/ov-java.cc,
libinterp/parse-tree/lex.ll, libinterp/parse-tree/oct-parse.in.yy,
libinterp/parse-tree/octave.gperf, liboctave/Makefile.am,
liboctave/array/Array.cc, liboctave/array/module.mk,
liboctave/cruft/daspk/datv.f, liboctave/cruft/daspk/dcnst0.f,
liboctave/cruft/daspk/dcnstr.f, liboctave/cruft/daspk/ddasic.f,
liboctave/cruft/daspk/ddasid.f, liboctave/cruft/daspk/ddasik.f,
liboctave/cruft/daspk/ddaspk.f, liboctave/cruft/daspk/ddstp.f,
liboctave/cruft/daspk/ddwnrm.f, liboctave/cruft/daspk/dfnrmd.f,
liboctave/cruft/daspk/dfnrmk.f, liboctave/cruft/daspk/dhels.f,
liboctave/cruft/daspk/dheqr.f, liboctave/cruft/daspk/dinvwt.f,
liboctave/cruft/daspk/dlinsd.f, liboctave/cruft/daspk/dlinsk.f,
liboctave/cruft/daspk/dmatd.f, liboctave/cruft/daspk/dnedd.f,
liboctave/cruft/daspk/dnedk.f, liboctave/cruft/daspk/dnsd.f,
liboctave/cruft/daspk/dnsid.f, liboctave/cruft/daspk/dnsik.f,
liboctave/cruft/daspk/dnsk.f, liboctave/cruft/daspk/dorth.f,
liboctave/cruft/daspk/dslvd.f, liboctave/cruft/daspk/dslvk.f,
liboctave/cruft/daspk/dspigm.f, liboctave/cruft/daspk/dyypnw.f,
liboctave/cruft/dasrt/ddasrt.f, liboctave/cruft/dasrt/drchek.f,
liboctave/cruft/dassl/ddaslv.f, liboctave/cruft/dassl/ddassl.f,
liboctave/cruft/misc/blaswrap.c, liboctave/cruft/misc/module.mk,
liboctave/cruft/odepack/cfode.f, liboctave/cruft/odepack/dlsode.f,
liboctave/cruft/odepack/ewset.f, liboctave/cruft/odepack/intdy.f,
liboctave/cruft/odepack/prepj.f, liboctave/cruft/odepack/sintdy.f,
liboctave/cruft/odepack/slsode.f, liboctave/cruft/odepack/solsy.f,
liboctave/cruft/odepack/ssolsy.f, liboctave/cruft/odepack/stode.f,
liboctave/cruft/odepack/vnorm.f, liboctave/cruft/ranlib/Basegen.doc,
liboctave/cruft/ranlib/README, liboctave/cruft/ranlib/genbet.f,
liboctave/cruft/ranlib/genexp.f, liboctave/cruft/ranlib/gennch.f,
liboctave/cruft/ranlib/gennf.f, liboctave/cruft/ranlib/gennor.f,
liboctave/cruft/ranlib/getsd.f, liboctave/cruft/ranlib/initgn.f,
liboctave/cruft/ranlib/phrtsd.f, liboctave/cruft/ranlib/randlib.fdoc,
liboctave/cruft/ranlib/setsd.f, liboctave/cruft/ranlib/tstgmn.for,
liboctave/cruft/ranlib/tstmid.for, liboctave/cruft/slatec-fn/atanh.f,
liboctave/cruft/slatec-fn/datanh.f,
liboctave/cruft/slatec-fn/xgmainc.f,
liboctave/cruft/slatec-fn/xsgmainc.f, liboctave/numeric/module.mk,
liboctave/operators/mk-ops.awk, liboctave/operators/mx-ops,
liboctave/operators/sparse-mk-ops.awk,
liboctave/operators/sparse-mx-ops, liboctave/operators/vx-ops,
liboctave/util/module.mk, run-octave.in, scripts/@ftp/ftp.m,
scripts/audio/wavread.m, scripts/deprecated/java_convert_matrix.m,
scripts/deprecated/java_debug.m, scripts/deprecated/java_invoke.m,
scripts/deprecated/java_new.m,
scripts/deprecated/java_unsigned_conversion.m,
scripts/deprecated/javafields.m, scripts/deprecated/javamethods.m,
scripts/deprecated/shell_cmd.m, scripts/general/accumarray.m,
scripts/general/display.m, scripts/general/fieldnames.m,
scripts/general/interp1.m, scripts/general/interp2.m,
scripts/general/interp3.m, scripts/general/isa.m,
scripts/general/methods.m, scripts/general/sortrows.m,
scripts/geometry/convhull.m, scripts/geometry/delaunay.m,
scripts/geometry/delaunay3.m, scripts/geometry/delaunayn.m,
scripts/geometry/griddata.m, scripts/geometry/griddatan.m,
scripts/geometry/voronoi.m, scripts/geometry/voronoin.m,
scripts/gui/guihandles.m, scripts/gui/inputdlg.m,
scripts/gui/listdlg.m, scripts/gui/msgbox.m, scripts/gui/questdlg.m,
scripts/gui/uigetfile.m, scripts/gui/waitbar.m, scripts/gui/warndlg.m,
scripts/help/doc.m, scripts/help/help.m, scripts/help/type.m,
scripts/image/bone.m, scripts/image/cmpermute.m,
scripts/image/cmunique.m, scripts/image/colorcube.m,
scripts/image/colormap.m, scripts/image/contrast.m,
scripts/image/gray2ind.m, scripts/image/image.m,
scripts/image/imshow.m, scripts/image/ind2gray.m, scripts/image/jet.m,
scripts/image/rgb2ntsc.m, scripts/image/spinmap.m,
scripts/io/importdata.m, scripts/io/strread.m, scripts/io/textread.m,
scripts/io/textscan.m, scripts/java/java_get.m,
scripts/java/java_set.m, scripts/java/javaaddpath.m,
scripts/java/javaclasspath.m, scripts/java/javamem.m,
scripts/linear-algebra/linsolve.m, scripts/linear-algebra/qzhess.m,
scripts/miscellaneous/debug.m, scripts/miscellaneous/desktop.m,
scripts/miscellaneous/dir.m, scripts/miscellaneous/dos.m,
scripts/miscellaneous/edit.m, scripts/miscellaneous/fact.m,
scripts/miscellaneous/getappdata.m, scripts/miscellaneous/inputname.m,
scripts/miscellaneous/license.m, scripts/miscellaneous/ls_command.m,
scripts/miscellaneous/run.m, scripts/miscellaneous/setfield.m,
scripts/miscellaneous/unix.m, scripts/miscellaneous/ver.m,
scripts/mk-pkg-add, scripts/mkdoc.pl,
scripts/optimization/fminsearch.m, scripts/optimization/optimset.m,
scripts/optimization/sqp.m, scripts/pkg/pkg.m,
scripts/pkg/private/create_pkgadddel.m,
scripts/pkg/private/fix_depends.m, scripts/pkg/private/install.m,
scripts/plot/appearance/axis.m, scripts/plot/appearance/box.m,
scripts/plot/appearance/clabel.m, scripts/plot/appearance/daspect.m,
scripts/plot/appearance/datetick.m, scripts/plot/appearance/grid.m,
scripts/plot/appearance/legend.m, scripts/plot/appearance/orient.m,
scripts/plot/appearance/shading.m, scripts/plot/appearance/text.m,
scripts/plot/appearance/title.m, scripts/plot/appearance/xlabel.m,
scripts/plot/appearance/ylabel.m, scripts/plot/appearance/zlabel.m,
scripts/plot/draw/area.m, scripts/plot/draw/bar.m,
scripts/plot/draw/barh.m, scripts/plot/draw/colorbar.m,
scripts/plot/draw/contour.m, scripts/plot/draw/contour3.m,
scripts/plot/draw/contourf.m, scripts/plot/draw/ellipsoid.m,
scripts/plot/draw/errorbar.m, scripts/plot/draw/ezcontour.m,
scripts/plot/draw/ezcontourf.m, scripts/plot/draw/ezmesh.m,
scripts/plot/draw/ezpolar.m, scripts/plot/draw/fill.m,
scripts/plot/draw/fplot.m, scripts/plot/draw/hist.m,
scripts/plot/draw/meshc.m, scripts/plot/draw/meshz.m,
scripts/plot/draw/pareto.m, scripts/plot/draw/patch.m,
scripts/plot/draw/peaks.m, scripts/plot/draw/pie.m,
scripts/plot/draw/pie3.m, scripts/plot/draw/plot.m,
scripts/plot/draw/plotyy.m, scripts/plot/draw/private/__bar__.m,
scripts/plot/draw/private/__contour__.m,
scripts/plot/draw/private/__errplot__.m,
scripts/plot/draw/private/__ezplot__.m,
scripts/plot/draw/private/__patch__.m,
scripts/plot/draw/private/__stem__.m, scripts/plot/draw/rectangle.m,
scripts/plot/draw/ribbon.m, scripts/plot/draw/rose.m,
scripts/plot/draw/scatter.m, scripts/plot/draw/scatter3.m,
scripts/plot/draw/semilogx.m, scripts/plot/draw/shrinkfaces.m,
scripts/plot/draw/sombrero.m, scripts/plot/draw/sphere.m,
scripts/plot/draw/stairs.m, scripts/plot/draw/stem.m,
scripts/plot/draw/stemleaf.m, scripts/plot/draw/surf.m,
scripts/plot/draw/surface.m, scripts/plot/draw/surfc.m,
scripts/plot/draw/surfl.m, scripts/plot/draw/surfnorm.m,
scripts/plot/draw/tetramesh.m, scripts/plot/draw/trimesh.m,
scripts/plot/draw/triplot.m, scripts/plot/draw/trisurf.m,
scripts/plot/util/__gnuplot_drawnow__.m,
scripts/plot/util/__plt_get_axis_arg__.m, scripts/plot/util/axes.m,
scripts/plot/util/clf.m, scripts/plot/util/copyobj.m,
scripts/plot/util/figure.m, scripts/plot/util/gcbo.m,
scripts/plot/util/graphics_toolkit.m, scripts/plot/util/hggroup.m,
scripts/plot/util/meshgrid.m, scripts/plot/util/newplot.m,
scripts/plot/util/print.m,
scripts/plot/util/private/__add_default_menu__.m,
scripts/plot/util/private/__fltk_print__.m,
scripts/plot/util/private/__gnuplot_print__.m,
scripts/plot/util/private/__print_parse_opts__.m,
scripts/plot/util/refreshdata.m, scripts/plot/util/subplot.m,
scripts/polynomial/conv.m, scripts/polynomial/poly.m,
scripts/polynomial/polyeig.m, scripts/polynomial/polyfit.m,
scripts/polynomial/polyval.m,
scripts/polynomial/private/__splinefit__.m,
scripts/polynomial/spline.m, scripts/prefs/prefdir.m,
scripts/prefs/preferences.m, scripts/prefs/private/prefsfile.m,
scripts/prefs/rmpref.m, scripts/signal/freqz.m,
scripts/signal/module.mk, scripts/sparse/eigs.m, scripts/sparse/pcg.m,
scripts/sparse/private/__sprand_impl__.m, scripts/sparse/sprand.m,
scripts/sparse/sprandn.m, scripts/sparse/spy.m, scripts/sparse/svds.m,
scripts/specfun/expint.m, scripts/specfun/factor.m,
scripts/special-matrix/gallery.m, scripts/special-matrix/hankel.m,
scripts/special-matrix/toeplitz.m, scripts/startup/inputrc,
scripts/statistics/base/kurtosis.m, scripts/statistics/base/moment.m,
scripts/statistics/base/qqplot.m, scripts/statistics/base/var.m,
scripts/statistics/distributions/betarnd.m,
scripts/statistics/distributions/binoinv.m,
scripts/statistics/distributions/binopdf.m,
scripts/statistics/distributions/binornd.m,
scripts/statistics/distributions/cauchy_rnd.m,
scripts/statistics/distributions/chi2rnd.m,
scripts/statistics/distributions/discrete_pdf.m,
scripts/statistics/distributions/discrete_rnd.m,
scripts/statistics/distributions/empirical_rnd.m,
scripts/statistics/distributions/exprnd.m,
scripts/statistics/distributions/frnd.m,
scripts/statistics/distributions/gamrnd.m,
scripts/statistics/distributions/geornd.m,
scripts/statistics/distributions/hygernd.m,
scripts/statistics/distributions/kolmogorov_smirnov_cdf.m,
scripts/statistics/distributions/laplace_cdf.m,
scripts/statistics/distributions/laplace_pdf.m,
scripts/statistics/distributions/logistic_cdf.m,
scripts/statistics/distributions/logistic_pdf.m,
scripts/statistics/distributions/lognrnd.m,
scripts/statistics/distributions/nbincdf.m,
scripts/statistics/distributions/nbininv.m,
scripts/statistics/distributions/nbinpdf.m,
scripts/statistics/distributions/nbinrnd.m,
scripts/statistics/distributions/normrnd.m,
scripts/statistics/distributions/poissinv.m,
scripts/statistics/distributions/poissrnd.m,
scripts/statistics/distributions/tinv.m,
scripts/statistics/distributions/trnd.m,
scripts/statistics/distributions/unidcdf.m,
scripts/statistics/distributions/unidpdf.m,
scripts/statistics/distributions/unidrnd.m,
scripts/statistics/distributions/unifrnd.m,
scripts/statistics/distributions/wblrnd.m,
scripts/statistics/models/module.mk,
scripts/statistics/tests/kruskal_wallis_test.m,
scripts/strings/base2dec.m, scripts/strings/deblank.m,
scripts/strings/dec2base.m, scripts/strings/dec2bin.m,
scripts/strings/dec2hex.m, scripts/strings/mat2str.m,
scripts/strings/ostrsplit.m, scripts/strings/regexptranslate.m,
scripts/strings/str2num.m, scripts/strings/strcat.m,
scripts/strings/strjoin.m, scripts/strings/strsplit.m,
scripts/strings/strtok.m, scripts/strings/strtrim.m,
scripts/strings/strtrunc.m, scripts/strings/substr.m,
scripts/testfun/__run_test_suite__.m, scripts/testfun/speed.m,
scripts/testfun/test.m, scripts/time/asctime.m,
scripts/time/datenum.m, scripts/time/datevec.m,
scripts/time/weekday.m, src/Makefile.am, test/Makefile.am,
test/build-bc-overload-tests.sh, test/build-sparse-tests.sh,
test/jit.tst, test/line-continue.tst: Strip trailing whitespace.
author | John W. Eaton <jwe@octave.org> |
---|---|
date | Tue, 20 Jan 2015 08:26:57 -0500 |
parents | d63878346099 |
children | 0e1f5a750d00 |
line wrap: on
line source
## Copyright (C) 2005-2013 John W. Eaton ## ## This file is part of Octave. ## ## Octave 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. ## ## Octave 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 Octave; see the file COPYING. If not, see ## <http://www.gnu.org/licenses/>. ## -*- texinfo -*- ## @deftypefn {Function File} {[@var{x}, @var{obj}, @var{info}, @var{iter}, @var{nf}, @var{lambda}] =} sqp (@var{x0}, @var{phi}) ## @deftypefnx {Function File} {[@dots{}] =} sqp (@var{x0}, @var{phi}, @var{g}) ## @deftypefnx {Function File} {[@dots{}] =} sqp (@var{x0}, @var{phi}, @var{g}, @var{h}) ## @deftypefnx {Function File} {[@dots{}] =} sqp (@var{x0}, @var{phi}, @var{g}, @var{h}, @var{lb}, @var{ub}) ## @deftypefnx {Function File} {[@dots{}] =} sqp (@var{x0}, @var{phi}, @var{g}, @var{h}, @var{lb}, @var{ub}, @var{maxiter}) ## @deftypefnx {Function File} {[@dots{}] =} sqp (@var{x0}, @var{phi}, @var{g}, @var{h}, @var{lb}, @var{ub}, @var{maxiter}, @var{tol}) ## Solve the nonlinear program ## @tex ## $$ ## \min_x \phi (x) ## $$ ## @end tex ## @ifnottex ## ## @example ## @group ## min phi (x) ## x ## @end group ## @end example ## ## @end ifnottex ## subject to ## @tex ## $$ ## g(x) = 0 \qquad h(x) \geq 0 \qquad lb \leq x \leq ub ## $$ ## @end tex ## @ifnottex ## ## @example ## @group ## g(x) = 0 ## h(x) >= 0 ## lb <= x <= ub ## @end group ## @end example ## ## @end ifnottex ## @noindent ## using a sequential quadratic programming method. ## ## The first argument is the initial guess for the vector @var{x0}. ## ## The second argument is a function handle pointing to the objective ## function @var{phi}. The objective function must accept one vector ## argument and return a scalar. ## ## The second argument may also be a 2- or 3-element cell array of ## function handles. The first element should point to the objective ## function, the second should point to a function that computes the ## gradient of the objective function, and the third should point to a ## function that computes the Hessian of the objective function. If the ## gradient function is not supplied, the gradient is computed by finite ## differences. If the Hessian function is not supplied, a BFGS update ## formula is used to approximate the Hessian. ## ## When supplied, the gradient function @code{@var{phi}@{2@}} must accept ## one vector argument and return a vector. When supplied, the Hessian ## function @code{@var{phi}@{3@}} must accept one vector argument and ## return a matrix. ## ## The third and fourth arguments @var{g} and @var{h} are function ## handles pointing to functions that compute the equality constraints ## and the inequality constraints, respectively. If the problem does ## not have equality (or inequality) constraints, then use an empty ## matrix ([]) for @var{g} (or @var{h}). When supplied, these equality ## and inequality constraint functions must accept one vector argument ## and return a vector. ## ## The third and fourth arguments may also be 2-element cell arrays of ## function handles. The first element should point to the constraint ## function and the second should point to a function that computes the ## gradient of the constraint function: ## @tex ## $$ ## \Bigg( {\partial f(x) \over \partial x_1}, ## {\partial f(x) \over \partial x_2}, \ldots, ## {\partial f(x) \over \partial x_N} \Bigg)^T ## $$ ## @end tex ## @ifnottex ## ## @example ## @group ## [ d f(x) d f(x) d f(x) ] ## transpose ( [ ------ ----- ... ------ ] ) ## [ dx_1 dx_2 dx_N ] ## @end group ## @end example ## ## @end ifnottex ## The fifth and sixth arguments, @var{lb} and @var{ub}, contain lower ## and upper bounds on @var{x}. These must be consistent with the ## equality and inequality constraints @var{g} and @var{h}. If the ## arguments are vectors then @var{x}(i) is bound by @var{lb}(i) and ## @var{ub}(i). A bound can also be a scalar in which case all elements ## of @var{x} will share the same bound. If only one bound (lb, ub) is ## specified then the other will default to (-@var{realmax}, ## +@var{realmax}). ## ## The seventh argument @var{maxiter} specifies the maximum number of ## iterations. The default value is 100. ## ## The eighth argument @var{tol} specifies the tolerance for the ## stopping criteria. The default value is @code{sqrt (eps)}. ## ## The value returned in @var{info} may be one of the following: ## ## @table @asis ## @item 101 ## The algorithm terminated normally. ## Either all constraints meet the requested tolerance, or the stepsize, ## @tex ## $\Delta x,$ ## @end tex ## @ifnottex ## delta @var{x}, ## @end ifnottex ## is less than @code{@var{tol} * norm (x)}. ## ## @item 102 ## The BFGS update failed. ## ## @item 103 ## The maximum number of iterations was reached. ## @end table ## ## An example of calling @code{sqp}: ## ## @example ## function r = g (x) ## r = [ sumsq(x)-10; ## x(2)*x(3)-5*x(4)*x(5); ## x(1)^3+x(2)^3+1 ]; ## endfunction ## ## function obj = phi (x) ## obj = exp (prod (x)) - 0.5*(x(1)^3+x(2)^3+1)^2; ## endfunction ## ## x0 = [-1.8; 1.7; 1.9; -0.8; -0.8]; ## ## [x, obj, info, iter, nf, lambda] = sqp (x0, @@phi, @@g, []) ## ## x = ## ## -1.71714 ## 1.59571 ## 1.82725 ## -0.76364 ## -0.76364 ## ## obj = 0.053950 ## info = 101 ## iter = 8 ## nf = 10 ## lambda = ## ## -0.0401627 ## 0.0379578 ## -0.0052227 ## @end example ## ## @seealso{qp} ## @end deftypefn function [x, obj, info, iter, nf, lambda] = sqp (x0, objf, cef, cif, lb, ub, maxiter, tolerance) globals = struct (); # data and handles, needed and changed by subfunctions if (nargin < 2 || nargin > 8 || nargin == 5) print_usage (); endif if (!isvector (x0)) error ("sqp: X0 must be a vector"); endif if (rows (x0) == 1) x0 = x0'; endif have_hess = 0; if (iscell (objf)) switch (numel (objf)) case 1 obj_fun = objf{1}; obj_grd = @ (x) fd_obj_grd (x, obj_fun); case 2 obj_fun = objf{1}; obj_grd = objf{2}; case 3 obj_fun = objf{1}; obj_grd = objf{2}; obj_hess = objf{3}; have_hess = 1; otherwise error ("sqp: invalid objective function specification"); endswitch else obj_fun = objf; # No cell array, only obj_fun set obj_grd = @ (x) fd_obj_grd (x, obj_fun); endif ce_fun = @empty_cf; ce_grd = @empty_jac; if (nargin > 2) if (iscell (cef)) switch (numel (cef)) case 1 ce_fun = cef{1}; ce_grd = @ (x) fd_ce_jac (x, ce_fun); case 2 ce_fun = cef{1}; ce_grd = cef{2}; otherwise error ("sqp: invalid equality constraint function specification"); endswitch elseif (! isempty (cef)) ce_fun = cef; # No cell array, only constraint equality function set ce_grd = @ (x) fd_ce_jac (x, ce_fun); endif endif ci_fun = @empty_cf; ci_grd = @empty_jac; if (nargin > 3) ## constraint function given by user with possible gradient globals.cif = cif; ## constraint function given by user without gradient globals.cifcn = @empty_cf; if (iscell (cif)) if (length (cif) > 0) globals.cifcn = cif{1}; endif elseif (! isempty (cif)) globals.cifcn = cif; endif if (nargin < 5 || (nargin > 5 && isempty (lb) && isempty (ub))) ## constraint inequality function only without any bounds ci_grd = @ (x) fd_ci_jac (x, globals.cifcn); if (iscell (cif)) switch (length (cif)) case 1 ci_fun = cif{1}; case 2 ci_fun = cif{1}; ci_grd = cif{2}; otherwise error ("sqp: invalid inequality constraint function specification"); endswitch elseif (! isempty (cif)) ci_fun = cif; # No cell array, only constraint inequality function set endif else ## constraint inequality function with bounds present lb_idx = ub_idx = true (size (x0)); ub_grad = - (lb_grad = eye (rows (x0))); if (isvector (lb)) globals.lb = tmp_lb = lb(:); lb_idx(:) = tmp_idx = (lb != -Inf); globals.lb = globals.lb(tmp_idx, 1); lb_grad = lb_grad(lb_idx, :); elseif (isempty (lb)) if (isa (x0, "single")) globals.lb = tmp_lb = -realmax ("single"); else globals.lb = tmp_lb = -realmax; endif else error ("sqp: invalid lower bound"); endif if (isvector (ub)) globals.ub = tmp_ub = ub(:); ub_idx(:) = tmp_idx = (ub != Inf); globals.ub = globals.ub(tmp_idx, 1); ub_grad = ub_grad(ub_idx, :); elseif (isempty (ub)) if (isa (x0, "single")) globals.ub = tmp_ub = realmax ("single"); else globals.ub = tmp_ub = realmax; endif else error ("sqp: invalid upper bound"); endif if (any (tmp_lb > tmp_ub)) error ("sqp: upper bound smaller than lower bound"); endif bounds_grad = [lb_grad; ub_grad]; ci_fun = @ (x) cf_ub_lb (x, lb_idx, ub_idx, globals); ci_grd = @ (x) cigrad_ub_lb (x, bounds_grad, globals); endif endif # if (nargin > 3) iter_max = 100; if (nargin > 6 && ! isempty (maxiter)) if (isscalar (maxiter) && maxiter > 0 && fix (maxiter) == maxiter) iter_max = maxiter; else error ("sqp: invalid number of maximum iterations"); endif endif tol = sqrt (eps); if (nargin > 7 && ! isempty (tolerance)) if (isscalar (tolerance) && tolerance > 0) tol = tolerance; else error ("sqp: invalid value for TOLERANCE"); endif endif ## Initialize variables for search loop ## Seed x with initial guess and evaluate objective function, constraints, ## and gradients at initial value x0. ## ## obj_fun -- objective function ## obj_grad -- objective gradient ## ce_fun -- equality constraint functions ## ci_fun -- inequality constraint functions ## A == [grad_{x_1} cx_fun, grad_{x_2} cx_fun, ..., grad_{x_n} cx_fun]^T x = x0; obj = feval (obj_fun, x0); globals.nfun = 1; c = feval (obj_grd, x0); ## Choose an initial NxN symmetric positive definite Hessian approximation B. n = length (x0); if (have_hess) B = feval (obj_hess, x0); else B = eye (n, n); endif ce = feval (ce_fun, x0); F = feval (ce_grd, x0); ci = feval (ci_fun, x0); C = feval (ci_grd, x0); A = [F; C]; ## Choose an initial lambda (x is provided by the caller). lambda = 100 * ones (rows (A), 1); qp_iter = 1; alpha = 1; info = 0; iter = 0; # report (); # Called with no arguments to initialize reporting # report (iter, qp_iter, alpha, __sqp_nfun__, obj); while (++iter < iter_max) ## Check convergence. This is just a simple check on the first ## order necessary conditions. nr_f = rows (F); lambda_e = lambda((1:nr_f)'); lambda_i = lambda((nr_f+1:end)'); con = [ce; ci]; t0 = norm (c - A' * lambda); t1 = norm (ce); t2 = all (ci >= 0); t3 = all (lambda_i >= 0); t4 = norm (lambda .* con); if (t2 && t3 && max ([t0; t1; t4]) < tol) info = 101; break; endif ## Compute search direction p by solving QP. g = -ce; d = -ci; [p, obj_qp, INFO, lambda] = qp (x, B, c, F, g, [], [], d, C, Inf (size (d))); info = INFO.info; ## FIXME -- check QP solution and attempt to recover if it has ## failed. For now, just warn about possible problems. id = "Octave:SQP-QP-subproblem"; switch (info) case 2 warning (id, "sqp: QP subproblem is non-convex and unbounded"); case 3 warning (id, "sqp: QP subproblem failed to converge in %d iterations", INFO.solveiter); case 6 warning (id, "sqp: QP subproblem is infeasible"); endswitch ## Choose mu such that p is a descent direction for the chosen ## merit function phi. [x_new, alpha, obj_new, globals] = ... linesearch_L1 (x, p, obj_fun, obj_grd, ce_fun, ci_fun, lambda, ... obj, globals); ## Evaluate objective function, constraints, and gradients at x_new. c_new = feval (obj_grd, x_new); ce_new = feval (ce_fun, x_new); F_new = feval (ce_grd, x_new); ci_new = feval (ci_fun, x_new); C_new = feval (ci_grd, x_new); A_new = [F_new; C_new]; ## Set ## ## s = alpha * p ## y = grad_x L (x_new, lambda) - grad_x L (x, lambda}) y = c_new - c; if (! isempty (A)) t = ((A_new - A)'*lambda); y -= t; endif delx = x_new - x; if (norm (delx) < tol * norm (x)) info = 101; break; endif if (have_hess) B = feval (obj_hess, x); else ## Update B using a quasi-Newton formula. delxt = delx'; ## Damped BFGS. Or maybe we would actually want to use the Hessian ## of the Lagrangian, computed directly? d1 = delxt*B*delx; t1 = 0.2 * d1; t2 = delxt*y; if (t2 < t1) theta = 0.8*d1/(d1 - t2); else theta = 1; endif r = theta*y + (1-theta)*B*delx; d2 = delxt*r; if (d1 == 0 || d2 == 0) info = 102; break; endif B = B - B*delx*delxt*B/d1 + r*r'/d2; endif x = x_new; obj = obj_new; c = c_new; ce = ce_new; F = F_new; ci = ci_new; C = C_new; A = A_new; # report (iter, qp_iter, alpha, __sqp_nfun__, obj); endwhile if (iter >= iter_max) info = 103; endif nf = globals.nfun; endfunction function [merit, obj, globals] = phi_L1 (obj, obj_fun, ce_fun, ci_fun, ... x, mu, globals) ce = feval (ce_fun, x); ci = feval (ci_fun, x); idx = ci < 0; con = [ce; ci(idx)]; if (isempty (obj)) obj = feval (obj_fun, x); globals.nfun++; endif merit = obj; t = norm (con, 1) / mu; if (! isempty (t)) merit += t; endif endfunction function [x_new, alpha, obj, globals] = ... linesearch_L1 (x, p, obj_fun, obj_grd, ce_fun, ci_fun, lambda, ... obj, globals) ## Choose parameters ## ## eta in the range (0, 0.5) ## tau in the range (0, 1) eta = 0.25; tau = 0.5; delta_bar = sqrt (eps); if (isempty (lambda)) mu = 1 / delta_bar; else mu = 1 / (norm (lambda, Inf) + delta_bar); endif alpha = 1; c = feval (obj_grd, x); ce = feval (ce_fun, x); [phi_x_mu, obj, globals] = phi_L1 (obj, obj_fun, ce_fun, ci_fun, x, ... mu, globals); D_phi_x_mu = c' * p; d = feval (ci_fun, x); ## only those elements of d corresponding ## to violated constraints should be included. idx = d < 0; t = - norm ([ce; d(idx)], 1) / mu; if (! isempty (t)) D_phi_x_mu += t; endif while (1) [p1, obj, globals] = phi_L1 ([], obj_fun, ce_fun, ci_fun, ... x+alpha*p, mu, globals); p2 = phi_x_mu+eta*alpha*D_phi_x_mu; if (p1 > p2) ## Reset alpha = tau_alpha * alpha for some tau_alpha in the ## range (0, tau). tau_alpha = 0.9 * tau; # ?? alpha = tau_alpha * alpha; else break; endif endwhile x_new = x + alpha * p; endfunction function grd = fdgrd (f, x) if (! isempty (f)) y0 = feval (f, x); nx = length (x); grd = zeros (nx, 1); deltax = sqrt (eps); for i = 1:nx t = x(i); x(i) += deltax; grd(i) = (feval (f, x) - y0) / deltax; x(i) = t; endfor else grd = zeros (0, 1); endif endfunction function jac = fdjac (f, x) nx = length (x); if (! isempty (f)) y0 = feval (f, x); nf = length (y0); nx = length (x); jac = zeros (nf, nx); deltax = sqrt (eps); for i = 1:nx t = x(i); x(i) += deltax; jac(:,i) = (feval (f, x) - y0) / deltax; x(i) = t; endfor else jac = zeros (0, nx); endif endfunction function grd = fd_obj_grd (x, obj_fun) grd = fdgrd (obj_fun, x); endfunction function res = empty_cf (x) res = zeros (0, 1); endfunction function res = empty_jac (x) res = zeros (0, length (x)); endfunction function jac = fd_ce_jac (x, ce_fun) jac = fdjac (ce_fun, x); endfunction function jac = fd_ci_jac (x, cifcn) ## cifcn = constraint function without gradients and lb or ub jac = fdjac (cifcn, x); endfunction function res = cf_ub_lb (x, lbidx, ubidx, globals) ## combine constraint function with ub and lb if (isempty (globals.cifcn)) res = [x(lbidx,1)-globals.lb; globals.ub-x(ubidx,1)]; else res = [feval(globals.cifcn,x); x(lbidx,1)-globals.lb; globals.ub-x(ubidx,1)]; endif endfunction function res = cigrad_ub_lb (x, bgrad, globals) cigradfcn = @ (x) fd_ci_jac (x, globals.cifcn); if (iscell (globals.cif) && length (globals.cif) > 1) cigradfcn = globals.cif{2}; endif if (isempty (cigradfcn)) res = bgrad; else res = [feval(cigradfcn,x); bgrad]; endif endfunction # Utility function used to debug sqp function report (iter, qp_iter, alpha, nfun, obj) if (nargin == 0) printf (" Itn ItQP Step Nfun Objective\n"); else printf ("%5d %4d %8.1g %5d %13.6e\n", iter, qp_iter, alpha, nfun, obj); endif endfunction %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% Test Code %!function r = __g (x) %! r = [sumsq(x)-10; %! x(2)*x(3)-5*x(4)*x(5); %! x(1)^3+x(2)^3+1 ]; %!endfunction %! %!function obj = __phi (x) %! obj = exp (prod (x)) - 0.5*(x(1)^3 + x(2)^3 + 1)^2; %!endfunction %! %!test %! %! x0 = [-1.8; 1.7; 1.9; -0.8; -0.8]; %! %! [x, obj, info, iter, nf, lambda] = sqp (x0, @__phi, @__g, []); %! %! x_opt = [-1.717143501952599; %! 1.595709610928535; %! 1.827245880097156; %! -0.763643103133572; %! -0.763643068453300]; %! %! obj_opt = 0.0539498477702739; %! %! assert (x, x_opt, 8*sqrt (eps)); %! assert (obj, obj_opt, sqrt (eps)); %% Test input validation %!error sqp () %!error sqp (1) %!error sqp (1,2,3,4,5,6,7,8,9) %!error sqp (1,2,3,4,5) %!error sqp (ones (2,2)) %!error sqp (1, cell (4,1)) %!error sqp (1, cell (3,1), cell (3,1)) %!error sqp (1, cell (3,1), cell (2,1), cell (3,1)) %!error sqp (1, cell (3,1), cell (2,1), cell (2,1), ones (2,2),[]) %!error sqp (1, cell (3,1), cell (2,1), cell (2,1),[], ones (2,2)) %!error sqp (1, cell (3,1), cell (2,1), cell (2,1),1,-1) %!error sqp (1, cell (3,1), cell (2,1), cell (2,1),[],[], ones (2,2)) %!error sqp (1, cell (3,1), cell (2,1), cell (2,1),[],[],-1) %!error sqp (1, cell (3,1), cell (2,1), cell (2,1),[],[],1.5) %!error sqp (1, cell (3,1), cell (2,1), cell (2,1),[],[],[], ones (2,2)) %!error sqp (1, cell (3,1), cell (2,1), cell (2,1),[],[],[],-1)