@c Copyright (C) 1996-2024 The Octave Project Developers
@c
@c This file is part of Octave.
@c
@c Octave is free software: you can redistribute it and/or modify it
@c under the terms of the GNU General Public License as published by
@c the Free Software Foundation, either version 3 of the License, or
@c (at your option) any later version.
@c
@c Octave is distributed in the hope that it will be useful, but
@c WITHOUT ANY WARRANTY; without even the implied warranty of
@c MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
@c GNU General Public License for more details.
@c
@c You should have received a copy of the GNU General Public License
@c along with Octave; see the file COPYING.  If not, see
@c <https://www.gnu.org/licenses/>.

@node Optimization
@chapter Optimization

Octave comes with support for solving various kinds of optimization
problems.  Specifically Octave can solve problems in Linear Programming,
Quadratic Programming, Nonlinear Programming, and Linear Least Squares
Minimization.

@menu
* Linear Programming::
* Quadratic Programming::
* Nonlinear Programming::
* Linear Least Squares::
@end menu

@c @cindex linear programming
@cindex quadratic programming
@cindex nonlinear programming
@cindex optimization
@cindex LP
@cindex QP
@cindex NLP

@node Linear Programming
@section Linear Programming

Octave can solve Linear Programming problems using the @code{glpk}
function.  That is, Octave can solve

@tex
$$
  \min_x c^T x
$$
@end tex
@ifnottex

@example
min C'*x
@end example

@end ifnottex
subject to the linear constraints
@tex
$Ax = b$ where $x \geq 0$.
@end tex
@ifnottex
@math{A*x = b} where @math{x @geq{} 0}.
@end ifnottex

@noindent
The @code{glpk} function also supports variations of this problem.

@DOCSTRING(glpk)

@node Quadratic Programming
@section Quadratic Programming

Octave can also solve Quadratic Programming problems, this is
@tex
$$
 \min_x {1 \over 2} x^T H x + x^T q
$$
@end tex
@ifnottex

@example
min 0.5 x'*H*x + x'*q
@end example

@end ifnottex
subject to
@tex
$$
 A x = b \qquad lb \leq x \leq ub \qquad A_{lb} \leq A_{in} x \leq A_{ub}
$$
@end tex
@ifnottex

@example
@group
     A*x = b
     lb <= x <= ub
     A_lb <= A_in*x <= A_ub
@end group
@end example

@end ifnottex

@DOCSTRING(qp)

@DOCSTRING(pqpnonneg)

@node Nonlinear Programming
@section Nonlinear Programming

Octave can also perform general nonlinear minimization using a
successive quadratic programming solver.

@DOCSTRING(sqp)

@node Linear Least Squares
@section Linear Least Squares

Octave also supports linear least squares minimization.  That is,
Octave can find the parameter @math{b} such that the model
@tex
$y = xb$
@end tex
@ifnottex
@math{y = x*b}
@end ifnottex
fits data @math{(x,y)} as well as possible, assuming zero-mean
Gaussian noise.  If the noise is assumed to be isotropic the problem
can be solved using the @samp{\} or @samp{/} operators, or the @code{ols}
function.  In the general case where the noise is assumed to be anisotropic
the @code{gls} is needed.

@DOCSTRING(ols)

@DOCSTRING(gls)

@DOCSTRING(lsqnonneg)

@DOCSTRING(lscov)

@DOCSTRING(optimset)

@DOCSTRING(optimget)