STrustr

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Purpose

Solve optimization problems constrained by a convex feasible region.

sTrustr solves problems of the form

${\begin{array}{cccccc}\min \limits _{x}&f(x)&&&&\\s/t&x_{L}&\leq &x&\leq &x_{U}\\&b_{L}&\leq &Ax&\leq &b_{U}\\&c_{L}&\leq &c(x)&\leq &c_{U}\\\end{array}}$

where Failed to parse (unknown function "\MATHSET"): {\displaystyle x,x_{L},x_{U}\in \MATHSET{R}^{n}} , Failed to parse (unknown function "\MATHSET"): {\displaystyle c(x),c_{L},c_{U}\in \MATHSET{R}^{m_{1}}} , Failed to parse (unknown function "\MATHSET"): {\displaystyle A\in \MATHSET{R}^{m_{2}\times n}} and Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b_{L},b_{U}\in \MATHSET{R}^{m_{2}}} .

Calling Syntax

Result = sTrustr(Prob, varargin)

Description of Inputs

Prob	Problem description structure. The following fields are used:
	A	Constraint matrix for linear constraints.
	b_L	Lower bounds on the linear constraints.
	b_U	Upper bounds on the linear constraints.
	c_L	Lower bounds on the general constraints.
	c_U	Upper bounds on the general constraints.
	x_L	Lower bounds on the variables.
	x_U	Upper bounds on the variables.
	x_0	Starting point.
	FUNCS.f	Name of m-file computing the objective function f (x).
	FUNCS.g	Name of m-file computing the gradient vector g(x).
	FUNCS.H	Name of m-file computing the Hessian matrix H (x).
	FUNCS.c	Name of m-file computing the vector of constraint functions c(x).
	FUNCS.dc	Name of m-file computing the matrix of constraint normals ?c(x)/dx.
	optParam	Structure with special fields for optimization parameters, see Table 141.
		Fields used are: eps f, eps g, eps c, eps x, eps Rank, MaxIter, wait, size x, size f, xTol, LowIts, PriLev, method and QN InitMatrix.
	PartSep	Structure with special fields for partially separable functions, see Table 142.
	varargin	Other parameters directly sent to low level routines.

Description of Outputs

Result	Structure with result from optimization. The following fields are changed:
	x_k	Optimal point.
	f_k	Function value at optimum.
	g_k	Gradient value at optimum.
	c_k	Value of constraints at optimum.
	H_k	Hessian value at optimum.
	v_k	Lagrange multipliers.
	x_0	Starting point.
	f_0	Function value at start.
	cJac	Constraint Jacobian at optimum.
	xState	State of each variable, described in Table 150.
	Iter	Number of iterations.
	ExitFlag	Flag giving exit status.
	Inform	Binary code telling type of convergence:
		1: Iteration points are close.
		2: Projected gradient small.
		3: Iteration points are close and projected gradient small.
		4: Relative function value reduction low for LowIts iterations.
		5: Iteration points are close and relative function value reduction low for LowIts iterations.
		6: Projected gradient small and relative function value reduction low for LowIts iterations.
		7: Iteration points are close, projected gradient small and relative function value reduction low for LowIts iterations.
		8: Too small trust region.
		9: Trust region small. Iteration points close.
		10: Trust region and projected gradient small.
		11: Trust region and projected gradient small, iterations close.
		12: Trust region small, Relative f(x) reduction low.
		13: Trust region small, Relative f(x) reduction low. Iteration points are close.
		14: Trust region small, Relative f(x) reduction low. Projected gradient small.
		15: Trust region small, Relative f(x) reduction low. Iteration points close, Projected gradient small.
		101: Maximum number of iterations reached.
		102: Function value below given estimate.
		103: Convergence to saddle point (eigenvalues computed).
	Solver	Solver used.
	SolverAlgorithm	Solver algorithm used.
	Prob	Problem structure used.

Description

The routine sTrustr is a solver for general constrained optimization, which uses a structural trust region algorithm combined with an initial trust region radius algorithm (itrr). The feasible region defined by the constraints must be convex. The code is based on the algorithms in \[13\] and \[67\]. BFGS or DFP is used for the Quasi-Newton update, if the analytical Hessian is not used. sTrustr calls internal routine itrr.

M-files Used

qpSolve.m, tomSolve.m, iniSolve.m, endSolve.m

See Also

• conSolve
• nlpSolve
• clsSolve