UcSolve

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Purpose

Solve unconstrained nonlinear optimization problems with simple bounds on the variables.

ucSolve solves problems of the form

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

where Failed to parse (unknown function "\MATHSET"): {\displaystyle x,x_{L},x_{U}\in \MATHSET{R} ^{n}} .

Calling Syntax

Result = ucSolve(Prob, varargin)

Description of Inputs

Prob	Problem description structure. The following fields are used:
	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).
	f_Low	Lower bound on function value.
	Solver.Alg	Solver algorithm to be run:
		0: Gives default, either Newton or BFGS.
		1: Newton with subspace minimization, using SVD.
		2: Safeguarded BFGS with inverse Hessian updates (standard).
		3: Safeguarded BFGS with Hessian updates.
		4: Safeguarded DFP with inverse Hessian updates.
		5: Safeguarded DFP with Hessian updates.
		6: Fletcher-Reeves CG.
		7: Polak-Ribiere CG.
		8: Fletcher conjugate descent CG-method.
	Solver.Method	Method used to solve equation system:
		0: SVD (default).
		1: LU-decomposition.
		2: LU-decomposition with pivoting.
		3: Matlab built in QR.
		4: Matlab inversion.
		5: Explicit inverse.
	Solver.Method	Restart or not for C-G method:
		0: Use restart in CG-method each n:th step.
		1: Use restart in CG-method each n:th step.
	LineParam	Structure with line search parameters, see routine LineSearch and Table 140.
	optParam	Structure with special fields for optimization parameters, see Table 141.
		Fields used are: eps absf, eps f, eps g, eps x, eps Rank, MaxIter, wait, size x, xTol, size f, LineSearch, LineAlg, xTol, IterPrint and QN InitMatrix.
	PriLevOpt	Print level.
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.
	H_k	Hessian value at optimum.
	B_k	Quasi-Newton approximation of the Hessian at optimum.
	v_k	Lagrange multipliers.
	x_0	Starting point.
	f_0	Function value at start.
	xState	State of each variable, described in Table 150.
	Iter	Number of iterations.
	ExitFlag	0 if convergence to local min. Otherwise errors.
	Inform	Binary code telling type of convergence:
		1: Iteration points are close.
		2: Projected gradient small.
		4: Relative function value reduction low for LowIts iterations.
		101: Maximum number of iterations reached.
		102: Function value below given estimate.
		104: Convergence to a saddle point.
	Solver	Solver used.
	SolverAlgorithm	Solver algorithm used.
	Prob	Problem structure used.

Description

The solver ucSolve includes several of the most popular search step methods for unconstrained optimization. The search step methods included in ucSolve are: the Newton method, the quasi-Newton BFGS and DFP methods, the Fletcher-Reeves and Polak-Ribiere conjugate-gradient method, and the Fletcher conjugate descent method. The quasi-Newton methods may either update the inverse Hessian (standard) or the Hessian itself. The Newton method and the quasi-Newton methods updating the Hessian are using a subspace minimization technique to handle rank problems, see Lindstr¨om \[53\]. The quasi-Newton algorithms also use safe guarding techniques to avoid rank problem in the updated matrix. The line search algorithm in the routine LineSearch is a modified version of an algorithm by Fletcher \[20\]. Bound constraints are treated as described in Gill, Murray and Wright \[28\]. The accuracy in the line search is critical for the performance of quasi-Newton BFGS and DFP methods and for the CG methods. If the accuracy parameter Prob.LineParam.sigma is set to the default value 0.9, ucSolve changes it automatically according to:

Prob.Solver.Alg	Prob.LineParam.sigma
4,5 (DFP)	0.2
6,7,8 (CG)	0.01

M-files Used

ResultDef.m, LineSearch.m, iniSolve.m, tomSolve.m, endSolve.m

See Also

• clsSolve