ConSolve

Purpose

Solve general constrained nonlinear optimization problems.

conSolve 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 $x,x_{L}$, $x_{U}\in \mathbb {R} ^{n}$, $c(x)$, $c_{L}$, $c_{U}\in \mathbb {R} ^{m_{1}}$, $A\in \mathbb {R} ^{m_{2}\times n}$ and $b_{L},b_{U}\in \mathbb {R} ^{m_{2}}$.

Calling Syntax

Result = conSolve(Prob, varargin) 
Result = tomRun('conSolve', Prob);

Inputs

Prob	Problem description structure. The following fields are used:
	Solver.Alg	Choice of algorithm. Also affects how derivatives are obtained. See following fields. 0,1,2: Schittkowski SQP. 3,4: Han-Powell SQP.
	x_0	Starting point.
	x_L	Lower bounds on the variables.
	x_U	Upper bounds on the variables.
	b_L	Lower bounds on the linear constraints.
	b_U	Upper bounds on the linear constraints.
	A	Constraint matrix for linear constraints.
	c_L	Lower bounds on the general constraints.
	c_U	Upper bounds on the general constraints.
	NumDiff	How to obtain derivatives (gradient, Hessian).
	ConsDiff	How to obtain the constraint derivative matrix.
	SolverQP	Name of the solver used for QP subproblems. If empty, the default solver is used. See GetSolver.m and tomSolve.m.
	f_Low	A lower bound on the optimal function value, see LineParam.fLowBnd below.
		Used in convergence tests, f_k(x_k) <= f_Low. Only a feasible point x_k is accepted.
	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 δ(x)/dx.
	PriLevOpt	Print level.
	optParam	Structure with optimization parameters, see TOMLAB Appendix A. Fields that are used: bTol, cTol, eps_absf, eps_g, eps_x, eps_Rank, IterPrint, MaxIter, QN InitMatrix, size_f, size_x, xTol and wait.
	LineParam	Structure with line search parameters. See TOMLAB Appendix A.
	fLowBnd	A lower bound on the global optimum of f(x). The user might also give lower bound estimate in Prob.f_Low conSolve computes LineParam.fLowBnd as: LineParam.fLowBnd = max(Prob.f_Low,Prob.LineParam.fLowBnd).
varargin	Other parameters directly sent to low level routines.

Outputs

Result	Structure with result from optimization. The following fields are changed:
	x_k	Optimal point.
	v_k	Lagrange multipliers.
	f_k	Function value at optimum.
	g_k	Gradient value at optimum.
	H_k	Hessian value at optimum.
	x_0	Starting point.
	f_0	Function value at start.
	c_k	Value of constraints at optimum.
	cJac	Constraint Jacobian at optimum.
	xState	State of each variable, described in TOMLAB Appendix B.
	bState	State of each linear constraint, described in TOMLAB Appendix B.
	cState	State of each nonlinear constraint.
	Iter	Number of iterations.
	ExitFlag	Flag giving exit status.
	ExitText	'Text string giving ExitFlag and Inform information.
	Inform	Code telling type of convergence: 1: Iteration points are close. Result Structure with result from optimization. The following fields are changed:, continued 2: Small search direction. 3: Iteration points are close and Small search direction. 4: Gradient of merit function small. 5: Iteration points are close and gradient of merit function small. 6: Small search direction and gradient of merit function small. 7: Iteration points are close, small search direction and gradient of merit func-tion small. 8: Small search direction p and constraints satisfied. 101: Maximum number of iterations reached. 102: Function value below given estimate. 103: Iteration points are close, but constraints not fulfilled. Too large penalty weights to be able to continue. Problem is maybe infeasible. 104: Search direction is zero and infeasible constraints. The problem is very likely infeasible. 105: Merit function is infinity. 106: Penalty weights too high.
	Solver	Solver used.
	SolverAlgorithm	Solver algorithm used.
	Prob	Problem structure used.

Description

The routine conSolve implements two SQP algorithms for general constrained minimization problems. One imple- mentation, Solver.Alg = 0, 1, 2, is based on the SQP algorithm by Schittkowski with Augmented Lagrangian merit function. The other, Solver.Alg = 3, 4, is an implementation of the Han-Powell SQP method.

The Hessian in the QP subproblems are determined in one of several ways, dependent on the input parameters. The following table shows how the algorithm and Hessian method is selected.

Solver.Alg	NumDiff	AutoDiff	isempty(FUNCS.H)	Hessian computation	Algorithm
0	0	0	0	Analytic Hessian	Schittkowski SQP
0	any	any	any	BFGS	Schittkowski SQP
1	0	0	0	Analytic Hessian	Schittkowski SQP
1	0	0	1	Numerical differences H	Schittkowski SQP
1	> 0	0	any	Numerical differences g,H	Schittkowski SQP
1	< 0	0	any	Numerical differences H	Schittkowski SQP
1	any	1	any	Automatic differentiation	Schittkowski SQP
2	0	0	any	BFGS	Schittkowski SQP
2	= 0	0	any	BFGS, numerical gradient g	Schittkowski SQP
2	any	1	any	BFGS, automatic diff gradient	Schittkowski SQP
3	0	0	0	Analytic Hessian	Han-Powell SQP
3	0	0	1	Numerical differences H	Han-Powell SQP
3	> 0	0	any	Numerical differences g,H	Han-Powell SQP
3	< 0	0	any	Numerical differences H	Han-Powell SQP
3	any	1	any	Automatic differentiation	Han-Powell SQP
4	0	0	any	BFGS	Han-Powell SQP
4	= 0	0	any	BFGS, numerical gradient g	Han-Powell SQP
4	any	1	any	BFGS, automatic diff gradient	Han-Powell SQP

M-files Used

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

See Also

nlpSolve, sTrustr