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6  TOMLAB /DFNLP Solver Reference

A detailed description of the TOMLAB /DFNLP[3] solver interface is given below. Also see the M-file help for dfnlpTL.m.

6.1  dfnlpTL

Purpose
Solves nonlinear data fitting problems.

DFNLP solves problems of the form where x,x_L,x_U Rⁿ,  A R^{m₁ × n},  b_L,b_U R^m₁ and c(x), c_L, c_U R^m₂.
L is the number of objective functions. For details on the objective function see that different methods below.

Calling Syntax
Prob = clsAssign( ... );
Result = tomRun('dfnlp',Prob,...);


Description of Inputs

Prob	Problem description structure. The following fields are used:
	A	Linear constraints coefficient matrix.
	x_L, x_U	Bounds on variables.
	b_L, b_U	Bounds on linear constraints.
	c_L, c_U	Bounds on nonlinear constraints. For equality constraints (or fixed variables), set e.g. b_L(k) == b_U(k).
	PriLevOpt	Print level in MEX interface.
	DFNLP	Structure with special fields for the DFNLP solver:
	model	Desired scalar transformation as indicated below.
	1	L1 - DATA FITTING: Minimize |F(1,X)| + ... + |F(L,X)| by introducing L additional variables Z(1),...,Z(L) and L + L additional inequality constraints, the above problem is transformed into a smooth nonlinear programming problem, that is then solved by a sequential quadratic programming algorithm.
	2	L2 - OR LEAST SQUARES DATA FITTING: Minimize F(1,X)2 + ... + F(L,X)2 The algorithm transform the above problem into an equivalent nonlinear programming problem by introducing L additional variables Z(1),...,Z(L). The new objective function is H(X,Z) = 0.5*(Z(1)2 + ... + Z(L)2) and L equality constraints of the form F(J,X) − Z(J) = 0 are formulated, J = 1,...,L.
	3	MAXIMUM-NORM DATA FITTING: Minimize Maximum |F(I,X)| : I=1,...,L The problem is transformed into a smooth nonlinear programming problem by introducing one additional variable Z yielding the objective function H(X,Z) = Z and L + L additional inequality constraints of the form −F(J,X) + Z >= 0, J=1,...,L, F(J,X) + Z >= 0 , J=1,...,L.
	4	MAXIMUM FUNCTION: Minimize Maximum F(I,X) : I=1,...,L Similar to the model above, one additional variable X is introduced to get a simple objective function of the type H(X,Z) = Z and L additional restrictions −F(J,X) + Z >= 0 , J=1,...,L.
	maxfun	The integer variable defines an upper bound for the number of function calls during the line search.
	maxit	Maximum number of outer iterations, where one iteration corresponds to one formulation and solution of the quadratic programming subproblem, or, alternatively, one evaluation of gradients.
	acc	The user has to specify the desired final accuracy (e.g. 1.0e-7). The termination accuracy should not be smaller than the accuracy by which gradients are computed.
	ressiz	The user must indicate a guess for the approximate size of the least squares residual, i.e. a low positive real number if the residual is supposed to be small, and a large one in the order of 1 if the residual is supposed to be large. If model is not equal to 2, ressiz must not be set by the user.
	PrintFile	Name of DFNLP Print file. Amount and type of printing determined by PriLevOpt.

Description of Outputs

Result	Structure with result from optimization. The following fields are set:
	f_k	Function value at optimum.
	g_k	Gradient of the function.
	x_k	Solution vector.
	x_0	Initial solution vector.
	c_k	Nonlinear constraint residuals.
	cJac	Nonlinear constraint gradients.
	xState	State of variables. Free == 0; On lower == 1; On upper == 2; Fixed == 3;
	bState	State of linear constraints. Free == 0; Lower == 1; Upper == 2; Equality == 3;
	cState	State of nonlinear constraints. Free == 0; Lower == 1; Upper == 2; Equality == 3;
	ExitFlag	Exit status from DFNLP MEX.
	ExitText	Exit text from DFNLP MEX.
	Inform	DFNLP information parameter.
	FuncEv	Number of function evaluations.
	GradEv	Number of gradient evaluations.
	ConstrEv	Number of constraint evaluations.
	QP.B	Basis vector in TOMLAB QP standard.
	Solver	Name of the solver (DFNLP).
	SolverAlgorithm	Description of the solver.
	DFNLP.u	Contains the multipliers with respect to the actual iterate stored in X. The first M locations contain the multipliers of the nonlinear constraints, the subsequent N locations the multipliers of the lower bounds, and the final N locations the multipliers of the upper bounds subject to the scalar subproblem chosen. At an optimal solution, all multipliers with respect to inequality constraints should be nonnegative.
	DFNLP.act	The logical array indicates constraints, which DFNLP considers to be active at the last computed iterate.

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