PdscoTL

Purpose

pdscoTL solves linearly constrained convex nonlinear optimization problems.

<equation id="eqn:pdsco"> ${\begin{array}{cccccc}\min \limits _{x}&f(x)\\\mathrm {s/t} &x_{L}&\leq &x&\leq &x_{U}\\{}&b_{L}&\leq &Ax&\leq &b_{U}\\\end{array}}$ </equation>

where $f(x)$ is a convex separable nonlinear function, $x,x_{L},x_{U}\in \mathbb {R} ^{n}$ , $A\in \mathbb {R} ^{m\times n}$ and $b_{L},b_{U}\in \mathbb {R} ^{m}$.

Calling Syntax

Result=tomRun('pdsco',Prob,...);

Inputs

Prob	Problem description structure. The following fields are used:
	x_0	Initial x vector, used if non-empty.
	A	The linear constraints coefficient matrix.
	b_L,b_U	Lower and upper bounds for the linear constraints.
	HessPattern	Non-zero pattern for the objective function. Only the diagonal is needed. Default if empty is the unit matrix.
	PriLevOpt	Print level in pdsco solver. If > 0: prints summary information.
	SOL	Structure with SOL special parameters:
	pdco	Options structure with fields as defined by pdscoSet.
	gamma	Primal regularization parameter.
	delta	Dual regularization parameter.
	y0	Initial dual parameters for linear constraints (default 0)
	z0	Initial dual parameters for simple bounds (default 1/N ) xsize,zsize are used to scale (x, y, z). Good estimates should improve the performance of the barrier method.
	xsize	Estimate of the biggest x at the solution. (default 1/N )
	zsize	Estimate of the biggest z at the solution. (default 1/N )
	optParam	Structure with optimization parameters. The following fields are used:
	MaxIter	Maximum number of iterations. (Prob.SOL.pdco.MaxIter).
	MinorIter	Maximum number of iterations in LSQR (Prob.SOL.pdco.LSQRMaxIter).
	eps_x	Accuracy for satisfying x. * z = 0
	bTol	Accuracy for satisfying Ax + r = b, ATy + z = ∇f (x) and x - x1 = bL , x + x2 =bU, where x1 , x2 > 0. (Prob.SOL.pdco.FeaTol)
	wait	0 - solve the problem with default internal parameters; 1 - pause: allows interactive resetting of parameters. (Prob.SOL.pdco.wait)

Outputs

Result	Structure with result from optimization. The following fields are set by pdscoTL:
	x_k	Solution vector
	f_k	Function value at optimum
	g_k	Gradient of the function at the solution
	H_k	Hessian of the function at the solution, diagonal only
	x_0	Initial solution vector
	f_0	Function value at start, x = x_0
	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;
	v_k	Lagrangian multipliers (orignal bounds + constraints )
	y_k	Lagrangian multipliers (for bounds + dual solution vector) The full [z; y] vector as returned from pdsco, including slacks and extra linear constraints after rewriting constraints: -inf < b_L < A * x < b_U < inf ; non-inf lower AND upper bounds
	ExitFlag	Tomlab Exit status from pdsco MEX
	Inform	pdsco information parameter: 0 == Solution found 1 == Too many iterations 2 == Linesearch failed too often
	Iter	Number of iterations
	FuncEv	Number of function evaluations
	GradEv	Number of gradient evaluations
	HessEv	Number of Hessian evaluations
	Solver	Name of the solver ('pdsco')
	SolverAlgorithm	Description of the solver

Description

pdsco implements an primal-dual barrier method developed at Stanford Systems Optimization Laboratory (SOL). The problem (#Equation: pdsco) is first reformulated into SOL PDSCO form:

Equation: pdsco

${\begin{array}{clll}\min \limits _{x}&f(x)\\\mathrm {s/t} &x&\geq &x_{U}\\{}&Ax&=&b.\\\end{array}}$

The problem actually solved by pdsco is

${\begin{array}{lllcll}\min \limits _{x,r}&{f(x)+{\frac {1}{2}}\|\gamma x\|^{2}+{\frac {1}{2}}\|r/\delta \|^{2}}\\\\\mathrm {s/t} &&&x&\geq &0\\{}&Ax&+&r&=&b\\{}&{r\mathrm {\ unconstrained} }\\\end{array}}$

where $\gamma$ is the primal regularization parameter, typically small but 0 is allowed. Furthermore, $\delta$ is the dual regularization parameter, typically small or 1; must be strictly greater than zero.

With positive $\gamma ,\delta$ the primal-dual solution $(x,y,z)$ is bounded and unique.

See pdsco.m for a detailed discussion of $\gamma$ and $\delta$. Note that in pdsco.m, the objective $f(x)$ is denoted $\phi (x)$, $bl==x\_L$ and $bu==x\_U$.

Examples

Problem 14 and 15 in tomlab/testprob/con prob.m are good examples of the use of pdscoTL.

M-files Used

pdscoSet.m, pdsco.m, Tlsqrmat.m

See Also

pdcoTL.m