PdcoTL

Purpose

pdcoTL solves linearly constrained convex nonlinear optimization problems of the kind

Equation: pdcoTL

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

where $f(x)$ is a convex 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('pdco',Prob,...);

Inputs

Prob	Problem description structure. The following fields are used:
	x_0	Initial x vector, used if non-empty.
	A	The linear constraint matrix.
	b_L,b_U	Lower and upper bounds for the linear constraints.
	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 pdcoSet.
	d1	Primal regularization vector. Must be a positive vector (length n) or scalar, in which case D1 = diag(d1) is used. Default: 10-4 .
	d2	Dual regularization vector. Must be a positive vector (length m) or a scalar value, in which case D2 = diag(d2) is used. Default: 10-4 .
	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 x1 . * z1 = 0, x2 .z1 = 0, where z = z1 - z2 and z1 , z2 > 0.(Prob.SOL.pdco.OptTol)
	bTol	Accuracy for satisfying Ax + D2r = b, AT y + 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 pdcoTL
	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 pdco, 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 pdco MEX
	Inform	pdcoinformation parameter: 0 = Solution found;
	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 ('pdco')
	SolverAlgorithm	Description of the solver

Description

pdco implements an primal-dual barrier method developed at Stanford Systems Optimization Laboratory (SOL).

The problem (#Equation: pdcoTL) is first reformulated into SOL PDCO form:

${\begin{array}{cccccc}\min \limits _{x}&f(x)\\\mathrm {s/t} &x_{L}&\leq &x&\leq &x_{U}\\{}&&&Ax&=&b\\{}&{r\mathrm {\ unconstrained} }\\\end{array}}$

The problem actually solved by pdco is

${\begin{array}{lllcll}\min \limits _{x,r}&{\phi (x)+{\frac {1}{2}}\|D_{1}x\|^{2}+{\frac {1}{2}}\|r\|^{2}}\\\end{array}}$

${\begin{array}{lllcll}\mathrm {s/t} &x_{L}&\leq &x&\leq &x_{U}\\{}&Ax&+&D_{2}r&=&b\\\end{array}}$

where $D_{1}$ and $D_{2}$ are positive-definite diagonal matrices defined from $d_{1}$, $d_{2}$ given in Prob.SOL.d1 and Prob.SOL.d2.

In particular, $d_{2}$ indicates the accuracy required for satisfying each row of $Ax=b$. See pdco.m for a detailed discussion of $D_{1}$ and $D_{2}$. Note that in pdco, 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 pdcoTL.

M-files Used

pdcoSet.m, pdco.m, Tlsqrmat.m

See Also

pdscoTL.m