CpTransf

Purpose

Transform general convex programs on the form

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

where $x,x_{L},x_{U}\in \mathbb {R} ^{n}$, $A\in \mathbb {R} ^{m\times n}$ and $b_{L},b_{U}\in \mathbb {R} ^{m}$, to other forms.

Calling Syntax

[AA, bb, meq] = cpTransf(Prob, TransfType, makeEQ, LowInf)

Inputs

Input	Description
Prob	Problem description structure. The following fields are used: QP.c Constant vector c in cTx. A Constraint matrix for linear constraints. b_L Lower bounds on the linear constraints. b U Upper bounds on the linear constraints. x_L Lower bounds on the variables. x_U Upper bounds on the variables.
TransfType	Type of transformation, see the description below.
MakeEQ	Flag, if set true, make standard form (all equalities).
LowInf	Variables equal to -Inf or variables < LowInf are set to LowInf before transforming the problem. Default -10-4 . |LowInf| are limit if upper bound variables are to be used.

Outputs

Output	Description
AA	The expanded linear constraint matrix.
bb	The expanded upper bounds for the linear constraints.
meq	The first meq equations are equalities.

Description

If TransType = 1 the program is transformed into the form

${\begin{array}{cccc}\min \limits _{x}&f(x-x_{L})&&\\s/t&AA(x-x_{L})&\leq &bb\\&x-x_{L}&\geq &0\\\end{array}}$

where the first meq constraints are equalities. Translate back with (fixed variables do not change their

values):

x(~x_L==x_U) = (x-x_L) + x_L(~x_L==x_U)

If TransType = 2 the program is transformed into the form

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

where the first meq constraints are equalities.

If TransType = 3 the program is transformed into the form

${\begin{array}{cccc}\min \limits _{x}&f(x)&&\\s/t&AAx&\leq &bb\\&x&\geq &x_{L}\\\end{array}}$

where the first meqconstraints are equalities.