defcx

Help specify c^Tx objectives for mincx solver

Syntax

[V1,...,Vk] = defcx(lmisys,m,X1,...,Xk)

Description

[V1,...,Vk] = defcx(lmisys,m,X1,...,Xk) computes the values [V1,...,Vk] of the matrix variables X1,...,Xk of an LMI system when the decision variable having index m is set to 1 and all other decision variables are set to 0. These values are useful for deriving the c vector that you need to express the objective in terms of the matrix variables for mincx.

Examples

Compute c for mincx

The LMI solver mincx minimizes linear objectives of the form c^Tx, where x is the vector of decision variables. In most control problems, however, such objectives are expressed in terms of the matrix variables X rather than the decision variables x. Use defcx to compute the c vector for input into mincx.

Suppose that you want to minimize a linear objective given by

$$\text{Trace}\left(X\right)+x_0^{T}P{x}_0$$

Suppose further that you have already defined x0 = [1;1], specified the LMI variables X and P using lmivar, and specified the LMI relationships using lmiterm. Get the internal representation of the system.

lmisys = getlmis

For mincx, you must obtain a vector c such that

$$c^{T}x=\text{Trace}\left(X\right)+x_0^{T}P{x}_0$$

Use a for loop to get the values of each matrix variable with each decision variable set to one and the rest set to zero. The loop also evaluates the objective for X = Xm and P = Pm. This evaluation gives the mth entry of c by definition.

n = decnbr(lmisys) 
c = zeros(n,1)

for m = 1:n, 
	[Xm,Pm] = defcx(lmisys,m,X,P) 
	c(m) = trace(Xm) + x0'*Pm*x0 
end

You can now use c as an input to mincx with the system lmisys.

Input Arguments

lmisys — Optimization problem

vector

Optimization problem to solve, specified as a vector that encodes the internal representation of an LMI system. To obtain this vector, use setlmis, lmivar, lmiterm, and getlmis, as described in Specify LMI System at the Command Line.

m — Index

integer

Index of decision variable to set to 1, specified as an integer. Typically, you use defcx in a loop that steps through all n decision variables in the problem, where n = decnbr(lmisys).

X1,...,Xk — Matrix variable identifiers

integer

Matrix variable identifiers, specified as integer values. You obtain these identifiers when you define the matrix variables using lmivar.

Output Arguments

V1,...,Vk — Values of matrix variables

matrices

Values of matrix variables, returned as matrices. defcx computes the values of each matrix variable in lmisys with decision variable m set to 1 and all other decision variables set to 0.

Version History

Introduced before R2006a