decinfo

Describe how entries of matrix variable X relate to decision variables

Syntax

decX = decinfo(lmisys,X)
decinfo(lmisys)


Description

decinfo expresses the entries of a matrix variable X in terms of the decision variables x₁, . . ., x_N. Recall that the decision variables are the free scalar variables of the problem, or equivalently, the free entries of all matrix variables described in lmisys. Each entry of X is either a hard zero, some decision variable x_n, or its opposite –x_n.

If X is the identifier of X supplied by lmivar, the command decX = decinfo(lmisys,X) returns an integer matrix decX of the same dimensions as X whose (i, j) entry is

• 0 if X(i, j) is a hard zero

• n if X(i, j) = x_n (the n-th decision variable)

• –n if X(i, j) = –x_n

decX clarifies the structure of X as well as its entry-wise dependence on x₁, . . ., x_N. This is useful to specify matrix variables with atypical structures (see lmivar).

decinfo can also be used in interactive mode by invoking it with a single argument, as decinfo(lmisys). It then prompts the user for a matrix variable and displays in return the decision variable content of this variable.

Examples

Example 1

Consider an LMI with two matrix variables X and Y with structure:

• X = x I₃ with x scalar

• Y rectangular of size 2-by-1

If these variables are defined by

setlmis([]) 
X = lmivar(1,[3 0]) 
Y = lmivar(2,[2 1]) 
	: 
	: 
lmis = getlmis

The decision variables in X and Y are given by

dX = decinfo(lmis,X)

dX = 
	1 	0 	0 
	0 	1 	0 
	0 	0 	1

dY = decinfo(lmis,Y)

dY = 
	2 
	3

This indicates a total of three decision variables x₁, x₂, x₃ that are related to the entries of X and Y by

$$X=\left(\begin{array}{ccc}{x}_1& 0& 0\\ 0& x_1& 0\\ 0& 0& x_1\end{array}\right),Y=\left(\begin{array}{c}{x}_2\\ x3\end{array}\right)$$

Note that the number of decision variables corresponds to the number of free entries in X and Y when taking structure into account.

Example 2

Suppose that the matrix variable X is symmetric block diagonal with one 2-by-2 full block and one 2-by-2 scalar block, and is declared by

setlmis([]) 
X = lmivar(1,[2 1;2 0]) 
		: 
lmis = getlmis

The decision variable distribution in X can be visualized interactively as follows:

decinfo(lmis)

decinfo(lmis)

There are 4 decision variables labeled x1 to x4 in this problem.

Matrix variable Xk of interest (enter k between 1 and 1, or 0 to quit):

?> 1

The decision variables involved in X1 are among {-x1,...,x4}.
Their entry-wise distribution in X1 is as follows
        (0,j>0,-j<0 stand for 0,xj,-xj, respectively):

X1 :

	1 	2 	0 	0 
	2 	3 	0 	0 
	0 	0 	4 	0 
	0 	0 	0 	4
	
			 *********

Matrix variable Xk of interest (enter k between 1 and 1, or 0 to quit):

?> 0

Version History

Introduced before R2006a