## Minimize Linear Objectives Under LMI Constraints

Consider the optimization problem:

Minimize Trace(*X*) subject to

A + ^{T}XXA + XBB + ^{T}XQ < 0 | (1) |

with data

$$A=\left(\begin{array}{ccc}-1& -2& 1\\ 3& 2& 1\\ 1& -2& -1\end{array}\right);\text{\hspace{1em}}B=\left(\begin{array}{c}1\\ 0\\ 1\end{array}\right);\text{\hspace{1em}}Q=\left(\begin{array}{ccc}1& -1& 0\\ -1& -3& -12\\ 0& -12& -36\end{array}\right).$$

It can be shown that the minimizer *X** is simply the stabilizing
solution of the algebraic Riccati equation

*A*^{T}*X + XA
+
XBB*^{T}*X*
+ *Q* = 0

This solution can be computed directly with the Riccati solver `care`

and compared to the minimizer returned by `mincx`

.

From an LMI optimization standpoint, the problem specified in Equation 1 is equivalent to the following linear objective minimization problem:

Minimize Trace(*X*) subject to

$$\left(\begin{array}{cc}{A}^{T}X+XA+Q& XB\\ {B}^{T}X& -I\end{array}\right)<0.$$ | (2) |

Since Trace(*X*) is a linear function of the entries of
*X*, this problem falls within the scope of the `mincx`

solver and can be numerically solved as follows:

Define the LMI constraint of Equation 1 by the sequence of commands

setlmis([]) X = lmivar(1,[3 1]) % variable X, full symmetric lmiterm([1 1 1 X],1,a,'s') lmiterm([1 1 1 0],q) lmiterm([1 2 2 0],-1) lmiterm([1 2 1 X],b',1) LMIs = getlmis

Write the objective Trace(

*X*) as*c*^{T}*x*where*x*is the vector of free entries of*X*. Since*c*should select the diagonal entries of*X*, it is obtained as the decision vector corresponding to*X*=*I*, that is,c = mat2dec(LMIs,eye(3))

Note that the function

`defcx`

provides a more systematic way of specifying such objectives (see Specifying cTx Objectives for mincx for details).Call

`mincx`

to compute the minimizer`xopt`

and the global minimum`copt = c'*xopt`

of the objective:options = [1e-5,0,0,0,0] [copt,xopt] = mincx(LMIs,c,options)

Here

`1e–5`

specifies the desired relative accuracy on`copt`

.The following trace of the iterative optimization performed by

`mincx`

appears on the screen:Solver for linear objective minimization under LMI constraints Iterations : Best objective value so far

`1`

`2`

`-8.511476`

`3`

`-13.063640`

`***`

`new lower bound:`

`-34.023978`

`4`

`-15.768450`

`***`

`new lower bound:`

`-25.005604`

`5`

`-17.123012`

`***`

`new lower bound:`

`-21.306781`

`6`

`-17.882558`

`***`

`new lower bound:`

`-19.819471`

`7`

`-18.339853`

`***`

`new lower bound:`

`-19.189417`

`8`

`-18.552558`

`***`

`new lower bound:`

`-18.919668`

`9`

`-18.646811`

`***`

`new lower bound:`

`-18.803708`

`10`

`-18.687324`

`***`

`new lower bound:`

`-18.753903`

`11`

`-18.705715`

`***`

`new lower bound:`

`-18.732574`

`12`

`-18.712175`

`***`

`new lower bound:`

`-18.723491`

`13`

`-18.714880`

`***`

`new lower bound:`

`-18.719624`

`14`

`-18.716094`

`***`

`new lower bound:`

`-18.717986`

`15`

`-18.716509`

`***`

`new lower bound:`

`-18.717297`

`16`

`-18.716695`

`***`

`new lower bound:`

`-18.716873`

Result: feasible solution of required accuracy best objective value: -18.716695 guaranteed relative accuracy: 9.50e-06 f-radius saturation: 0.000% of R = 1.00e+09

The iteration number and the best value of

*c*^{T}*x*at the current iteration appear in the left and right columns, respectively. Note that no value is displayed at the first iteration, which means that a feasible*x*satisfying the constraint (Equation 2) was found only at the second iteration. Lower bounds on the global minimum of*c*^{T}*x*are sometimes detected as the optimization progresses. These lower bounds are reported by the message*** new lower bound: xxx

Upon termination,

`mincx`

reports that the global minimum for the objective Trace(*X*) is –18.716695 with relative accuracy of at least 9.5×10^{–6}. This is the value`copt`

returned by`mincx`

.`mincx`

also returns the optimizing vector of decision variables`xopt`

. The corresponding optimal value of the matrix variable*X*is given byXopt = dec2mat(LMIs,xopt,X)

which returns

$${X}_{opt}=\left(\begin{array}{ccc}-6.3542& -5.8895& 2.2046\\ -5.8895& -6.2855& 2.2201\\ 2.2046& 2.2201& -6.0771\end{array}\right).$$

This result can be compared with the stabilizing Riccati solution computed by

`care`

:Xst = care(a,b,q,-1) norm(Xopt-Xst) ans = 6.5390e-05