Scilab Function

lmisolver - linear matrix inequation solver

### Sequence d'appel

  `[XLISTF[,OPT]] = lmisolver(XLIST0,evalfunc [,options])`

### Parametres

- `XLIST0` : a list of containing initial guess (e.g. `XLIST0=list(X1,X2,..,Xn)`)
- `evalfunc` : a Scilab function ("external" function with specific syntax)
- `XLISTF` : a list of matrices (e.g. `XLIST0=list(X1,X2,..,Xn)`)
- `options` : optional parameter. If given, `options` is a real row vector with 5 components `[Mbound,abstol,nu,maxiters,reltol]`

The syntax the function `evalfunc` must be as follows:

`[LME,LMI,OBJ]=evalfunct(X)` where `X` is a list of matrices, `LME, LMI` are lists and `OBJ` a real scalar.

### Description

`lmisolver` solves the following problem:

minimize `f(X1,X2,...,Xn)` a linear function of Xi's

under the linear constraints: `Gi(X1,X2,...,Xn)=0` for i=1,...,p and LMI (linear matrix inequalities) constraints:

`Hj(X1,X2,...,Xn) > 0` for j=1,...,q

The functions f, G, H are coded in the Scilab function `evalfunc` and the set of matrices Xi's in the list X (i.e. `X=list(X1,...,Xn)`).

The function `evalfun` must return in the list `LME` the matrices `G1(X),...,Gp(X)` (i.e. `LME(i)=Gi(X1,...,Xn),` i=1,...,p). `evalfun` must return in the list `LMI` the matrices `H1(X0),...,Hq(X)` (i.e. `LMI(j)=Hj(X1,...,Xn)`, j=1,...,q). `evalfun` must return in `OBJ` the value of `f(X)` (i.e. `OBJ=f(X1,...,Xn)`).

`lmisolver` returns in `XLISTF`, a list of real matrices, i. e. `XLIST=list(X1,X2,..,Xn)` where the Xi's solve the LMI problem:

Defining `Y,Z` and `cost` by:

`[Y,Z,cost]=evalfunc(XLIST)`, `Y` is a list of zero matrices, `Y=list(Y1,...,Yp)`, `Y1=0, Y2=0, ..., Yp=0`.

`Z` is a list of square symmetric matrices, `Z=list(Z1,...,Zq)` , which are semi positive definite `Z1>0, Z2>0, ..., Zq>0` (i.e. `spec(Z(j))` > 0),

`cost` is minimized.

`lmisolver` can also solve LMI problems in which the `Xi's` are not matrices but lists of matrices. More details are given in the documentation of LMITOOL.

### Exemples

    //Find diagonal matrix X (i.e. X=diag(diag(X), p=1) such that
    //A1'*X+X*A1+Q1 < 0, A2'*X+X*A2+Q2 < 0 (q=2) and trace(X) is maximized
    n=2;A1=rand(n,n);A2=rand(n,n);
    Xs=diag(1:n);Q1=-(A1'*Xs+Xs*A1+0.1*eye());
    Q2=-(A2'*Xs+Xs*A2+0.2*eye());
    deff('[LME,LMI,OBJ]=evalf(Xlist)','X=Xlist(1),LME=X-diag(diag(X));...
    LMI=list(-(A1''*X+X*A1+Q1),-(A2''*X+X*A2+Q2)),OBJ= -sum(diag(X))  ');
    X=lmisolver(list(zeros(A1)),evalf);X=X(1)
    [Y,Z,c]=evalf(X)

### Voir aussi

`lmitool`