# COMPUTATION OF THE MIXED S. S. V

1.2 INTRODUCTION TO LMIs

A Linear Matrix Inequality (LMI) is an inequality constraint A(x) < 0, where A(x) is an hermitian square matrix (A(x) = A*(x)) depending linearly on the vector x of parameters:

N

A(x) = A0 + ^2 АіХі

i=i

Matrices Ao and A are fixed. Various techniques and softwares (Boyd et al., 1994; Gahinet et al., 1995) are available for solving the feasibility problem: does there exist a value of vector x for which the relation A(x) <

0 holds true ?

_ Let two hermitian matrices A and B, with В > 0. the quantities (A, B) andry(i4, B) are defined as:

A(A, В) = sup(7 Є R : det(A — 7B) = 0)

p(A, B) = sup(j Є R : (A — jB) > 0) (5-7)

where A(M) denotes the maximal eigenvalue of a matrix M. r](A, B) is almost always equal to the maximal generalized eigenvalue A(A, B), except possibly when r](A, B) = 00 (see Proposition 5.1.c of (Fan and Tits, 1992)).

Let A(x) and B(x) matrices which linearly depend on vector x. The minimization of A(A(x),B(x)) is a quasi-convex optimization problem, which can be recast as the minimization of scalar 7 under the LMI constraint A(x) — yB(x) < 0. A first simple solution is a dichotomy

search over 7: for a given value of 7, the feasibility problem "does

there exist a value of vector x satisfying A ( x ) — jB ( x ( < 0 ?’ 1 is to be solved. More sophisticated solutions, involving the direct minimization of A ( A(—) , A(—(), are also available (Gahinet et al., 1995).

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