Application to Extended Semidefinite Optimization

Abstract

In semidefinite optimization one investigates nonlinear optimization problems in finite dimensions with a constraint requiring that a certain matrix-valued function is negative semidefinite. This type of problems arises in convex optimization, approximation theory, control theory, combinatorial optimization and engineering. In system and control theory so-called linear matrix inequalities (LMI’s) and extensions like bilinear matrix inequalities (BMI’s) fit into this class of constraints. Our investigations include various partial orderings for the description of the matrix constraint and in this way we extend the standard semidefinite case to other types of constraints. We apply the theory on optimality conditions developed in Chap. 5 and the duality theory of Chap. 6 to these extended semidefinite optimization problems.