STRONG DUALITY FOR SEMIDEFINITE PROGRAMMING

It is well known that the duality theory for linear programming (LP) i s powerful and elegant and lies behind algorithms such as simplex and interior-point methods. However, the standard Lagrangian for nonlinear programs requires constraint qualifications to avoid duality gaps. Se midefinite linear programming (SDP) is a generalization of LP where th e nonnegativity constraints are replaced by a semidefiniteness constra int on the matrix variables. There are many applications, e.g., in sys tems and control theory and combinatorial optimization. However, the L agrangian dual for SDP can have a duality gap. We discuss the relation ships among various duals and give a unified treatment for strong dual ity in semidefinite programming. These duals guarantee strong duality, i.e., a zero duality gap and dual attainment. This paper is motivated by the recent paper by Ramana where one of these duals is introduced.