Foreword: for Special Issue on Semide nite Programming

Semideenite programming (denoted SDP) is an extension of linear programming (LP), with vector variables replaced by matrix variables and nonnegativity elementwise replaced by positive semideeniteness. Let us express a (primal) SDP as min C X s.t. AX = b X 0; where: C; X are in S n , the space of symmetric n n matrices; the inner product C X = trace CX; denotes nonnegativity in the LL owner partial order, i.e. A B if A ? B 0; i.e. The close similarity between SDP and LP is apparent, the essential diierence being that P; the cone of positive semideenite matrices, replaces < n + , the nonnegative orthant. Unlike LP, SDP is a nonlinear convex programming problem, because the boundary of the cone P is nonlinear. Nonetheless, LP and SDP share two key aspects: Duality theory: much of the theory of duality extends directly from LP to SDP. Indeed, this was recognized as early as 1963 by Bellman and Fan 2]. They also observed that duality theorems require somewhat stronger assumptions for SDP than for LP: speciically, they used a Slater-type constraint qualiication to guarantee a zero duality gap. Interior-point methods: SDP can be very eeectively solved by generalizing interior-point methods developed originally for LP. The credit for this important discovery goes primarily to Nesterov and Nemirovski 9] (who gave a far-reaching theory for general convex programs) and Alizadeh 1] (who argued that many interior-point methods could be extended from LP to SDP in a simple, uniied way). While the question of superiority of simplex versus interior-point methods for LP remains a controversial question, there is no contest for SDP, because the nonlinearity of the feasible set makes a simplex method impractical. The fact that SDP can be eeciently solved by interior-point methods, both in practice and, from a theoretical point of view, in polynomial time, has driven the recent surge of interest in the subject. In this volume, a wide variety of aspects of SDP are studied. One way to organize these is as follows: 1. THEORETICAL