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Introduction to Semidefinite Programming - MIT …

Introduction to Semidefinite Programming (SDP) Robert M. Freund 1 Introduction Semidefinite Programming (SDP) is the most exc iting development in math ematical Programming in the 1990 s. SDP has applications in such diverse fields as traditional convex constrained optimization, control theory, and combinatorial optimization. Because SDP is solvable vi a interior point methods, most of these applications can usually be solved very efficiently in practice as well as in theory. 2 Revi ew of Linear Programming Consider the linear Programming problem in standard form: LP : minimize c x ai x = bi, i = 1.

Given a feasible solution x of LP and a feasible solution (y,s) of LD, the duality gap is simply c x− P m i=1 y ib i = (c− P m · i=1 y ia i)·x = s·x ≥ 0, because x ≥ 0 and s ≥ 0. We know from LP duality theory that so long as the pri­ mal problem LP is feasible and has bounded optimal objective value, then

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