Introduction to Semidefinite Programming

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.

We will write the standard linear programming dual problem as: m LD : maximize y ib i i=1 m s.t. y ia i + s = c i=1 n s ∈ℜ +. 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.

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