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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, .. , m n +.x Here x is a vector of n variables, and we write c x for the inner-product P jn =1 cjxj , etc. Also, n + n x 0}, and we call n + the nonnegative orthant. n := {x |In fact, is a closed convex cone, where K is called a closed a convex cone + if K satisfies the following two conditions: 1 P P P3 If x, w K, then x+ w K for all nonnegative scalars and.

Introduction to Semidefinite Programming (SDP) Robert M. Freund 1 Introduction Semidefinite programming (SDP) is the most exciting 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.

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