By Jiri Matousek, Bernd Gärtner

Semidefinite courses represent one of many biggest sessions of optimization difficulties that may be solved with moderate potency - either in conception and perform. They play a key function in various examine components, resembling combinatorial optimization, approximation algorithms, computational complexity, graph concept, geometry, genuine algebraic geometry and quantum computing. This publication is an creation to chose facets of semidefinite programming and its use in approximation algorithms. It covers the fundamentals but additionally an important quantity of modern and extra complicated material. there are numerous computational difficulties, reminiscent of MAXCUT, for which one can't quite count on to acquire a precise answer successfully, and in such case, one has to accept approximate recommendations. For MAXCUT and its kinfolk, intriguing fresh effects recommend that semidefinite programming is among the final instrument. certainly, assuming the original video games Conjecture, a believable yet as but unproven speculation, it used to be proven that for those difficulties, identified algorithms according to semidefinite programming convey the very best approximation ratios between all polynomial-time algorithms. This ebook follows the “semidefinite side” of those advancements, featuring the various major principles at the back of approximation algorithms in accordance with semidefinite programming. It develops the elemental conception of semidefinite programming, offers one of many recognized effective algorithms intimately, and describes the foundations of a few others. it's also purposes, concentrating on approximation algorithms.

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7. 1, a semideﬁnite program in equational form is an optimization problem of the form maximize C • X subject to Ai • X = bi , X 0. i = 1, 2, . . 1) Here X is an unknown n × n symmetric real matrix (which we write as X ∈ SYMn ). The input data are a matrix C ∈ SYMn specifying the objective function, a vector b ∈ Rm , and matrices Ai ∈ SYMn , i = 1, 2, . . , m. For the purposes of this chapter, we summarize the m equality constraints in the form A(X) = b, where A: SYMn → Rm is the linear mapping A(X) = (A1 • X, A2 • X, .

1 Definition. 1 K is called a closed convex cone if the following two conditions hold. (i) For all x ∈ K and all nonnegative real numbers λ, we have λx ∈ K. (ii) For all x, y ∈ K, we have x + y ∈ K. Condition (i) ensures that K is a cone, while condition (ii) guarantees convexity of K. Indeed, if x, y ∈ K and λ ∈ [0, 1], then (1 − λ)x and λy are both in K by (i), and then (ii) shows that (1 − λ)x + λy ∈ K, as required by convexity. 2 Lemma. The set PSDn ⊆ SYMn of positive semideﬁnite matrices is a closed convex cone.

This is called the strong product to distinguish it from the usual graph product G×H in which there is an edge between (v, w) and (v , w ) if {v, v } ∈ E and {w, w } ∈ F ; see Fig. 2. In fact, the strong product is obtained from the usual product by adding the edges of the square product (or grid) of G and H. In this latter product, there is an edge between (v, w) and (v , w ) if they diﬀer in exactly one component, and the two involved vertices form an edge in the respective graph. 2 Lemma. For all graphs G and H, ϑ(G · H) ≤ ϑ(G)ϑ(H).