By E. de Klerk
Semidefinite programming has been defined as linear programming for the yr 2000. it really is a thrilling new department of mathematical programming, because of vital functions up to speed thought, combinatorial optimization and different fields. additionally, the winning inside aspect algorithms for linear programming might be prolonged to semidefinite programming.
In this monograph the fundamental thought of inside aspect algorithms is defined. This comprises the newest effects at the homes of the relevant direction in addition to the research of an important periods of algorithms. numerous "classic" functions of semidefinite programming also are defined intimately. those comprise the Lovász theta functionality and the MAX-CUT approximation set of rules via Goemans and Williamson.
Audience: Researchers or graduate scholars in optimization or similar fields, who desire to examine extra concerning the thought and purposes of semidefinite programming.
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Additional info for Aspects of Semidefinite Programming. Interior Point Algorithms and Selected Applications
P)) is feasible, then perfect duality holds and (resp. 4 in Appendix B). This theorem essentially states that two convex sets in can be separated by a hyperplane if and only if their relative interiors are disjoint. 2 (Strong duality) Assume that (resp. assume that (D) (resp. (P)) is strictly feasible. It now holds that and Proof: We will first consider the case where is trivial if since then Further (resp. and (D) is strictly feasible. The proof is optimal for (P). We can therefore assume Let us define the (nonempty) convex set The relative interiors of and are disjoint, by construction.
To this end, define auxiliary variables and and consider the problem: Note that the optimal value of this problem is zero if and only if (P) is either feasible or weakly infeasible. 12) is empty. 11) does. 12) is indeed zero. We can give an alternative characterization of weak infeasibility by introducing the concept of a weakly improving ray. Whereas an improving ray in (P) causes strict infeasibility in (D) (and vice versa), weakly improving rays cause weak infeasibility. 5 The problem (P) (resp.
2 (Perfect duality) The problems (P) and (D) are said to be in perfect duality if Note that this definition does not imply that and are nonempty. If nonempty, then we say that strong duality holds for (P) and its dual (D). 1 (Adapted from Vandenberghe and Boyd ) This example shows a pair of dual problems for which perfect duality holds but Consider the following problem in the standard dual form: 26 Aspects of Semidefinite Programming subject to This problem is not solvable but sup standard primal form) takes the form: Its dual problem (which is in the subject to Note that implies so that is the unique optimal solution with optimal value 1.