Abstract

AbstractPacking and covering semidefinite programs (SDPs) appear in natural relaxations of many combinatorial optimization problems as well as a number of other applications. Recently, several techniques have been proposed that utilize the particular structure of this class of problems in order to obtain more efficient algorithms than those offered by general SDP solvers. For certain applications, it may be necessary to deal with SDPs with a very large number of (e.g., exponentially or even infinitely many) constraints. In this chapter, we give an overview of some of the techniques that can be used to solve this class of problems, focusing on multiplicative weight updates and logarithmic-potential methods.

Highlights

  • Packing and covering semidefinite programs (SDPs) appear in natural relaxations of many combinatorial optimization problems as well as a number of other applications

  • And “ ” is the Löwner order on matrices: A B if and only if A − B is psd. This type of SDP arises in many applications

  • When C = I = In and b = 1m, we say that the packing-covering SDPs are in normalized form

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Summary

Packing and Covering Semidefinite Programs

We denote by Sn the set of all n × n real symmetric matrices and by Sn+ ⊆ Sn the set of all n × n positive semidefinite (psd) matrices.

Elbassioni (B)
SDP relaxation for Robust MaxCut
Mahalanobis Distance Learning
Related Work
General Framework for Packing-Covering SDPs
Scalar MWU Algorithm for ( PAC K I N G - I)-( C OV ERING - I)
Scalar Logarithmic Potential Algorithm For ( PAC K I N G - I)–( C OV ERING - I)
Matrix MWU Algorithm For ( C OV ERING - I I)-( PAC K I N G - I I)
Matrix Logarithmic Potential Algorithm For ( PAC K I N G - I)-( C OV ERING - I)
Matrix Logarithmic Potential Algorithm For ( PAC K I N G - I I)-( C OV ERING - I I)
Full Text
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