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
Summary
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.
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