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Semidefinite Programming

In: Design and Analysis of Approximation Algorithms

Author

Listed:
  • Ding-Zhu Du

    (University of Texas at Dallas)

  • Ker-I Ko

    (State University of New York at Stony Brook)

  • Xiaodong Hu

    (Academy of Mathematics and Systems Science Chinese Academy of Sciences)

Abstract

Semidefinite programming studies optimization problems with a linear objective function over semidefinite constraints. It shares many interesting properties with linear programming. In particular, a semidefinite program can be solved in polynomial time. Moreover, an integer quadratic programcan be transformed into a semidefinite programthrough relaxation. Therefore, if a combinatorial optimization problem can be formulated as an integer quadratic program, then we can approximate it using the semidefinite programming relaxation and other related techniques such as the primal’dual schema. As the semidefinite programming relaxation is a higher-order relaxation, it often produces better results than the linear programming relaxation, even if the underlying problem can be formulated as an integer linear program. In this chapter, we introduce the fundamental concepts of semidefinite programming, and demonstrate its application to the approximation of NP-hard combinatorial optimization problems, with various rounding techniques.

Suggested Citation

  • Ding-Zhu Du & Ker-I Ko & Xiaodong Hu, 2012. "Semidefinite Programming," Springer Optimization and Its Applications, in: Design and Analysis of Approximation Algorithms, chapter 9, pages 339-370, Springer.
    • Handle: RePEc:spr:spochp:978-1-4614-1701-9_9
    DOI: 10.1007/978-1-4614-1701-9_9
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