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Semidefinite Representation of Convex Sets and Convex Hulls

In: Handbook on Semidefinite, Conic and Polynomial Optimization

Author

Listed:
  • J. William Helton

    (University of California at San Diego)

  • Jiawang Nie

    (University of California at San Diego)

Abstract

Semidefinite programming (SDP) (Ben-Tal and Nemirovski: Lectures on Modern Convex Optimization: Analysis, Algorithms, and Engineering Applications. MPS-SIAM Series on Optimization, SIAM, Philadelphia (2001); Nesterov and Nemirovskii: Interior-point polynomial algorithms in convex programming. SIAM Studies in Applied Mathematics 13. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA (1994); Nemirovskii, A.: Advances in convex optimization: conic programming. Plenary Lecture at the International Congress of Mathematicians (ICM), Madrid, Spain, 22-30 August 2006; Wolkowicz et al.: Handbook of semidefinite programming. Kluwer’s, Boston, MA (2000)) is one of the main advances in convex optimization theory and applications. It has a profound effect on combinatorial optimization, control theory and nonconvex optimization as well as many other disciplines. There are effective numerical algorithms for solving problems presented in terms of Linear Matrix Inequalities (LMIs). A fundamental problem in semidefinite programming concerns the range of its applicability. What convex sets S can be represented by using LMI? This question has several natural variants; the main one is to represent S as the projection of a higher dimensional set that is representable by LMI; such S is called SDP representable (SDr). Here we shall describe these variants and what is known about them. The chapter is a survey of the existing results of (Helton and Nie: Mathematical Programming 122(1):21–64 (2010); Helton and Nie: SIAM Journal on Optimization 20(2):759–791 (2009); Lasserre: Mathematical Programming 120:457–477 (2009)) and related work.

Suggested Citation

  • J. William Helton & Jiawang Nie, 2012. "Semidefinite Representation of Convex Sets and Convex Hulls," International Series in Operations Research & Management Science, in: Miguel F. Anjos & Jean B. Lasserre (ed.), Handbook on Semidefinite, Conic and Polynomial Optimization, chapter 0, pages 77-112, Springer.
    • Handle: RePEc:spr:isochp:978-1-4614-0769-0_4
    DOI: 10.1007/978-1-4614-0769-0_4
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