The convex and monotone functions associated with second-order cone

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31 Citations (Scopus)

Abstract

Like the matrix-valued functions used in solutions methods for semidefinite programs (SDPs) and semidefinite complementarity problems (SDCPs), the vector-valued functions associated with second-order cones are defined analogously and also used in solutions methods for second-order-cone programs (SOCPs) and second-order-cone complementarity problems (SOCCPs). In this article, we study further about these vector-valued functions associated with second-order cones (SOCs). In particular, we define the so-called SOC-convex and SOC-monotone functions for any given function f: ℝ → ℝ. We discuss the SOC-convexity and SOC-monotonicity for some simple functions, e.g., f(t) = t2, t3 1/t, t1/2, |t|, and [t] +. Some characterizations of SOC-convex and SOC-monotone functions are studied, and some conjectures about the relationship between SOC-convex and SOC-monotone functions are proposed.

Original languageEnglish
Pages (from-to)363-385
Number of pages23
JournalOptimization
Volume55
Issue number4
DOIs
Publication statusPublished - 2006 Aug 1

Keywords

  • Complementarity
  • Convex function
  • Monotone function
  • Second-order cone
  • Spectral decomposition

ASJC Scopus subject areas

  • Control and Optimization
  • Management Science and Operations Research
  • Applied Mathematics

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