Equivalent Lipschitz surrogates for zero-norm and rank optimization problems

Yulan Liu (), Shujun Bi () and Shaohua Pan ()
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Yulan Liu: GuangDong University of Technology
Shujun Bi: South China University of Technology
Shaohua Pan: South China University of Technology

Journal of Global Optimization, 2018, vol. 72, issue 4, No 4, 679-704

Abstract: Abstract This paper proposes a mechanism to produce equivalent Lipschitz surrogates for zero-norm and rank optimization problems by means of the global exact penalty for their equivalent mathematical programs with an equilibrium constraint (MPECs). Specifically, we reformulate these combinatorial problems as equivalent MPECs by the variational characterization of the zero-norm and rank function, show that their penalized problems, yielded by moving the equilibrium constraint into the objective, are the global exact penalization, and obtain the equivalent Lipschitz surrogates by eliminating the dual variable in the global exact penalty. These surrogates, including the popular SCAD function in statistics, are also difference of two convex functions (D.C.) if the function and constraint set involved in zero-norm and rank optimization problems are convex. We illustrate an application by designing a multi-stage convex relaxation approach to the rank plus zero-norm regularized minimization problem.

Keywords: Zero-norm; Rank; Global exact penalty; Equivalent Lipschitz surrogates; 90C27; 90C33; 49M20 (search for similar items in EconPapers)
Date: 2018
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Citations: View citations in EconPapers (8)

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DOI: 10.1007/s10898-018-0675-5

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