Complexity analysis of interior point algorithms for non-Lipschitz and nonconvex minimization

Wei Bian, Xiaojun Chen, Yinyu Ye

Research output: Journal article publicationJournal articleAcademic researchpeer-review

45 Citations (Scopus)

Abstract

We propose a first order interior point algorithm for a class of non-Lipschitz and nonconvex minimization problems with box constraints, which arise from applications in variable selection and regularized optimization. The objective functions of these problems are continuously differentiable typically at interior points of the feasible set. Our first order algorithm is easy to implement and the objective function value is reduced monotonically along the iteration points. We show that the worst-case iteration complexity for finding an ϵ scaled first order stationary point is (Formula Presented). Furthermore, we develop a second order interior point algorithm using the Hessian matrix, and solve a quadratic program with a ball constraint at each iteration. Although the second order interior point algorithm costs more computational time than that of the first order algorithm in each iteration, its worst-case iteration complexity for finding an ϵ scaled second order stationary point is reduced to (Formula Presented.). Note that an ϵ scaled second order stationary point must also be an ϵ scaled first order stationary point.
Original languageEnglish
Pages (from-to)301-327
Number of pages27
JournalMathematical Programming
Volume149
Issue number1-2
DOIs
Publication statusPublished - 1 Jan 2014

Keywords

  • Interior point method
  • Complexity analysis
  • Constrained non-Lipschitz optimization
  • First order algorithm
  • Second order algorithm

ASJC Scopus subject areas

  • Software
  • Mathematics(all)

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