Full Length Paper Series A

Mathematical Programming

, Volume 149, Issue 1, pp 301-327

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

  • Wei BianAffiliated withDepartment of Mathematics, Harbin Institute of Technology
  • , Xiaojun ChenAffiliated withDepartment of Applied Mathematics, The Hong Kong Polytechnic University Email author 
  • , Yinyu YeAffiliated withDepartment of Management Science and Engineering, Stanford University

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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 \(\epsilon \) scaled first order stationary point is \(O(\epsilon ^{-2})\). 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 \(\epsilon \) scaled second order stationary point is reduced to \(O(\epsilon ^{-3/2})\). Note that an \(\epsilon \) scaled second order stationary point must also be an \(\epsilon \) scaled first order stationary point.

Keywords

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

Mathematics Subject Classifcation (2010)

90C30 90C26 65K05 49M37