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

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 typical...

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Vydáno v:Mathematical programming Ročník 149; číslo 1-2; s. 301 - 327
Hlavní autoři: Bian, Wei, Chen, Xiaojun, Ye, Yinyu
Médium: Journal Article
Jazyk:angličtina
Vydáno: Berlin/Heidelberg Springer Berlin Heidelberg 01.02.2015
Springer Nature B.V
Témata:
Algorithms
Analysis
Applied mathematics
Calculus of Variations and Optimal Control; Optimization
Combinatorics
Complexity
Complexity theory
Feature selection
First order algorithms
Full Length Paper
Iterative methods
Mathematical analysis
Mathematical and Computational Physics
Mathematical Methods in Physics
Mathematical models
Mathematical programming
Mathematics
Mathematics and Statistics
Mathematics of Computing
Minimization
Numerical Analysis
Optimization
Optimization techniques
Regularization methods
Studies
Texts
Theoretical
Complexity analysis
Second order algorithm
90C30
65K05
Constrained non-Lipschitz optimization
Interior point method
90C26
49M37
First order algorithm
ISSN:0025-5610, 1436-4646
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Shrnutí: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 O ( ϵ - 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 ϵ scaled second order stationary point is reduced to O ( ϵ - 3 / 2 ) . Note that an ϵ scaled second order stationary point must also be an ϵ scaled first order stationary point.
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ISSN:0025-5610
1436-4646
DOI:10.1007/s10107-014-0753-5