Lifted stationary points of sparse optimization with complementarity constraints

We aim to compute lifted stationary points of a sparse optimization problem ( P 0 ) with complementarity constraints. We define a continuous relaxation problem ( R ν ) that has the same global minimizers and optimal value with problem ( P 0 ). Problem ( R ν ) is a mathematical program with complemen...

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Bibliographic Details
Published in:Computational optimization and applications Vol. 84; no. 3; pp. 973 - 1003
Main Authors: Liu, Shisen, Chen, Xiaojun
Format: Journal Article
Language:English
Published: New York Springer US 01.04.2023
Springer Nature B.V
Subjects:
Algorithms
Convex and Discrete Geometry
Management Science
Mathematics
Mathematics and Statistics
Operations Research
Operations Research/Decision Theory
Optimization
Statistics
Lifted stationary point
folded concave function
Sparse solution
Complementarity constraints
Capped
Vertical linear complementarity constraints
ISSN:0926-6003, 1573-2894
Online Access:Get full text
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Summary:We aim to compute lifted stationary points of a sparse optimization problem ( P 0 ) with complementarity constraints. We define a continuous relaxation problem ( R ν ) that has the same global minimizers and optimal value with problem ( P 0 ). Problem ( R ν ) is a mathematical program with complementarity constraints (MPCC) and a difference-of-convex objective function. We define MPCC lifted-stationarity of ( R ν ) and show that it is weaker than directional stationarity, but stronger than Clarke stationarity for local optimality. Moreover, we propose an approximation method to solve ( R ν ) and an augmented Lagrangian method to solve its subproblem ( R ν , σ ), which relaxes the equality constraint in ( R ν ) with a tolerance σ . We prove the convergence of our algorithm to an MPCC lifted-stationary point of problem ( R ν ) and use a sparse optimization problem with vertical linear complementarity constraints to demonstrate the efficiency of our algorithm on finding sparse solutions in practice.
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ISSN:0926-6003
1573-2894
DOI:10.1007/s10589-022-00444-1