Splitting and successively solving augmented Lagrangian method for optimization with semicontinuous variables and cardinality constraint

TL;DR

This paper introduces a novel splitting and successively solving augmented Lagrangian (SSAL) method for efficiently tackling NP-hard optimization problems with semicontinuous variables and cardinality constraints, demonstrating superior performance in real-world applications.

Contribution

The paper develops the SSAL method, providing convergence analysis and showing significant computational advantages over existing methods in large-scale problems.

Findings

SSAL outperforms CPLEX 12.6 by nearly 200 times in portfolio selection.

SSAL is more than 40 times faster than the penalty decomposition method in compressed sensing.

SSAL remains effective and efficient as problem size increases.

Abstract

We propose a new splitting and successively solving augmented Lagrangian (SSAL) method for solving an optimization problem with both semicontinuous variables and a cardinality constraint. This optimization problem arises in several contexts such as the portfolio selection problem, the compressed sensing problem and the unit commitment problem, etc. The problem is in general NP-hard. We derive an optimality condition for this optimization problem, under some suitable assumptions. By introducing an auxiliary variable and using an augmented Lagrangian function, the constraints are decomposed into two parts. By fixing particular variables, the optimization problem is split into two subproblems, which are solved alternatively. Furthermore, we prove the convergence of SSAL, under some suitable assumptions. Finally, we implement our method for the portfolio selection problem and the compressed sensing problem, respectively. Real world data and simulation study show that SSAL outperforms the well-known CPLEX 12.6 and the penalty decomposition (PD) method, while enjoying a similar cardinality of decision variable. For example, the numerical results are even nearly 200 times faster than that of CPLEX 12.6 for the portfolio selection problem, and more than 40 times faster than PD method for the compressed sensing problem, respectively. In particularly, SSAL is powerful when the size of problem increase largely.