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Sequential Difference-of-Convex Programming

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Abstract

Optimization methods for difference-of-convex programs iteratively solve convex subproblems to define iterates. Although convex, depending on the problem’s structure, these subproblems are very often challenging and require specialized solvers. This work investigates a new methodology that defines iterates as approximate critical points of significantly easier difference-of-convex subproblems approximating the original one. Since there is considerable freedom to choose such more accessible subproblems, several algorithms can be designed from the given approach. In some cases, the resulting algorithm boils down to a straightforward process with iterates given in an analytic form. In other situations, decomposable subproblems can be chosen, opening the way for parallel computing even when the original program is not decomposable. Depending on the problem’s assumptions, a possible variant of the given approach is the Josephy–Newton method applied to the system of (necessary) optimality conditions of the original difference-of-convex program. In such a setting, local convergence with superlinear and even quadratic rates can be achieved under certain conditions.

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Acknowledgements

The author is grateful to four anonymous reviewers for their remarks and constructive suggestions that considerably improved the first version of this article. The author also acknowledges financial support from the Gaspard-Monge program for Optimization and Operations Research (PGMO) project “Models for planning energy investment under uncertainty.”

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  1. CMA – Centre de Mathématiques Appliquées, MINES ParisTech, PSL – Research University, Sophia Antipolis, France

    Welington de Oliveira

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de Oliveira, W. Sequential Difference-of-Convex Programming. J Optim Theory Appl 186, 936–959 (2020). https://doi.org/10.1007/s10957-020-01721-x

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