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DC Semidefinite programming and cone constrained DC optimization I: theory

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Abstract

In this two-part study, we discuss possible extensions of the main ideas and methods of constrained DC optimization to the case of nonlinear semidefinite programming problems and more general nonlinear cone constrained optimization problems. In the first paper, we analyse two different approaches to the definition of DC matrix-valued functions (namely, order-theoretic and componentwise), study some properties of convex and DC matrix-valued mappings and demonstrate how to compute DC decompositions of some nonlinear semidefinite constraints appearing in applications. We also compute a DC decomposition of the maximal eigenvalue of a DC matrix-valued function. This DC decomposition can be used to reformulate DC semidefinite constraints as DC inequality constrains. Finally, we study local optimality conditions for general cone constrained DC optimization problems.

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The author is sincerely grateful to the anonymous referees and the Coordinating Editor for carefully reading the paper and providing many thoughtful comments and suggestions that helped to significantly improve the overall quality of the article.

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    M. V. Dolgopolik

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This work was performed in IPME RAS and supported by the Russian Science Foundation (Grant No. 20-71-10032).

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Dolgopolik, M.V. DC Semidefinite programming and cone constrained DC optimization I: theory. Comput Optim Appl 82, 649–671 (2022). https://doi.org/10.1007/s10589-022-00374-y

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