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Elementary Optimality Conditions for Nonlinear SDPs

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Handbook on Semidefinite, Conic and Polynomial Optimization

Part of the book series: International Series in Operations Research & Management Science ((ISOR,volume 166))

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Abstract

An increasing number of recent applications rely on the solution of nonlinear semidefinite programs. First and second order optimality conditions for nonlinear programs are widely known today. This chapter generalizes these optimality conditions to nonlinear semidefinite programs, highlighting some parallels and some differences. It starts by discussing a constraint qualification for both programs. First order optimality conditions are presented for the case where this constraint qualification is satisfied. For the second order conditions, in addition, strict complementarity is assumed.

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Notes

  1. 1.

    Assuming that G(x) is a symmetric matrix, and that the “off-diagonal inequalities” are thus listed twice among the inequalities G(x) ≤ 0, introduces some complication in the discussion of nondegeneracy for (16.2). Below, we will work with Lagrange multipliers being symmetric as well, thus “counting” the off-diagonal inequalities just once. To make things short, we may simply ignore the fact that formally, (16.2) contains redundant constraints.

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Acknowledgements

The author is thankful to two anonymous referees whose constructive criticism helped to improve the presentation of the chapter.

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Authors and Affiliations

  1. Deparment of Mathematics, University of Düsseldorf, Düsseldorf, Germany

    Florian Jarre

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  1. Florian Jarre
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Correspondence to Florian Jarre .

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Editors and Affiliations

  1. Department of Mathematics and Industrial, Ecole Polytechnique Montreal, Édouard-Montpetit Boulevard 2350, Montreal, QC, Canada, H3C 3A7

    Miguel F. Anjos

  2. LAAS-CNRS, Avenue du Colonel Roche 7, 31077, Toulouse Cedex 4, France

    Jean B. Lasserre

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Jarre, F. (2012). Elementary Optimality Conditions for Nonlinear SDPs. In: Anjos, M.F., Lasserre, J.B. (eds) Handbook on Semidefinite, Conic and Polynomial Optimization. International Series in Operations Research & Management Science, vol 166. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-0769-0_16

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