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Regularized Lagrangian duality for linearly constrained quadratic optimization and trust-region problems

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Abstract

In this paper we first establish a Lagrange multiplier condition characterizing a regularized Lagrangian duality for quadratic minimization problems with finitely many linear equality and quadratic inequality constraints, where the linear constraints are not relaxed in the regularized Lagrangian dual. In particular, in the case of a quadratic optimization problem with a single quadratic inequality constraint such as the linearly constrained trust-region problems, we show that the Slater constraint qualification (SCQ) is necessary and sufficient for the regularized Lagrangian duality in the sense that the regularized duality holds for each quadratic objective function over the constraints if and only if (SCQ) holds. A new theorem of the alternative for systems involving both equality constraints and two quadratic inequality constraints plays a key role. We also provide classes of quadratic programs, including a class of CDT-subproblems with linear equality constraints, where (SCQ) ensures regularized Lagrangian duality.

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Authors and Affiliations

  1. Department of Applied Mathematics, University of New South Wales, Sydney, NSW, 2052, Australia

    V. Jeyakumar & Guoyin Li

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  1. V. Jeyakumar
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  2. Guoyin Li
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Correspondence to V. Jeyakumar.

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The authors are grateful to the referees for their comments which have contributed to the final preparation of the paper. Research was partially supported by a grant from the Australian Research Council.

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Jeyakumar, V., Li, G. Regularized Lagrangian duality for linearly constrained quadratic optimization and trust-region problems. J Glob Optim 49, 1–14 (2011). https://doi.org/10.1007/s10898-009-9518-8

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  • DOI: https://doi.org/10.1007/s10898-009-9518-8

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