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Solutions and optimality criteria for nonconvex constrained global optimization problems with connections between canonical and Lagrangian duality

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Abstract

This paper presents a canonical duality theory for solving a general nonconvex quadratic minimization problem with nonconvex constraints. By using the canonical dual transformation developed by the first author, the nonconvex primal problem can be converted into a canonical dual problem with zero duality gap. A general analytical solution form is obtained. Both global and local extrema of the nonconvex problem can be identified by the triality theory associated with the canonical duality theory. Illustrative applications to quadratic minimization with multiple quadratic constraints, box/integer constraints, and general nonconvex polynomial constraints are discussed, along with insightful connections to classical Lagrangian duality. Criteria for the existence and uniqueness of optimal solutions are presented. Several numerical examples are provided.

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Authors and Affiliations

  1. Department of Mathematics, Virginia Tech, Blacksburg, VA, 24061, USA

    David Yang Gao & Ning Ruan

  2. Grado Department of Industrial and Systems Engineering, Virginia Tech, Blacksburg, VA, 24061, USA

    David Yang Gao, Ning Ruan & Hanif D. Sherali

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  1. David Yang Gao
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  2. Ning Ruan
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  3. Hanif D. Sherali
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Correspondence to David Yang Gao.

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Gao, D.Y., Ruan, N. & Sherali, H.D. Solutions and optimality criteria for nonconvex constrained global optimization problems with connections between canonical and Lagrangian duality. J Glob Optim 45, 473–497 (2009). https://doi.org/10.1007/s10898-009-9399-x

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  • DOI: https://doi.org/10.1007/s10898-009-9399-x

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