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Extensions of the Standard Quadratic Optimization Problem: Strong Duality, Optimality, Hidden Convexity and S-lemma

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Abstract

Many formulations of quadratic allocation problems, portfolio optimization problems, the maximum weight clique problem, among others, take the form as the well-known standard quadratic optimization problem, which consists in minimizing a homogeneous quadratic function on the usual simplex in the non negative orthant. We propose to analyze the same problem when the simplex is substituted by a convex and compact base of any pointed, closed, convex cone (so, the cone of positive semidefinite matrices or the cone of copositive matrices are particular instances). Three main duals (for which a semi-infinite formulation of the primal problem is required) are associated, and we establish some characterizations of strong duality with respect to each of the three duals in terms of copositivity of the Hessian of the quadratic objective function on suitable cones. Such a problem reveals a hidden convexity and the validity of S-lemma. In case of bidimensional quadratic optimization problems, copositivity of the Hessian of the objective function is characterized, and the case when every local solution is global.

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Acknowledgements

The authors want to express their gratitud to both referees for the valuable suggestion and comments which led to the present version.

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Authors and Affiliations

  1. Departamento de Ingeniería Matemática, Universidad de Concepción, Casilla 160-C, Concepción, Chile

    Fabián Flores-Bazán & Stephanie Caro

  2. Departamento de Ingeniería Matemática, CI2MA, Universidad de Concepción, Casilla 160-C, Concepción, Chile

    Gabriel Cárcamo

Authors
  1. Fabián Flores-Bazán
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  2. Gabriel Cárcamo
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  3. Stephanie Caro
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Corresponding author

Correspondence to Fabián Flores-Bazán.

Additional information

The research for the first author was supported in part by CONICYT-Chile through FONDECYT 1181316, and PIA/Concurso Apoyo a Centros Científicos y Tecnológicos de Excelencia con Financiamiento Basal AFB170001. Support by the Programa Regional MATHAMSUD, 17MATH-06 is greatly acknowledged.

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Flores-Bazán, F., Cárcamo, G. & Caro, S. Extensions of the Standard Quadratic Optimization Problem: Strong Duality, Optimality, Hidden Convexity and S-lemma. Appl Math Optim 81, 383–408 (2020). https://doi.org/10.1007/s00245-018-9502-0

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