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Global optimization of concave functions subject to quadratic constraints: An application in nonlinear bilevel programming

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Abstract

When the follower's optimality conditions are both necessary and sufficient, the nonlinear bilevel program can be solved as a global optimization problem. The complementary slackness condition is usually the complicating constraint in such problems. We show how this constraint can be replaced by an equivalent system of convex and separable quadratic constraints. In this paper, we propose different methods for finding the global minimum of a concave function subject to quadratic separable constraints. The first method is of the branch and bound type, and is based on rectangular partitions to obtain upper and lower bounds. Convergence of the proposed algorithm is also proved. For computational purposes, different procedures that accelerate the convergence of the proposed algorithm are analysed. The second method is based on piecewise linear approximations of the constraint functions. When the constraints are convex, the problem is reduced to global concave minimization subject to linear constraints. In the case of non-convex constraints, we use zero-one integer variables to linearize the constraints. The number of integer variables depends only on the concave parts of the constraint functions.

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Authors and Affiliations

  1. School of Industrial and Systems Engineering, Georgia Institute of Technology, 30332, Atlanta, GA, USA

    Faiz A. Al-Khayyal

  2. Fachbereich IV - Mathematik, Universität Trier, Trier, Germany

    Reiner Horst

  3. Department of Industrial and Systems Engineering, University of Florida, 32611, Gainesville, FL, USA

    Panos M. Pardalos

Authors
  1. Faiz A. Al-Khayyal
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  2. Reiner Horst
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  3. Panos M. Pardalos
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Additional information

Parts of the present paper were prepared while the second author was visiting Georgia Tech and the University of Florida.

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Al-Khayyal, F.A., Horst, R. & Pardalos, P.M. Global optimization of concave functions subject to quadratic constraints: An application in nonlinear bilevel programming. Ann Oper Res 34, 125–147 (1992). https://doi.org/10.1007/BF02098176

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