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The extended supporting hyperplane algorithm for convex mixed-integer nonlinear programming

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Abstract

A new deterministic algorithm for solving convex mixed-integer nonlinear programming (MINLP) problems is presented in this paper: The extended supporting hyperplane (ESH) algorithm uses supporting hyperplanes to generate a tight overestimated polyhedral set of the feasible set defined by linear and nonlinear constraints. A sequence of linear or quadratic integer-relaxed subproblems are first solved to rapidly generate a tight linear relaxation of the original MINLP problem. After an initial overestimated set has been obtained the algorithm solves a sequence of mixed-integer linear programming or mixed-integer quadratic programming subproblems and refines the overestimated set by generating more supporting hyperplanes in each iteration. Compared to the extended cutting plane algorithm ESH generates a tighter overestimated set and unlike outer approximation the generation point for the supporting hyperplanes is found by a simple line search procedure. In this paper it is proven that the ESH algorithm converges to a global optimum for convex MINLP problems. The ESH algorithm is implemented as the supporting hyperplane optimization toolkit (SHOT) solver, and an extensive numerical comparison of its performance against other state-of-the-art MINLP solvers is presented.

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Acknowledgments

Financial support from the Foundation of Åbo Akademi University and the Center of Excellence in Optimization and Systems Engineering, is gratefully acknowledged, as is the support from GAMS Development Corporation. JK is grateful for the financial support from the Finnish Graduate School in Chemical Engineering.

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Authors and Affiliations

  1. Optimization and Systems Engineering, Åbo Akademi University, Turku, Finland

    Jan Kronqvist, Andreas Lundell & Tapio Westerlund

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  1. Jan Kronqvist
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  2. Andreas Lundell
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  3. Tapio Westerlund
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Correspondence to Jan Kronqvist.

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Kronqvist, J., Lundell, A. & Westerlund, T. The extended supporting hyperplane algorithm for convex mixed-integer nonlinear programming. J Glob Optim 64, 249–272 (2016). https://doi.org/10.1007/s10898-015-0322-3

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