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

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...

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Vydané v:	Journal of global optimization Ročník 64; číslo 2; s. 249 - 272
Hlavní autori:	Kronqvist, Jan, Lundell, Andreas, Westerlund, Tapio
Médium:	Journal Article
Jazyk:	English
Vydavateľské údaje:	New York Springer US 01.02.2016 Springer Springer Nature B.V
Predmet:	Algorithms Approximation Computer Science Hyperplanes Integer programming Libraries Linear programming Mathematical analysis Mathematics Mathematics and Statistics Nonlinear programming Operations Research/Decision Theory Optimization Performance evaluation Quadratic programming Real Functions Searching Shot Solvers Studies Variables Extended cutting plane (ECP) algorithm Convex MINLP Supporting hyperplanes Cutting planes Supporting hyperplane optimization toolkit (SHOT) Extended supporting hyperplane (ESH) algorithm
ISSN:	0925-5001, 1573-2916
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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.
AbstractList	Issue Title: Special Issue on MAGO 2014 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. 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.
Audience	Academic
Author	Lundell, Andreas Westerlund, Tapio Kronqvist, Jan
Author_xml	– sequence: 1 givenname: Jan surname: Kronqvist fullname: Kronqvist, Jan email: jan.kronqvist@abo.fi organization: Optimization and Systems Engineering, Åbo Akademi University – sequence: 2 givenname: Andreas surname: Lundell fullname: Lundell, Andreas organization: Optimization and Systems Engineering, Åbo Akademi University – sequence: 3 givenname: Tapio surname: Westerlund fullname: Westerlund, Tapio organization: Optimization and Systems Engineering, Åbo Akademi University
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Keywords	Extended cutting plane (ECP) algorithm Convex MINLP Supporting hyperplanes Cutting planes Supporting hyperplane optimization toolkit (SHOT) Extended supporting hyperplane (ESH) algorithm
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Snippet	A new deterministic algorithm for solving convex mixed-integer nonlinear programming (MINLP) problems is presented in this paper: The extended supporting... Issue Title: Special Issue on MAGO 2014 A new deterministic algorithm for solving convex mixed-integer nonlinear programming (MINLP) problems is presented in...
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SubjectTerms	Algorithms Approximation Computer Science Hyperplanes Integer programming Libraries Linear programming Mathematical analysis Mathematics Mathematics and Statistics Nonlinear programming Operations Research/Decision Theory Optimization Performance evaluation Quadratic programming Real Functions Searching Shot Solvers Studies Variables
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Title	The extended supporting hyperplane algorithm for convex mixed-integer nonlinear programming
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