A reformulation framework for global optimization

In this paper, we present a global optimization method for solving nonconvex mixed integer nonlinear programming (MINLP) problems. A convex overestimation of the feasible region is obtained by replacing the nonconvex constraint functions with convex underestimators. For signomial functions single-va...

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Bibliographic Details
Published in:Journal of global optimization Vol. 57; no. 1; pp. 115 - 141
Main Authors: Lundell, Andreas, Skjäl, Anders, Westerlund, Tapio
Format: Journal Article
Language:English
Published: Boston Springer US 01.09.2013
Springer Nature B.V
Subjects:
Algorithms
Approximation
Computer Science
Integer programming
Mathematical programming
Mathematics
Mathematics and Statistics
Mixed integer
Nonlinear programming
Operations Research/Decision Theory
Optimization
Optimization algorithms
Real Functions
Solvers
Studies
Transformations
Variables
Finland
SGO-algorithm
Global optimization
Signomial functions
Reformulation technique
Convex underestimators
Mixed integer nonlinear programming
Twice-differentiable functions
Piecewise linear functions
BB-underestimator
ISSN:0925-5001, 1573-2916
Online Access:Get full text
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Summary:In this paper, we present a global optimization method for solving nonconvex mixed integer nonlinear programming (MINLP) problems. A convex overestimation of the feasible region is obtained by replacing the nonconvex constraint functions with convex underestimators. For signomial functions single-variable power and exponential transformations are used to obtain the convex underestimators. For more general nonconvex functions two versions of the so-called α BB-underestimator, valid for twice-differentiable functions, are integrated in the actual reformulation framework. However, in contrast to what is done in branch-and-bound type algorithms, no direct branching is performed in the actual algorithm. Instead a piecewise convex reformulation is used to convexify the entire problem in an extended variable-space, and the reformulated problem is then solved by a convex MINLP solver. As the piecewise linear approximations are made finer, the solution to the convexified and overestimated problem will form a converging sequence towards a global optimal solution. The result is an easily-implementable algorithm for solving a very general class of optimization problems.
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ISSN:0925-5001
1573-2916
DOI:10.1007/s10898-012-9877-4