Computing tight bounds via piecewise linear functions through the example of circle cutting problems

This paper discusses approximations of continuous and mixed-integer non-linear optimization problems via piecewise linear functions. Various variants of circle cutting problems are considered, where the non-overlap of circles impose a non-convex feasible region. While the paper is written in an “eas...

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Bibliographic Details
Published in:Mathematical methods of operations research (Heidelberg, Germany) Vol. 84; no. 1; pp. 3 - 57
Main Author: Rebennack, Steffen
Format: Journal Article
Language:English
Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.08.2016
Springer Nature B.V
Subjects:
Approximation
Business and Management
Calculus of Variations and Optimal Control; Optimization
Computation
Cutting
Cutting parameters
Didacticism
Electronics
Integer programming
Linear programming
Mathematical analysis
Mathematical models
Mathematics
Mathematics and Statistics
Operations research
Operations Research/Decision Theory
Optimization
Optimization algorithms
Or in the Classroom
Quadratic programming
Studies
Circle cutting
Non-linear programming (NLP)
Quadratically constrained programming (QCP)
Logarithmic formulation
Non-convex optimization
Global optimization
Mixed-integer linear programming (MILP)
Incremental formulation
Outer approximation
Piecewise linear functions
Inner approximation
ISSN:1432-2994, 1432-5217
Online Access:Get full text
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Summary:This paper discusses approximations of continuous and mixed-integer non-linear optimization problems via piecewise linear functions. Various variants of circle cutting problems are considered, where the non-overlap of circles impose a non-convex feasible region. While the paper is written in an “easy-to-understand” and “hands-on” style which should be accessible to graduate students, also new ideas are presented. Specifically, piecewise linear functions are employed to yield mixed-integer linear programming problems which provide lower and upper bounds on the original problem, the circle cutting problem. The piecewise linear functions are modeled by five different formulations, containing the incremental and logarithmic formulations. Another variant of the cutting problem involves the assignment of circles to pre-defined rectangles. We introduce a new global optimization algorithm, based on piecewise linear function approximations, which converges in finitely many iterations to a globally optimal solution. The discussed formulations are implemented in GAMS. All GAMS-files are available for download in the Electronic supplementary material. Extensive computational results are presented with various illustrations.
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ISSN:1432-2994
1432-5217
DOI:10.1007/s00186-016-0546-0