Piecewise-Linear Approximations of Multidimensional Functions

We develop explicit, piecewise-linear formulations of functions f ( x ):ℝ n ↦ ℝ, n ≤3, that are defined on an orthogonal grid of vertex points. If mixed-integer linear optimization problems (MILPs) involving multidimensional piecewise-linear functions can be easily and efficiently solved to global o...

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Bibliographic Details
Published in:Journal of optimization theory and applications Vol. 145; no. 1; pp. 120 - 147
Main Authors: Misener, R., Floudas, C. A.
Format: Journal Article
Language:English
Published: Boston Springer US 01.04.2010
Springer
Springer Nature B.V
Subjects:
Algorithms
Applications of Mathematics
Applied sciences
Approximation
Atmospheric pollution
Calculus of Variations and Optimal Control; Optimization
Engineering
Enhanced oil recovery
Exact sciences and technology
Linear programming
Mathematical programming
Mathematics
Mathematics and Statistics
Operational research and scientific management
Operational research. Management science
Operations Research/Decision Theory
Optimization
Pollution
Pollution sources. Measurement results
Studies
Theory of Computation
Linear interpolation
EPA Complex Emissions Model
Approximate optimization
Simplices
Global optimum
Linear programming
Mixed integer programming
Function approximation
Modeling
Mixed method
Constrained optimization
Linear approximation
Simplex method
Air pollution
Mixed problem
Objective function
Non linear effect
Analytical function
Piecewise linearization
Piecewise-linear techniques
ISSN:0022-3239, 1573-2878
Online Access:Get full text
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Description
Summary:We develop explicit, piecewise-linear formulations of functions f ( x ):ℝ n ↦ ℝ, n ≤3, that are defined on an orthogonal grid of vertex points. If mixed-integer linear optimization problems (MILPs) involving multidimensional piecewise-linear functions can be easily and efficiently solved to global optimality, then non-analytic functions can be used as an objective or constraint function for large optimization problems. Linear interpolation between fixed gridpoints can also be used to approximate generic, nonlinear functions, allowing us to approximately solve problems using mixed-integer linear optimization methods. Toward this end, we develop two different explicit formulations of piecewise-linear functions and discuss the consequences of integrating the formulations into an optimization problem.
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ISSN:0022-3239
1573-2878
DOI:10.1007/s10957-009-9626-0