Non-convex nested Benders decomposition

We propose a new decomposition method to solve multistage non-convex mixed-integer (stochastic) nonlinear programming problems (MINLPs). We call this algorithm non-convex nested Benders decomposition (NC-NBD). NC-NBD is based on solving dynamically improved mixed-integer linear outer approximations...

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Bibliographic Details
Published in:Mathematical programming Vol. 196; no. 1-2; pp. 987 - 1024
Main Authors: Füllner, Christian, Rebennack, Steffen
Format: Journal Article
Language:English
Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.11.2022
Springer
Springer Nature B.V
Subjects:
Algorithms
Approximation
Benders decomposition
Calculus of Variations and Optimal Control; Optimization
Combinatorics
Full Length Paper
Mathematical and Computational Physics
Mathematical Methods in Physics
Mathematics
Mathematics and Statistics
Mathematics of Computing
Mixed integer
Nonlinear programming
Numerical Analysis
Optimization
Regularization
Theoretical
Unit commitment
Mixed-integer nonlinear programming (MINLP)
Global optimization
Non-convexities
90C11
Nested Benders decomposition
90C26
49M27
Non-convex value functions
ISSN:0025-5610, 1436-4646
Online Access:Get full text
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Summary:We propose a new decomposition method to solve multistage non-convex mixed-integer (stochastic) nonlinear programming problems (MINLPs). We call this algorithm non-convex nested Benders decomposition (NC-NBD). NC-NBD is based on solving dynamically improved mixed-integer linear outer approximations of the MINLP, obtained by piecewise linear relaxations of nonlinear functions. Those MILPs are solved to global optimality using an enhancement of nested Benders decomposition, in which regularization, dynamically refined binary approximations of the state variables and Lagrangian cut techniques are combined to generate Lipschitz continuous non-convex approximations of the value functions. Those approximations are then used to decide whether the approximating MILP has to be dynamically refined and in order to compute feasible solutions for the original MINLP. We prove that NC-NBD converges to an ε -optimal solution in a finite number of steps. We provide promising computational results for some unit commitment problems of moderate size.
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ISSN:0025-5610
1436-4646
DOI:10.1007/s10107-021-01740-0