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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Vydáno v:	Mathematical programming Ročník 196; číslo 1-2; s. 987 - 1024
Hlavní autoři:	Füllner, Christian, Rebennack, Steffen
Médium:	Journal Article
Jazyk:	angličtina
Vydáno:	Berlin/Heidelberg Springer Berlin Heidelberg 01.11.2022 Springer Springer Nature B.V
Témata:	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
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Abstract	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.
AbstractList	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 $$\varepsilon $$ ε -optimal solution in a finite number of steps. We provide promising computational results for some unit commitment problems of moderate size. 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. 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 [Formula omitted]-optimal solution in a finite number of steps. We provide promising computational results for some unit commitment problems of moderate size. 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.
Audience	Academic
Author	Rebennack, Steffen Füllner, Christian
Author_xml	– sequence: 1 givenname: Christian orcidid: 0000-0003-2485-6199 surname: Füllner fullname: Füllner, Christian email: christian.fuellner@kit.edu organization: Institute for Operations Research (IOR), Stochastic Optimization (SOP), Karlsruhe Institute of Technology (KIT) – sequence: 2 givenname: Steffen surname: Rebennack fullname: Rebennack, Steffen organization: Institute for Operations Research (IOR), Stochastic Optimization (SOP), Karlsruhe Institute of Technology (KIT)
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Snippet	We propose a new decomposition method to solve multistage non-convex mixed-integer (stochastic) nonlinear programming problems (MINLPs). We call this algorithm...
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SubjectTerms	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
Title	Non-convex nested Benders decomposition
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