Dynamic convexification within nested Benders decomposition using Lagrangian relaxation: An application to the strategic bidding problem

2017 ◽  
Vol 257 (2) ◽  
pp. 669-686 ◽  
Author(s):  
Gregory Steeger ◽  
Steffen Rebennack
Keyword(s):  
Lagrangian Relaxation ◽  
Benders Decomposition ◽  
Strategic Bidding ◽  
Nested Benders Decomposition

scholarly journals Flow Merging and Hub Route Optimization in Collaborative Transportation

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Kerui Weng ◽  
Zi-hao Xu
Keyword(s):  
Integer Programming ◽  
Lagrangian Relaxation ◽  
Benders Decomposition ◽  
Programming Model ◽  
Route Optimization ◽  
Mixed Integer ◽  
Routing Problem ◽  
Fixed Charges ◽  
Postal Services ◽  
Cost Discount

This paper studies the optimal hub routing problem of merged tasks in collaborative transportation. This problem allows all carriers’ transportation tasks to reach the destinations optionally passing through 0, 1, or 2 hubs within limited distance, while a cost discount on arcs in the hub route could be acquired after paying fixed charges. The problem arises in the application of logistics, postal services, airline transportation, and so forth. We formulate the problem as a mixed-integer programming model, and provide two heuristic approaches, respectively, based on Lagrangian relaxation and Benders decomposition. Computational experiments show that the algorithms work well.

scholarly journals Non-convex nested Benders decomposition

2022 ◽  
Author(s):  
Christian Füllner ◽  
Steffen Rebennack
Keyword(s):  
Benders Decomposition ◽  
Unit Commitment ◽  
Piecewise Linear ◽  
Optimal Solution ◽  
Mixed Integer ◽  
Global Optimality ◽  
Value Functions ◽  
State Variables ◽  
Lipschitz Continuous ◽  
Nested Benders Decomposition

AbstractWe 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.

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