scholarly journals Benders decomposition and column-and-row generation for solving large-scale linear programs with column-dependent-rows

2018 ◽  
Vol 264 (1) ◽  
pp. 29-45 ◽  
Author(s):  
İbrahim Muter ◽  
Ş. İlker Birbil ◽  
Kerem Bülbül
Keyword(s):  
Large Scale ◽  
Benders Decomposition ◽  
Linear Programs

The Star Degree Centrality Problem: A Decomposition Approach

2021 ◽  
Author(s):  
Mustafa C. Camur ◽  
Thomas Sharkey ◽  
Chrysafis Vogiatzis
Keyword(s):  
Integer Programming ◽  
Large Scale ◽  
Benders Decomposition ◽  
Open Neighborhood ◽  
Decomposition Algorithm ◽  
Solution Time ◽  
Set Cover ◽  
Degree Centrality ◽  
Decomposition Approach ◽  
Acceleration Techniques

We consider the problem of identifying the induced star with the largest cardinality open neighborhood in a graph. This problem, also known as the star degree centrality (SDC) problem, is shown to be [Formula: see text]-complete. In this work, we first propose a new integer programming (IP) formulation, which has a smaller number of constraints and nonzero coefficients in them than the existing formulation in the literature. We present classes of networks in which the problem is solvable in polynomial time and offer a new proof of [Formula: see text]-completeness that shows the problem remains [Formula: see text]-complete for both bipartite and split graphs. In addition, we propose a decomposition framework that is suitable for both the existing and our formulations. We implement several acceleration techniques in this framework, motivated by techniques used in Benders decomposition. We test our approaches on networks generated based on the Barabási–Albert, Erdös–Rényi, and Watts–Strogatz models. Our decomposition approach outperforms solving the IP formulations in most of the instances in terms of both solution time and quality; this is especially true for larger and denser graphs. We then test the decomposition algorithm on large-scale protein–protein interaction networks, for which SDC is shown to be an important centrality metric. Summary of Contribution: In this study, we first introduce a new integer programming (NIP) formulation for the star degree centrality (SDC) problem in which the goal is to identify the induced star with the largest open neighborhood. We then show that, although the SDC can be efficiently solved in tree graphs, it remains [Formula: see text]-complete in both split and bipartite graphs via a reduction performed from the set cover problem. In addition, we implement a decomposition algorithm motivated by Benders decomposition together with several acceleration techniques to both the NIP formulation and the existing formulation in the literature. Our experimental results indicate that the decomposition implementation on the NIP is the best solution method in terms of both solution time and quality.

scholarly journals Optimal Torpedo Scheduling

2018 ◽  
Vol 63 ◽  
pp. 955-986 ◽  
Author(s):  
Adrian Goldwaser ◽  
Andreas Schutt
Keyword(s):  
Large Scale ◽  
Benders Decomposition ◽  
Steel Production ◽  
Planning Horizon ◽  
Mixed Integer ◽  
Hot Metal ◽  
Rail Network ◽  
Sulfur Level ◽  
First Time ◽  
Resource Capacity

We consider the torpedo scheduling problem in steel production, which is concerned with the transport of hot metal from a blast furnace to an oxygen converter. A schedule must satisfy, amongst other considerations, resource capacity constraints along the path and the locations traversed as well as the sulfur level of the hot metal. The goal is first to minimize the number of torpedo cars used during the planning horizon and second to minimize the time spent desulfurizing the hot metal. We propose an exact solution method based on Logic based Benders Decomposition using Mixed-Integer and Constraint Programming, which optimally solves and proves, for the first time, the optimality of all instances from the ACP Challenge 2016 within 10 minutes. In addition, we adapted our method to handle large-scale instances and instances with a more general rail network. This adaptation optimally solved all challenge instances within one minute and was able to solve instances of up to 100,000 hot metal pickups.

Latest Advances on Benders Decomposition

2018 ◽  
pp. 5411-5421 ◽  
Author(s):  
Antonios Fragkogios ◽  
Georgios K. D. Saharidis
Keyword(s):  
Mathematical Programming ◽  
Operations Research ◽  
Decomposition Method ◽  
Large Scale ◽  
Benders Decomposition ◽  
Information Science ◽  
Solution Process ◽  
Cpu Time ◽  
Mathematical Problems ◽  
Modern Economic

Operations Research and Mathematical Programming together with Information Science and Technology are tools used to solve various problems in the modern economic environment. This chapter addresses the Benders Decomposition method, which is used for the solution of problems of Operations Research. This method, applied to certain large-scale mathematical problems, can make their solution feasible (if they cannot be solved with another procedure) or can accelerate the solution process in terms of CPU time. The authors provide a thorough presentation of how the decomposition of a problem is made and the Benders algorithm is applied for its solution. Main purpose of this chapter is to analyze the recent studies that address the method's weaknesses and accelerate its application for the faster solution of mathematical problems. A large number of papers is presented and the contribution of each one of them to the improvement of the method is described.

scholarly journals Stabilized Benders Methods for Large-Scale Combinatorial Optimization, with Application to Data Privacy

2020 ◽  
Vol 66 (7) ◽  
pp. 3051-3068 ◽  
Author(s):  
Daniel Baena ◽  
Jordi Castro ◽  
Antonio Frangioni
Keyword(s):  
Data Privacy ◽  
Large Scale ◽  
Benders Decomposition ◽  
Mixed Integer ◽  
Master Problem ◽  
Continuous Variables ◽  
Typical Structure ◽  
Disclosure Control ◽  
Current State ◽  
Statistical Disclosure

The cell-suppression problem (CSP) is a very large mixed-integer linear problem arising in statistical disclosure control. However, CSP has the typical structure that allows application of the Benders decomposition, which is known to suffer from oscillation and slow convergence, compounded with the fact that the master problem is combinatorial. To overcome this drawback, we present a stabilized Benders decomposition whose master is restricted to a neighborhood of successful candidates by local-branching constraints, which are dynamically adjusted, and even dropped, during the iterations. Our experiments with synthetic and real-world instances with up to 24,000 binary variables, 181 million (M) continuous variables, and 367M constraints show that our approach is competitive with both the current state-of-the-art code for CSP and the Benders implementation in CPLEX 12.7. In some instances, stabilized Benders provided a very good solution in less than 1 minute, whereas the other approaches found no feasible solution in 1 hour. This paper was accepted by Yinyu Ye, optimization.

scholarly journals Benders decomposition with adaptive oracles for large scale optimization

2020 ◽  
Author(s):  
Nicolò Mazzi ◽  
Andreas Grothey ◽  
Ken McKinnon ◽  
Nagisa Sugishita
Keyword(s):  
Large Scale ◽  
Benders Decomposition ◽  
Optimization Problems ◽  
Cutting Plane ◽  
Computational Effort ◽  
Planning Problem ◽  
Illustrative Case ◽  
Investment Planning ◽  
Cutting Plane Algorithms ◽  
Inexact Information

AbstractThis paper proposes an algorithm to efficiently solve large optimization problems which exhibit a column bounded block-diagonal structure, where subproblems differ in right-hand side and cost coefficients. Similar problems are often tackled using cutting-plane algorithms, which allow for an iterative and decomposed solution of the problem. When solving subproblems is computationally expensive and the set of subproblems is large, cutting-plane algorithms may slow down severely. In this context we propose two novel adaptive oracles that yield inexact information, potentially much faster than solving the subproblem. The first adaptive oracle is used to generate inexact but valid cutting planes, and the second adaptive oracle gives a valid upper bound of the true optimal objective. These two oracles progressively “adapt” towards the true exact oracle if provided with an increasing number of exact solutions, stored throughout the iterations. These adaptive oracles are embedded within a Benders-type algorithm able to handle inexact information. We compare the Benders with adaptive oracles against a standard Benders algorithm on a stochastic investment planning problem. The proposed algorithm shows the capability to substantially reduce the computational effort to obtain an $$\epsilon $$ ϵ -optimal solution: an illustrative case is 31.9 times faster for a $$1.00\%$$ 1.00 % convergence tolerance and 15.4 times faster for a $$0.01\%$$ 0.01 % tolerance.

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