scholarly journals An exploratory computational analysis of dual degeneracy in mixed-integer programming

2020 ◽  
Vol 8 (3-4) ◽  
pp. 241-261 ◽  
Author(s):  
Gerald Gamrath ◽  
Timo Berthold ◽  
Domenico Salvagnin
Keyword(s):  
Integer Programming ◽  
Mixed Integer Programming ◽  
Computational Study ◽  
Solution Process ◽  
Mixed Integer ◽  
Different Types ◽  
Primal Heuristics ◽  
Individual Variables ◽  
The Individual ◽  
Constraint Ratio

Abstract Dual degeneracy, i.e., the presence of multiple optimal bases to a linear programming (LP) problem, heavily affects the solution process of mixed integer programming (MIP) solvers. Different optimal bases lead to different cuts being generated, different branching decisions being taken and different solutions being found by primal heuristics. Nevertheless, only a few methods have been published that either avoid or exploit dual degeneracy. The aim of the present paper is to conduct a thorough computational study on the presence of dual degeneracy for the instances of well-known public MIP instance collections. How many instances are affected by dual degeneracy? How degenerate are the affected models? How does branching affect degeneracy: Does it increase or decrease by fixing variables? Can we identify different types of degenerate MIPs? As a tool to answer these questions, we introduce a new measure for dual degeneracy: the variable–constraint ratio of the optimal face. It provides an estimate for the likelihood that a basic variable can be pivoted out of the basis. Furthermore, we study how the so-called cloud intervals—the projections of the optimal face of the LP relaxations onto the individual variables—evolve during tree search and the implications for reducing the set of branching candidates.

scholarly journals Conflict-Driven Heuristics for Mixed Integer Programming

2020 ◽  
Author(s):  
Jakob Witzig ◽  
Ambros Gleixner
Keyword(s):  
Integer Programming ◽  
Mixed Integer Programming ◽  
State Of The Art ◽  
Computational Study ◽  
Search Tree ◽  
General Purpose ◽  
Conflict Analysis ◽  
Mixed Integer ◽  
Linear Programs ◽  
Feasible Solutions

Two essential ingredients of modern mixed-integer programming solvers are diving heuristics, which simulate a partial depth-first search in a branch-and-bound tree, and conflict analysis, which learns valid constraints from infeasible subproblems. So far, these techniques have mostly been studied independently: primal heuristics for finding high-quality feasible solutions early during the solving process and conflict analysis for fathoming nodes of the search tree and improving the dual bound. In this paper, we pose the question of whether and how the orthogonal goals of proving infeasibility and generating improving solutions can be pursued in a combined manner such that a state-of-the-art solver can benefit. To do so, we integrate both concepts in two different ways. First, we develop a diving heuristic that simultaneously targets the generation of valid conflict constraints from the Farkas dual and the generation of improving solutions. We show that, in the primal, this is equivalent to the optimistic strategy of diving toward the best bound with respect to the objective function. Second, we use information derived from conflict analysis to enhance the search of a diving heuristic akin to classic coefficient diving. In a detailed computational study, both methods are evaluated on the basis of an implementation in the source-open-solver SCIP. The experimental results underline the potential of combining both diving heuristics and conflict analysis. Summary of Contribution. This original article concerns the advancement of exact general-purpose algorithms for solving one of the largest and most prominent problem classes in optimization, mixed-integer linear programs. It demonstrates how methods for conflict analysis that learn from infeasible subproblems can be combined successfully with diving heuristics that aim at finding primal solutions. For two newly designed diving heuristics, this paper features a thoroughly computational study regarding their impact on the overall performance of a state-of-the-art MIP solver.

scholarly journals Mixed-Integer Programming Model for Ranking Universities: Letting Universities Choose the Weights

MENDEL ◽  
2021 ◽  
Vol 27 (1) ◽  
pp. 41-48
Author(s):  
Jakub Kudela
Keyword(s):  
Integer Programming ◽  
Mixed Integer Programming ◽  
Programming Model ◽  
Mixed Integer ◽  
Integer Programming Model ◽  
Mixed Integer Programming Model ◽  
World University Rankings ◽  
The Times ◽  
Talented Students ◽  
The Individual

Regardless of the shortcomings and criticisms of world university rankings, these metrics are still widely used by students and parents to select universities and by universities to attract talented students and researchers, as well as funding. This paper proposes a new mixed-integer programming model for ranking universities. The new approach alleviates one of the criticisms -- the issue of the ``arbitrariness'' of the weights used for aggregation of the individual criteria (or indicators) utilized in the contemporary rankings. Instead, the proposed model uses intervals of different sizes for the weights and lets the universities themselves ``choose'' the weights to optimize their position in the rankings. A numerical evaluation of the proposed ranking, based on the indicator values and weights from the Times Higher Education World University Ranking, is presented.

A Mixed Integer Programming Formulation for Generating Shared Press-Brake Setups

2001 ◽  
Author(s):  
Satyandra K. Gupta ◽  
Deepak Rajagopal
Keyword(s):  
Integer Programming ◽  
Mixed Integer Programming ◽  
Sheet Metal ◽  
Cost Effective ◽  
Mixed Integer ◽  
Actual Process ◽  
Planning Techniques ◽  
Different Types ◽  
Sheet Metal Bending ◽  
Different Parts

Abstract To enable cost-effective small batch manufacturing, we need to eliminate unnecessary setup operations, improve tool utilization, and thereby increase throughput. In sheet metal bending, the time taken for the actual process of bending is significantly less compared to the time taken for setup and tool changes. Press-brakes for sheet metal bending can be setup to produce more than one type of parts without requiring a setup change. To exploit this flexibility, we need setup planning techniques so that press-brake setups can be shared among many different parts. In this paper, we describe an approach based on mixed integer programming to generate a shared setup for a set of parts. We expect that by producing many different types of parts on the same setup, we can significantly reduce the number of setups required and enable cost-effective small-batch manufacturing.

scholarly journals Solving the Minimum Label Spanning Tree Problem by Mathematical Programming Techniques

2011 ◽  
Vol 2011 ◽  
pp. 1-38 ◽  
Author(s):  
Andreas M. Chwatal ◽  
Günther R. Raidl
Keyword(s):  
Integer Programming ◽  
Spanning Tree ◽  
Network Flows ◽  
Solution Process ◽  
Building Blocks ◽  
Branch And Cut ◽  
Mixed Integer ◽  
Programming Techniques ◽  
Speed Up ◽  
Primal Heuristics

We present exact mixed integer programming approaches including branch-and-cut and branch-and-cut-and-price for the minimum label spanning tree problem as well as a variant of it having multiple labels assigned to each edge. We compare formulations based on network flows and directed connectivity cuts. Further, we show how to use odd-hole inequalities and additional inequalities to strengthen the formulation. Label variables can be added dynamically to the model in the pricing step. Primal heuristics are incorporated into the framework to speed up the overall solution process. After a polyhedral comparison of the involved formulations, comprehensive computational experiments are presented in order to compare and evaluate the underlying formulations and the particular algorithmic building blocks of the overall branch-and-cut- (and-price) framework.

Non-blocking Supervisory Design Using Mixed Integer Programming

2009 ◽  
Vol 35 (2) ◽  
pp. 180-185
Author(s):  
Mi ZHAO ◽  
Zhi-Wu LI ◽  
Na WEI
Keyword(s):  
Integer Programming ◽  
Mixed Integer Programming ◽  
Mixed Integer
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