Higher-order cover cuts from zero–one knapsack constraints augmented by two-sided bounding inequalities

AbstractExtending our work on second-order cover cuts [F. Glover, H.D. Sherali, Second-order cover cuts, Mathematical Programming (ISSN: 0025-5610 1436-4646) (2007), doi:10.1007/s10107-007-0098-4. (Online)], we introduce a new class of higher-order cover cuts that are derived from the implications of a knapsack constraint in concert with supplementary two-sided inequalities that bound the sums of sets of variables. The new cuts can be appreciably stronger than the second-order cuts, which in turn dominate the classical knapsack cover inequalities. The process of generating these cuts makes it possible to sequentially utilize the second-order cuts by embedding them in systems that define the inequalities from which the higher-order cover cuts are derived. We characterize properties of these cuts, design specialized procedures to generate them, and establish associated dominance relationships. These results are used to devise an algorithm that generates all non-dominated higher-order cover cuts, and, in particular, to formulate and solve suitable separation problems for deriving a higher-order cut that deletes a given fractional solution to an underlying continuous relaxation. We also discuss a lifting procedure for further tightening any generated cut, and establish its polynomial-time operation for unit-coefficient cuts. A numerical example is presented that illustrates these procedures and the relative strength of the generated non-redundant, non-dominated higher-order cuts, all of which turn out to be facet-defining for this example. Some preliminary computational results are also presented to demonstrate the efficacy of these cuts in comparison with lifted minimal cover inequalities for the underlying knapsack polytope

Polyhedral techniques in combinatorial optimization II: computations

Combinatorial optimization problems appear in many disciplines ranging from management and logistics to mathematics, physics, and chemistry. These problems are usually relatively easy to formulate mathematically, but most of them are computationally hard due to the restriction that a subset of the variables have to take integral values. During the last two decades there has been a remarkable progress in techniques based on the polyhedral description of combinatorial problems. leading to a large increase in the size of several problem types that can be solved. The basic idea behind polyhedral techniques is to derive a good linear formulation of the set of solutions by identifying linear inequalities that can be proved to be necessary in the description of the convex hull of feasible solutions. Ideally we can then solve the problem as a linear programming problem, which can be done efficiently. The purpose of this manuscript is to give an overview of the developments in polyhedral theory, starting with the pioneering work by Dantzig, Fulkerson and Johnson on the traveling salesman problem, and by Gomory on integer programming. We also present some modern applications, and computational experience

The incremental connected facility location problem

We consider the incremental connected facility location problem (incremental ConFL), in which we are given a set of potential facilities, a set of interconnection nodes, a set of customers with demands, and a planning horizon. For each time period, we have to select a set of facilities to open, a set of customers to be served, the assignment of these customers to the open facilities, and a network that connects the open facilities. Once a customer is served, it must remain served in subsequent periods. Furthermore, in each time period the total demand of all customers served must be at least equal to a given minimum coverage requirement for that period. The objective is to minimize the total cost for building the network given by the investment and maintenance costs for the facilities and the network summed up over all time periods. We propose a mixed integer programming approach in which, in each time period, a single period ConFL with coverage restrictions has to be solved. For this latter problem, which is of particular interest in itself, new families of valid inequalities are proposed: these are set union knapsack cover (SUKC) inequalities, which are further enhanced by lifting and/or combined with cut-set inequalities, which are primarily used to ensure connectivity requirements. Details of an efficient branch-and-cut implementation are presented and computational results on a benchmark set of large instances are given, including examples of telecommunication networks in German

Strategic Surveillance System Design for Ports and Waterways

The purpose of this dissertation is to synthesize a methodology to prescribe a strategic design of a surveillance system to provide the required level of surveillance for ports and waterways. The method of approach to this problem is to formulate a linear integer programming model to prescribe a strategic surveillance system design (SSD) for ports or waterways, to devise branch-and-price decomposition (

Mixed Integer Programming Approaches for Group Decision Making

Group decision making problems are everywhere in our day-to-day lives and have great influence on the daily operation of companies and institutions. With the recent advances in computational technology, it's not surprising that some companies would want to harvest that power to aid their decision-making procedures. Ethelo, the company that we partnered with in this project, developed an online platform that aids decision-making procedures by formulating the decision-making problem as a mixed integer nonlinear program (MINLP), providing feedback by solving the MINLP in real-time, and allowing the general public to contribute their opinions. Since an interactive component is involved, it is the goal of this thesis to attempt to reduce the solve time of their MINLP by applying tools from Operational Research. The main contribution in this thesis is threefold: first, we noticed that a big proportion of the MINLPs can be easily reposed as linear integer programs, and that a runtime reduction of at least 87.9\% can be achieved by simply redirecting them to a linear solver. Second, we identified a knapsack-like polyhedral structure that, to the best of our knowledge, has not been studied before, and derived a sufficient condition to identify the cases for which all valid cuts can be derived by considering other knapsack or covering problems. Finally, for the more general case where the objective function is nonlinear and not continuous, we derived a few different formulations to get to different approximations of the nonlinear model, and tested all of the approximations computationally

Foundations of Software Science and Computation Structures

This open access book constitutes the proceedings of the 24th International Conference on Foundations of Software Science and Computational Structures, FOSSACS 2021, which was held during March 27 until April 1, 2021, as part of the European Joint Conferences on Theory and Practice of Software, ETAPS 2021. The conference was planned to take place in Luxembourg and changed to an online format due to the COVID-19 pandemic. The 28 regular papers presented in this volume were carefully reviewed and selected from 88 submissions. They deal with research on theories and methods to support the analysis, integration, synthesis, transformation, and verification of programs and software systems