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

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

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 (

Second-order cover inequalities Mathematics Subject Classification 90C10 · 90C27

Abstract We introduce a new class of second-order cover inequalities whose members are generally stronger than the classical knapsack cover inequalities that are commonly used to enhance the performance of branch-and-cut methods for 0-1 integer programming problems. These inequalities result by focusing attention on a single knapsack constraint in addition to an inequality that bounds the sum of all variables, or in general, that bounds a linear form containing only the coefficients 0, 1, and -1. We provide an algorithm that generates all non-dominated second-order cover inequalities, making use of theorems on dominance relationships to bypass the examination of many dominated alternatives. Furthermore, we derive conditions under which these non-dominated second-order cover inequalities would be facets of the convex hull of feasible solutions to the parent constraints, and demonstrate how they can be lifted otherwise. Numerical examples of applying the algorithm disclose its ability to generate valid inequalities that are sometimes significantly stronger than those derived from traditional knapsack covers. Our results can also be extended to incorporate multiple choice inequalities that limit sums over disjoint subsets of variables to be at most one