# Browsing by Subject "cutting planes"

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Item A new polyhedral approach to combinatorial designs(Texas A&M University, 2004-09-30) Arambula Mercado, IvetteShow more We consider combinatorial t-design problems as discrete optimization problems. Our motivation is that only a few studies have been done on the use of exact optimization techniques in designs, and that classical methods in design theory have still left many open existence questions. Roughly defined, t-designs are pairs of discrete sets that are related following some strict properties of size, balance, and replication. These highly structured relationships provide optimal solutions to a variety of problems in computer science like error-correcting codes, secure communications, network interconnection, design of hardware; and are applicable to other areas like statistics, scheduling, games, among others. We give a new approach to combinatorial t-designs that is useful in constructing t-designs by polyhedral methods. The first contribution of our work is a new result of equivalence of t-design problems with a graph theory problem. This equivalence leads to a novel integer programming formulation for t-designs, which we call GDP. We analyze the polyhedral properties of GDP and conclude, among other results, the associated polyhedron dimension. We generate new classes of valid inequalities to aim at approximating this integer program by a linear program that has the same optimal solution. Some new classes of valid inequalities are generated as Chv?atal-Gomory cuts, other classes are generated by graph complements and combinatorial arguments, and others are generated by the use of incidence substructures in a t-design. In particular, we found a class of valid inequalities that we call stable-set class that represents an alternative graph equivalence for the problem of finding a t-design. We analyze and give results on the strength of these new classes of valid inequalities. We propose a separation problem and give its integer programming formulation as a maximum (or minimum) edge-weight biclique subgraph problem. We implement a pure cutting-plane algorithm using one of the stronger classes of valid inequalities derived. Several instances of t-designs were solved efficiently by this algorithm at the root node of the search tree. Also, we implement a branch-and-cut algorithm and solve several instances of 2-designs trying different base formulations. Computational results are included.Show more Item Facets for Continuous Multi-Mixing Set and Its Generalizations: Strong Cuts for Multi-Module Capacitated Lot-Sizing Problem(2014-08-20) Bansal, ManishShow more The research objective of this dissertation is to develop new facet-defining valid inequalities for several new multi-parameter multi-constraint mixed integer sets. These valid inequalities result in cutting planes that significantly improve the efficiency of algorithms for solving mixed integer programming (MIP) problems involving multimodule capacity constraints. These MIPs arise in many classical and modern applications ranging from production planning to cloud computing. The research in this dissertation generalizes cut-generating methods such as mixed integer rounding (MIR), mixed MIR, continuous mixing, n-step MIR, mixed n-step MIR, migling, and n-step mingling, along with various well-known families of cuts for problems such as multi-module capacitated lot-sizing (MMLS), multi-module capacitated facility location (MMFL), and multi-module capacitated network design (MMND) problems. More specifically, in the first step, we introduce a new generalization of the continuous mixing set, referred to as the continuous multi-mixing set, where the coefficients satisfy certain conditions. For each n? ? {1; : : : ; n}, we develop a class of valid inequalities for this set, referred to as the n0-step cycle inequalities, and present their facet-defining properties. We also present a compact extended formulation for this set and an exact separation algorithm to separate over the set of all n?-step cycle inequalities for a given n? ? {1; : : : ; n}. In the next step, we extend the results of the first step to the case where conditions on the coefficients of the continuous multi-mixing set are relaxed. This leads to an extended formulation and a generalization of the n-step cycle inequalities, n ? N, for the continuous multi-mixing set with general coefficients. We also show that these inequalities are facet-defining in many cases. In the third step, we further generalize the continuous multi-mixing set (where no conditions are imposed on the coefficients) by incorporating upper bounds on the integer variables. We introduce a compact extended formulation and new families of multi-row cuts for this set, referred to as the mingled n-step cycle inequalities (n ? N), through a generalization of the n-step mingling. We also provide an exact separation algorithm to separate over a set of all these inequalities. Furthermore, we present the conditions under which a subset of the mingled n-step cycle inequalities are facet-defining for this set. Finally, in the fourth step, we utilize the results of first step to introduce new families of valid inequalities for MMLS, MMFL, and MMND problems. Our computational results show that the developed cuts are very effective in solving the MMLS instances with two capacity modules, resulting in considerable reduction in the integrality gap, the number of nodes, and total solution time.Show more