Browsing by Author "Lyu, Bochuan"
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Item Biclique Partitions, Biclique Covers, and Disjunctive Constraints(2023-04-20) Lyu, Bochuan; Hicks, Illya V.This thesis focuses on disjunctive constraints and related combinatorial optimization problems: minimum biclique partition problem and minimum biclique cover problem. We introduce techniques to build small ideal mixed-integer programming (MIP) formulations of combinatorial disjunctive constraints (CDCs) via the independent branching scheme. We present a pairwise IB-representable class of CDCs, CDCs admitting junction trees, and provide a combinatorial procedure to build MIP formulations for those constraints. An NP-complete combinatorial optimization problem, the minimum biclique cover problem, needs to be solved in the combinatorial procedure. It motivates us to study minimum biclique partition and biclique cover problems on co-chordal graphs, which is the complementary graph of chordal. We provide heuristics to find biclique partitions on co-chordal graphs and show that the heuristics can provide biclique partitions close to optimal or even optimal if the graphs satisfy further assumptions on the structures. In addition, we provide a tighter bound for minimum biclique covers on a subclass of co-chordal graphs and present a lower bound for minimum biclique covers of general graphs. We also investigate the disjunctive constraints for piecewise linear relaxations of univariate and high-dimensional nonlinear functions that could appear in optimization problems. In order to study those piecewise linear relaxations, we introduce a new class of combinatorial disjunctive constraints: generalized n-dimensional ordered CDCs and present logarithmically sized ideal formulations under the independent branching scheme.Item Modeling Disjunctive Constraints via Junction Trees(2021-12-22) Lyu, Bochuan; Hicks, Illya V.; Huchette, Joseph A.In this thesis, we study the independent-branching (IB) framework of disjunctive constraints and identify a class of pairwise IB-representable disjunctive constraints: disjunctive constraints with junction trees. For this class of constraints, the existence of junction trees can be recognized in polynomial time. We also present a polynomial-time heuristic algorithm for the minimum biclique cover problem on the associated conflict graphs to build small and strong mixed-integer programming (MIP) formulations. Additionally, we apply the heuristic to find a smaller MIP formulation of generalized special ordered set with less variables and constraints than Huchette and Vielma [2019]. In computational experiments, we compare the proposed heuristic with other methods on a large set of artificially generated disjunctive constraints with junction trees. The new method significantly reduces the numbers of binary variables and constraints required for the MIP formulations than those of vertex cover approach.