52 research outputs found

    Relaxations and Cutting Planes for Linear Programs with Complementarity Constraints

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    We study relaxations for linear programs with complementarity constraints, especially instances whose complementary pairs of variables are not independent. Our formulation is based on identifying vertex covers of the conflict graph of the instance and generalizes the extended reformulation-linearization technique of Nguyen, Richard, and Tawarmalani to instances with general complementarity conditions between variables. We demonstrate how to obtain strong cutting planes for our formulation from both the stable set polytope and the boolean quadric polytope associated with a complete bipartite graph. Through an extensive computational study for three types of practical problems, we assess the performance of our proposed linear relaxation and new cutting-planes in terms of the optimality gap closed

    Recognition of Facets for Knapsack Polytope is DP-complete

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    DP is a complexity class that is the class of all languages that are the intersection of a language in NP and a language in co-NP, as coined by Papadimitriou and Yannakakis. In this paper, we will establish that, recognizing a facet for the knapsack polytope is DP-complete, as conjectured by Hartvigsen and Zemel in 1992. Moreover, we show that the recognition problem of a supporting hyperplane for the knapsack polytope and the exact knapsack problem are both DP-complete, and the membership problem of knapsack polytope is NP-complete

    Large scale group network optimization

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    Every knapsack problem may be relaxed to a cyclic group problem. In 1969, Gomory found the subadditive characterization of facets of the master cyclic group problem. We simplify the subadditive relations by the substitution of complementarities and discover a minimal representation of the subadditive polytope for the master cyclic group problem. By using the minimal representation, we characterize the vertices of cardinality length 3 and implement the shooting experiment from the natural interior point. The shooting from the natural interior point is a shooting from the inside of the plus level set of the subadditive polytope. It induces the shooting for the knapsack problem. From the shooting experiment for the knapsack problem we conclude that the most hit facet is the knapsack mixed integer cut which is the 2-fold lifting of a mixed integer cut. We develop a cutting plane algorithm augmenting cutting planes generated by shooting, and implement it on Wong-Coppersmith digraphs observing that only small number of cutting planes are enough to produce the optimal solution. We discuss a relaxation of shooting as a clue to quick shooting. A max flow model on covering space is shown to be equivalent to the dual of shooting linear programming problem.Ph.D.Committee Chair: Ellis L. Johnson; Committee Member: Brady Hunsaker; Committee Member: George Nemhauser; Committee Member: Jozef Siran; Committee Member: Shabbir Ahmed; Committee Member: William Coo

    Reformulation and decomposition of integer programs

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    In this survey we examine ways to reformulate integer and mixed integer programs. Typically, but not exclusively, one reformulates so as to obtain stronger linear programming relaxations, and hence better bounds for use in a branch-and-bound based algorithm. First we cover in detail reformulations based on decomposition, such as Lagrangean relaxation, Dantzig-Wolfe column generation and the resulting branch-and-price algorithms. This is followed by an examination of Benders’ type algorithms based on projection. Finally we discuss in detail extended formulations involving additional variables that are based on problem structure. These can often be used to provide strengthened a priori formulations. Reformulations obtained by adding cutting planes in the original variables are not treated here.Integer program, Lagrangean relaxation, column generation, branch-and-price, extended formulation, Benders' algorithm

    Cardinality Constrained Optimization Problems

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    In this thesis, we examine optimization problems with a constraint that allows for only a certain number of variables to be nonzero. This constraint, which is called a cardinality constraint, has received considerable attention in a number of areas such as machine learning, statistics, computational finance, and operations management. Despite their practical needs, most optimization problems with a cardinality constraints are hard to solve due to their nonconvexity. We focus on constructing tight convex relaxations to such problems

    Explicit convex and concave envelopes through polyhedral subdivisions with Unstable Equilibria

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    In this paper, we derive explicit characterizations of convex and concave envelopes of several nonlinear functions over various subsets of a hyper-rectangle. These envelopes are obtained by identifying polyhedral subdivisions of the hyper-rectangle over which the envelopes can be constructed easily. In particular, we use these techniques to derive, in closed-form, the concave envelopes of concave-extendable supermodular functions and the convex envelopes of disjunctive convex functions.

    Mixed-integer Nonlinear Optimization: a hatchery for modern mathematics

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    The second MFO Oberwolfach Workshop on Mixed-Integer Nonlinear Programming (MINLP) took place between 2nd and 8th June 2019. MINLP refers to one of the hardest Mathematical Programming (MP) problem classes, involving both nonlinear functions as well as continuous and integer decision variables. MP is a formal language for describing optimization problems, and is traditionally part of Operations Research (OR), which is itself at the intersection of mathematics, computer science, engineering and econometrics. The scientific program has covered the three announced areas (hierarchies of approximation, mixed-integer nonlinear optimal control, and dealing with uncertainties) with a variety of tutorials, talks, short research announcements, and a special "open problems'' session

    Optimization with mixed-integer, complementarity and bilevel constraints with applications to energy and food markets

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    In this dissertation, we discuss three classes of nonconvex optimization problems, namely, mixed-integer programming, nonlinear complementarity problems, and mixed-integer bilevel programming. For mixed-integer programming, we identify a class of cutting planes, namely the class of cutting planes derived from lattice-free cross-polytopes, which are proven to provide good approximations to the problem while being efficient to compute. We show that the closure of these cuts gives an approximation that depends only on the ambient dimension and that the cuts can be computed efficiently by explicitly providing an algorithm to compute the cut coefficients in O(n2n)O(n2^n) time, as opposed to solving a nearest lattice-vector problem, which could be much harder. For complementarity problems, we develop a first-order approximation algorithm to efficiently approximate the covariance of the decision in a stochastic complementarity problem. The method can be used to approximate the covariance for large-scale problems by solving a system of linear equations. We also provide bounds to the error incurred in this technique. We then use the technique to analyze policies related to the North American natural gas market. Further, we use this branch of nonconvex problems in the Ethiopian food market to analyze the regional effects of exogenous shocks on the market. We develop a detailed model of the food production, transportation, trade, storage, and consumption in Ethiopia, and test it against exogenous shocks. These shocks are motivated by the prediction that teff, a food grain whose export is banned now, could become a super grain. We present the regional effects of different government policies in response to this shock. For mixed-integer bilevel programming, we develop algorithms that run in polynomial time, provided a subset of the input parameters are fixed. Besides the Σ2p\Sigma^p_2-hardness of the general version of the problem, we show polynomial solvability and NPNP-completeness of certain restricted versions of this problem. Finally, we completely characterize the feasible regions represented by each of these different types of nonconvex optimization problems. We show that the representability of linear complementarity problems, continuous bilevel programs, and polyhedral reverse-convex programs are the same, and they coincide with that of mixed-integer programs if the feasible region is bounded. We also show that the feasible region of any mixed-integer bilevel program is a union of the feasible regions of finitely many mixed-integer programs up to projections and closures
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