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24
On the Global Solution of Linear Programs with Linear Complementarity Constraints
, 2007
"... This paper presents a parameter-free integer-programming based algorithm for the global resolution of a linear program with linear complementarity constraints (LPEC). The cornerstone of the algorithm is a minimax integer program formulation that characterizes and provides certificates for the three ..."
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Cited by 20 (3 self)
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This paper presents a parameter-free integer-programming based algorithm for the global resolution of a linear program with linear complementarity constraints (LPEC). The cornerstone of the algorithm is a minimax integer program formulation that characterizes and provides certificates for the three outcomes—infeasibility, unboundedness, or solvability—of an LPEC. An extreme point/ray generation scheme in the spirit of Benders decomposition is developed, from which valid inequalities in the form of satisfiability constraints are obtained. The feasibility problem of these inequalities and the carefully guided linear programming relaxations of the LPEC are the workhorse of the algorithm, which also employs a specialized procedure for the sparsification of the satifiability cuts. We establish the finite termination of the algorithm and report computational results using the algorithm for solving randomly generated LPECs of reasonable sizes. The results establish that the algorithm can handle infeasible, unbounded, and solvable LPECs effectively.
IIS Branch-and-Cut for Joint Chance-Constrained Stochastic Programs and Application to Optimal Vaccine Allocation
- European Journal of Operational Research
"... We present a new method for solving stochastic programs with joint chance constraints with random technology matrices and discretely distributed random data. The problem can be reformulated as a large-scale mixed 0-1 integer program. We derive a new class of optimality cuts called IIS cuts and apply ..."
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Cited by 13 (0 self)
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We present a new method for solving stochastic programs with joint chance constraints with random technology matrices and discretely distributed random data. The problem can be reformulated as a large-scale mixed 0-1 integer program. We derive a new class of optimality cuts called IIS cuts and apply them to our problem. The cuts are based on irreducibly infeasible subsets (IIS) of an LP defined by requiring that all scenarios be satisfied. We propose an efficient method for improving the upper bound of the problem when no cut can be found. We derive and implement a branch-and-cut algorithm based on IIS cuts, and refer to this algorithm as the IIS Branch-and-Cut algorithm. We report on computational results with several test instances from optimal vaccine allocation and a production planning problem from the literature. The computational results are very promising as the IIS branch-and-cut algorithm gives significantly better results than a state-of-the-art commercial solver.
An LPCC Approach to Nonconvex Quadratic Programs
, 2008
"... Filling a gap in nonconvex quadratic programming, this paper shows that the global resolution of a feasible quadratic program (QP), which is not known a priori to be bounded or unbounded below, can be accomplished in finite time by solving a linear program with linear complementarity constraints, i. ..."
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Cited by 6 (2 self)
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Filling a gap in nonconvex quadratic programming, this paper shows that the global resolution of a feasible quadratic program (QP), which is not known a priori to be bounded or unbounded below, can be accomplished in finite time by solving a linear program with linear complementarity constraints, i.e., an LPCC. Alternatively, this task can be divided into two LPCCs: the first one confirms whether or not the QP is bounded below on the feasible set and computes a feasible ray on which the QP is unbounded if such a ray exists; the second LPCC computes a globally optimal solution if it exists, by identifying a stationary point that yields the best quadratic objective value. In turn, the global resolution of these LPCCs can be accomplished by a parameter-free, mixed integer-programming based, finitely terminating algorithm developed recently by the authors, which can be enhanced in this context by a new kind of valid cuts derived from the second-order conditions of the QP and by exploiting the special structure of the LPCCs. Throughout, our treatment makes no boundedness assumption of the QP; this is a significant departure from much of the existing literature which consistently employs the boundedness of the feasible set as a blanket assumption. The general theory is illustrated by 3 classes of indefinite problems: QPs with simple upper and lower bounds (existence of optimal solutions is guaranteed); same QPs with an additional inequality constraint (extending the case of simple bound constraints); and nonnegatively constrained copositive QPs (no guarantee of the existence of an optimal solution). 1
The branch and cut method in the PLATO-N project
- EngOpt 2008 – International Conference on Engineering Optimization, Rio de
, 2008
"... We present the multiple load structural topology design problems with discrete design variables which are considered in the PLATO-N project. For the considered class of problems a global optimization method based on the concept of branch and cut is developed and implemented. In the method a large nu ..."
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Cited by 1 (0 self)
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We present the multiple load structural topology design problems with discrete design variables which are considered in the PLATO-N project. For the considered class of problems a global optimization method based on the concept of branch and cut is developed and implemented. In the method a large number of continuous relaxations are solved. Strong continuous relaxations are obtained by removing certain complicating constraints. Using duality results these relaxations are reformulated into programs which can be efficiently solved within the branch and cut search tree. We also present an algorithm for generating cuts to strengthen the quality of the relaxations. The branch and cut method is used to solve a benchmark example which can be used to validate other methods and heuristics. 2. Keywords: Topology optimization, branch and cut, stress constraints, reformulations, relaxations. In this work we consider structural topology optimization problems in which the design variables are chosen from a finite set of given values. The optimal design problems are modeled as non-convex mixed 0–1 optimization problems. In the PLATO-N ∗ project we are interested in two particular problems from the field of optimal structural topology design. The first problem is a minimum weight problem with
Theoretical and Computational Advances for Network Diversion
, 2012
"... The network-diversion problem (ND) is defined on a directed or undirected graph G = (V, E) having non-negative edge weights, a source vertex s, a sink vertex t, and a “diversion edge ” e ′. This problem, with intelligence-gathering and war-fighting applications, seeks a minimum-weight, minimal s-t c ..."
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The network-diversion problem (ND) is defined on a directed or undirected graph G = (V, E) having non-negative edge weights, a source vertex s, a sink vertex t, and a “diversion edge ” e ′. This problem, with intelligence-gathering and war-fighting applications, seeks a minimum-weight, minimal s-t cut EC ⊆ E in G such that e ′ ∈ EC. We present (a) a new NP-completeness proof for ND on directed graphs, (b) the first polynomial-time solution algorithm for a special graph topology, (c) an improved mixed-integer programming formula-tion (MIP), and (d) useful valid inequalities for that MIP. The proof strengthens known results by showing, for instance, that ND is strongly NP-complete on a directed graph even when e ′ is incident from s or into t, but not both, and even when G is acyclic; a corollary shows
20:00 – 22:00 Social Dinner
"... There is a secretary office close to the conference rooms. Participants can get the badges and other material on • Sunday, 18:00- 20:00, ..."
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There is a secretary office close to the conference rooms. Participants can get the badges and other material on • Sunday, 18:00- 20:00,
Revenue Adequacy Constrained Optimal Transmission Switching
"... Financial Transmission Rights (FTRs) are used to hedge congestion risk and they are financed by congestion rents. The ISO may not collect enough congestion rents to cover its obligation to the FTR holders; this is known as revenue inadequacy. Revenue inadequacy may occur even though ISOs run a Simul ..."
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Financial Transmission Rights (FTRs) are used to hedge congestion risk and they are financed by congestion rents. The ISO may not collect enough congestion rents to cover its obligation to the FTR holders; this is known as revenue inadequacy. Revenue inadequacy may occur even though ISOs run a Simultaneous Feasibility Test (SFT), which ensures revenue adequacy. However, the SFT relies on the assumption that the grid topology is not modified. Recent research suggests that we should co-optimize generation with the network topology. There is the concern that optimizing the topology will cause revenue inadequacy. In this paper, we examine how transmission switching affects revenue adequacy of FTRs. We discuss how the optimal transmission switching problem can be modified in order to maximize the market surplus subject to maintaining revenue adequacy, if that is the desired motivation of the ISO. We also discuss the policy implications of adopting such a method.
A Combinatorial Benders' Cuts Algorithm for the Local Container Drayage Problem
"... This paper examines the local container drayage problem under a special operation mode in which tractors and trailers can be separated; that is, tractors can be assigned to a new task at another location while trailers with containers are waiting for packing or unpacking. Meanwhile, the strategy of ..."
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This paper examines the local container drayage problem under a special operation mode in which tractors and trailers can be separated; that is, tractors can be assigned to a new task at another location while trailers with containers are waiting for packing or unpacking. Meanwhile, the strategy of sharing empty containers between different customers is also considered to improve the efficiency and lower the operation cost. The problem is formulated as a vehicle routing and scheduling problem with temporal constraints. We adopt combinatorial benders' cuts algorithm to solve this problem. Numerical experiments are performed on a group of randomly generated instances to test the performance of the proposed algorithm.
