Recently an O*(n(4)) volume algorithm has been presented for convex bodies by Lovasz and Vempala, where n is the number of dimensions of the convex body. Essentially the algorithm is a series of Monte Carlo integratio...
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Recently an O*(n(4)) volume algorithm has been presented for convex bodies by Lovasz and Vempala, where n is the number of dimensions of the convex body. Essentially the algorithm is a series of Monte Carlo integrations. In this paper we describe a computer implementation of the volume algorithm, where we improved the computational aspects of the original algorithm by adding variance decreasing modifications: a stratified sampling strategy, double point integration and orthonormalised estimators. Formulas and methodology were developed so that the errors in each phase of the algorithm can be controlled. Some computational results for convex bodies in dimensions ranging from 2 to 10 are presented as well. (C) 2011 Elsevier B.V. All rights reserved.
Recently an volume algorithm has been presented for convex bodies by Lovasz and Vempala, where is the number of dimensions of the convex body. Essentially the algorithm consists of several, interlocked simulational st...
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Recently an volume algorithm has been presented for convex bodies by Lovasz and Vempala, where is the number of dimensions of the convex body. Essentially the algorithm consists of several, interlocked simulational steps of slightly different natures. A computer implementation was later developed to gather some information about the numerical aspects of the algorithm, the number of dimensions in the examples was at most 10, and the errors of the results were somewhat dissatisfying. Now we present a parallel version of the improved algorithm, where variance reducing was added to make the algorithm faster, and the use of a GPU with 480 processors made experimentation easier. Computational results for convex bodies in dimensions ranging from 2 to 20 are presented as well.
We employ the volume algorithm as a subgradient deflection strategy in a variable target value method for solving nondifferentiable optimization problems. Focusing on Lagrangian duals for LPs, we exhibit primal noncon...
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We employ the volume algorithm as a subgradient deflection strategy in a variable target value method for solving nondifferentiable optimization problems. Focusing on Lagrangian duals for LPs, we exhibit primal nonconvergence of the original method, establish convergence of the proposed algorithm in the dual space, and present related computational results. (C) 2004 Elsevier B.V. All rights reserved.
We revise the volume algorithm (VA) for linear programming and relate it to bundle methods. When first introduced, VA was presented as a subgradient-like method for solving the original problem in its dual form. In a ...
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We revise the volume algorithm (VA) for linear programming and relate it to bundle methods. When first introduced, VA was presented as a subgradient-like method for solving the original problem in its dual form. In a way similar to the serious/null steps philosophy of bundle methods, VA produces green, yellow or red steps. In order to give convergence results, we introduce in VA a precise measure for the improvement needed to declare a green or serious step. This addition yields a revised formulation (RVA) that is halfway between VA and a specific bundle method, that we call BVA. We analyze the convergence properties of both RVA and BVA. Finally, we compare the performance of the modified algorithms versus VA on a set of Rectilinear Steiner problems of various sizes and increasing complexity, derived from real world VLSI design instances.
The fixed-charge multicommodity capacitated network design (FCMC) problem remains challenging, particularly in large-scale contexts. In this particular case, the ability to produce good-quality solutions in a reasonab...
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The fixed-charge multicommodity capacitated network design (FCMC) problem remains challenging, particularly in large-scale contexts. In this particular case, the ability to produce good-quality solutions in a reasonable amount of time depends on the availability of efficient algorithms. Therefore, this paper proposes a volume-based branch-and-cut algorithm, which applies a relax-and-cut procedure to solve linear programs in each node of the enumeration tree. Moreover, a Lagrangian feasibility pump heuristic using the volume algorithm as a solver for linear programs was implemented to accelerate the search for good feasible solutions in large-scale cases. The obtained results showed that the proposed branch-and-cut scheme is competitive with other state-of-the-art algorithms, and presents better performance when solving large-scale instances.
The Bundle Method and the volume algorithm are among the most efficient techniques to obtain accurate Lagrangian dual bounds for hard combinatorial optimization problems. We propose here to compare their performance o...
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The Bundle Method and the volume algorithm are among the most efficient techniques to obtain accurate Lagrangian dual bounds for hard combinatorial optimization problems. We propose here to compare their performance on very large scale Fixed-Charge Multicommodity Capacitated Network Design problems. The motivation is not only the quality of the approximation of these bounds as a function of the computational time but also the ability to produce feasible primal solutions and thus to reduce the gap for very large instances for which optimal solutions are out of reach. Feasible solutions are obtained through the use of Lagrangian information in constructive and improving heuristic schemes. We show in particular that, if the Bundle implementation has provided great quality bounds in fewer iterations, the volume algorithm is able to reduce the gaps of the largest instances, taking profit of the low computational cost per iteration compared to the Bundle Method.
We consider the version of prize collecting Steiner tree problem (PCSTP) where each node of a given weighted graph is associated with a prize and where the objective is to find a minimum weight tree spanning a subset ...
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We consider the version of prize collecting Steiner tree problem (PCSTP) where each node of a given weighted graph is associated with a prize and where the objective is to find a minimum weight tree spanning a subset of nodes and collecting a total prize not less that a given quota Q. We present a lower bound and a genetic algorithm for the PCSTR The lower bound is based on a Lagrangian decomposition of a minimum spanning tree formulation of the problem. The volume algorithm is used to solve the Lagrangian dual. The genetic algorithm incorporates several enhancements. In particular, it fully exploits both primal and dual information produced by Lagrangian decomposition. The proposed lower and upper bounds are assessed through computational experiments on randomly generated instances with up to 500 nodes and 5000 edges. For these instances, the proposed lower and upper bounds exhibit consistently a tight gap: in 76% of the cases the gap is strictly less than 2%. (c) 2004 Elsevier Ltd. All rights reserved.
We study a realistic Bi-objective Multimodal Transportation Planning Problem (BMTPP) faced by logistics companies when trying to obtain cost advantages and improve the customer satisfaction in a competitive market. Th...
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We study a realistic Bi-objective Multimodal Transportation Planning Problem (BMTPP) faced by logistics companies when trying to obtain cost advantages and improve the customer satisfaction in a competitive market. The two objectives considered are: the minimization of total transportation cost and the maximization of service quality. Given a set of transportation orders described by an origin, a destination and a time window, solving BMTPP involves determining the delivery path for each order in a capacitated network as well as selecting the carrier with the best service quality for each edge of the path. The BMTPP is formulated as a novel bi-objective mixed integer linear programming model and an iterative $\epsilon$ -constraint method is applied to solve it. As the NP-hardness of the single-objective problems derived from BMTPP, a Lagrangian Relaxation (LR) heuristic which can not only provide a near-optimal solution but also a lower bound for each of the single-objective problems is developed. 100 randomly generated instances are tested and the computational results demonstrate the effectiveness of the heuristic in obtaining a tight lower bound and a high-quality near-optimal solution for the derived single-objective problem. Various performance indicators show the high-quality of the Pareto front of the bi-objective problem obtained by the heuristic. We also provide a case study for the proposed LR heuristic in a logistics network in China.
The notion of conductance introduced by Jerrum and Sinclair [8] has been widely used to prove rapid mixing of Markov chains. Here we introduce a bound that extends this in two directions. First, instead of measuring t...
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The notion of conductance introduced by Jerrum and Sinclair [8] has been widely used to prove rapid mixing of Markov chains. Here we introduce a bound that extends this in two directions. First, instead of measuring the conductance of the worst subset of states, we bound the mixing time by a formula that can be thought of as a weighted average of the Jerrum-Sinclair bound (where the average is taken over subsets of states with different sizes). Furthermore, instead of just the conductance, which in graph theory terms measures edge expansion, we also take into account node expansion. Our bound is related to the logarithmic Sobolev inequalities, but it appears to be more flexible and easier to compute. In the case of random walks in convex bodies, we show that this new bound is better than the known bounds for the worst case. This saves a factor of O(n) in the mixing time bound, which is incurred in all proofs as a 'penalty' for a 'bad start'. We show that in a convex body in R-n, with diameter D, random walk with steps in a ball with radius 6 mixes in O*(nD(2)/delta(2)) time (if idle steps at the boundary are not counted). This gives an O*(n(3)) sampling algorithm after appropriate preprocessing, improving the previous bound of O*(n(4)). The application of the general conductance bound in the geometric setting depends on an improved isoperimetric inequality for convex bodies.
We present a Lagrangean Decomposition approach for obtaining strong lower bounds on minimizing medium to large scale multistage stochastic mixed 0-1 problems. The problem is represented by a mixture of the splitting r...
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We present a Lagrangean Decomposition approach for obtaining strong lower bounds on minimizing medium to large scale multistage stochastic mixed 0-1 problems. The problem is represented by a mixture of the splitting representation up to a given stage, so-named break stage, and the compact representation for the other stages along the time horizon. The dualization of the nonanticipativity constraints for the variables up to the break stage results in a model that can be decomposed into a set of independent scenario cluster submodels. The nonanticipativity constraints for the 0-1 and continuous variables in the cluster submodels are implicitly satisfied. Four scenario cluster schemes are compared for Lagrangean multipliers updating such as the Subgradient Method, the volume algorithm, the Lagrangean Progressive Hedging algorithm and the Dynamic Constrained Cutting Plane scheme. We have observed in randomly generated instances that the smaller the number of clusters, the stronger the lower bound provided for the original problem (even, frequently, it is the solution value). (C) 2015 Elsevier Ltd. All rights reserved.
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