The linear complementarity problem (LCP) can be viewed as the problem of minimizing xTy subject to y = Mx + q and x,y ≥ 0. We are interested in finding a point with xTy 0. The algorithm proceeds by iteratively reduci...
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The linear complementarity problem (LCP) can be viewed as the problem of minimizing xTy subject to y = Mx + q and x,y ≥ 0. We are interested in finding a point with xTy 0. The algorithm proceeds by iteratively reducing the potential function f(x,y) = ρ ln xTy - Σ ln xjyj, where, for example, ρ = 2n. The direction of movement in the original space can be viewed as follows. First, apply a linear scaling transformation to make the coordinates of the current point all equal to 1. Take a gradient step in the transformed space using the gradient of the transformed potential function, where the step size is either predetermined by the algorithm or decided by line search to minimize the value of the potential. Finally, map the point back to the original space. A bound on the worst-case performance of the algorithm depends on the parameter λ* = λ*(M, Ε), which is defined as the minimum of the smallest eigenvalue of a matrix of the form (I + Y-1MX)(I + MTY-2MX)-1(I + XMTY-1) where X and Y vary over the nonnegative diagonal matrices such that eTXYe >Q Ε and XjjYjj &le n2. If M is a P-matrix, λ* is positive and the algorithm solves the problem in polynomial time in terms of the input size, |log g|, and 1/λ*. It is also shown that when M is positive semi-definite, the choice of ρ = 2n + √2n yields a polynomial-time algorithm. This covers the convex quadratic minimization problem.
We consider partial updating in Ye's affine potential reduction algorithm for linearprogramming. We show that using a Goldstein-Armijo rule to safeguard a linesearch of the potential function during primal steps ...
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We consider partial updating in Ye's affine potential reduction algorithm for linearprogramming. We show that using a Goldstein-Armijo rule to safeguard a linesearch of the potential function during primal steps is sufficient to control the number of updates. We also generalize the dual step construction to apply with partial updating. The result is the first O(n3L) algorithm for linearprogramming whose steps are not constrained by the need to remain approximately centered. The fact that the algorithm has a rigorous 'primal-only' initialization actually reduces the complexity to less than O(m1.5n1.5L).
We consider the class of linear programs with infinitely many variables and constraints having the property that every constraint contains at most finitely many variables while every variable appears in at most finite...
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We consider the class of linear programs with infinitely many variables and constraints having the property that every constraint contains at most finitely many variables while every variable appears in at most finitely many constraints. Examples include production planning and equipment replacement over an infinite horizon. We form the natural dual linearprogramming problem and prove strong duality under a transversality condition that dual prices are asymptotically zero. That is, we show, under this transversality condition, that optimal solutions are attained in both primal and dual problems and their optimal values are equal. The transversality condition, and hence strong duality, is established for an infinite horizon production planning problem.
The first two parts of this paper have developed a simplex algorithm for minimizing convex separable piecewise-linear functions subject to linear constraints. This concluding part argues that a direct piecewise-linear...
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The first two parts of this paper have developed a simplex algorithm for minimizing convex separable piecewise-linear functions subject to linear constraints. This concluding part argues that a direct piecewise-linear simplex implementation has inherent advantages over an indirect approach that relies on transformation to a linear program. The advantages are shown to be implicit in relationships between the linear and piecewise-linear algorithms, and to be independent of many details of implementation. Two sets of computational results serve to illustrate these arguments;the piecewise-linear simplex algorithm is observed to run 2-6 times faster than a comparable linear algorithm, not including any additional expense that might be incurred in setting up the equivalent linear program. Further support for the practical value of a good piecewise-linearprogramming algorithm is provided by a survey of many varied applications.
In this paper, new lower bounds for the asymmetric travelling salesman problem are presented, based on spanning arborescences. The new bounds are combined in an additive procedure whose theoretical performance is comp...
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In this paper, new lower bounds for the asymmetric travelling salesman problem are presented, based on spanning arborescences. The new bounds are combined in an additive procedure whose theoretical performance is compared with that of the Balas and Christofides procedure (1981). Both procedures have been imbedded in a simple branch and bound algorithm and experimentally evaluated on hard test problems.
We introduce an upper bound on the expectation of a special class of sublinear functions of multivariate random variables defined over the entire Euclidean space without an independence assumption. The bound can be ev...
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We introduce an upper bound on the expectation of a special class of sublinear functions of multivariate random variables defined over the entire Euclidean space without an independence assumption. The bound can be evaluated easily requiring only the solution of systems of linear equations thus permitting implementations in high-dimensional space. Only knowledge on the underlying distribution means and second moments is necessary. We discuss pertinent techniques on dominating general sublinear functions by using simpler sublinear and polyhedral functions and second order quadratic functions.
This paper discusses the maximization of a bilinear function over two independent polytopes. The maximization problem is converted into a max-min problem, using duality. This problem is then solved via a sequence of d...
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This paper discusses the maximization of a bilinear function over two independent polytopes. The maximization problem is converted into a max-min problem, using duality. This problem is then solved via a sequence of dual linear programmes, whose constraint vectors are successively determined by tth order optima of a master linear programme.
Anstreicher has proposed a variant of Karmarkar's projective algorithm that handles standard-form linearprogramming problems nicely. We suggest modifications to his method that we suspect will lead to better sear...
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Anstreicher has proposed a variant of Karmarkar's projective algorithm that handles standard-form linearprogramming problems nicely. We suggest modifications to his method that we suspect will lead to better search directions and a more useful algorithm. Much of the analysis depends on a two-constraint linearprogramming problem that is a relaxation of the sealed original problem.
Given a graph and a length function defined on its edge-set, the Traveling Salesman Problem can be described as the problem of finding a family of edges (an edge may be chosen several times) which forms a spanning Eul...
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Given a graph and a length function defined on its edge-set, the Traveling Salesman Problem can be described as the problem of finding a family of edges (an edge may be chosen several times) which forms a spanning Eulerian subgraph of minimum length. In this paper we characterize those graphs for which the convex hull of all solutions is given by the nonnegativity constraints and the classical cut constraints. This characterization is given in terms of excluded minors. A constructive characterization is also given which uses a small number of basic graphs.
In this paper we discuss the problem of how to divide the total cost of a round trip along several institutes among the institutes visited. We introduce two types of cooperative games - fixed-route traveling salesman ...
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In this paper we discuss the problem of how to divide the total cost of a round trip along several institutes among the institutes visited. We introduce two types of cooperative games - fixed-route traveling salesman games and traveling salesman games - as a tool to attack this problem. Under very mild conditions we prove that fixed-route traveling salesman games have non-empty cores if the fixed route is a solution of the classical traveling salesman problem. Core elements provide us with fair cost allocations. A traveling salesman game may have an empty core, even if the cost matrix satisfies the triangle inequality. In this paper we introduce a class of matrices defining TS-games with non-empty cores.
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