In this paper, we reformulate a nonlinear complementarity problem or a mixed complementarity problem as a system of piecewise almost linear equations. The problems arise, for example, from the obstacle problems with a...
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In this paper, we reformulate a nonlinear complementarity problem or a mixed complementarity problem as a system of piecewise almost linear equations. The problems arise, for example, from the obstacle problems with a nonlinear source term or some contact problems. Based on the reformulated systems of the piecewise almost linear equations, we propose a class of semi-iterative algorithms to find the exact solution of the problems. We prove that the semi-iterative algorithms enjoy a nice monotone convergence property in the sense that subsets of the indices consisting of the indices, for which the corresponding components of the iterates violate the constraints, become smaller and smaller. Then the algorithms converge monotonically to the exact solutions of the problems in a finite number of steps. Some numerical experiments are presented to show the effectiveness of the proposed algorithms.
LetG=(V,E) be an undirected graph with positive integer edge lengths. Letm be the number of edges inE, and letd be the sum of the edge lengths. We prove that the solution value to the continuousp-center location probl...
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LetG=(V,E) be an undirected graph with positive integer edge lengths. Letm be the number of edges inE, and letd be the sum of the edge lengths. We prove that the solution value to the continuousp-center location problem is a rationalp 1/p 2, where logp 1=O(m 5 logd+m 6 logp),i=1,2. This result is then used to construct a finite algorithm for the continuousp-center problem.
We review and extend previous work on the approximation of the linear l, estimator by the Huber M-estimator based on the algorithms proposed by Clark and Osborne [7], and Madsen and Nielsen [12]. Although the Madsen-N...
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We review and extend previous work on the approximation of the linear l, estimator by the Huber M-estimator based on the algorithms proposed by Clark and Osborne [7], and Madsen and Nielsen [12]. Although the Madsen-Nielsen algorithm is a promising one, it is guaranteed to terminate finitely under certain assumptions. We describe a variant of the Madsen-Nielsen algorithm to compute the l, estimator from the Huber M-estimator in a finite number of steps without any restrictive steps nor assumptions. Summary computational results are given.
This paper addresses a general class of two-stage stochastic programs with integer recourse and discrete distributions. We exploit the structure of the value function of the second-stage integer problem to develop a n...
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This paper addresses a general class of two-stage stochastic programs with integer recourse and discrete distributions. We exploit the structure of the value function of the second-stage integer problem to develop a novel global optimization algorithm. The proposed scheme departs from those in the current literature in that it avoids explicit enumeration of the search space while guaranteeing finite termination. Computational experiments on standard test problems indicate superior performance of the proposed algorithm in comparison to those in the existing literature.
We describe a new finite continuation algorithm for linear programming. The dual of the linear programming problem with unit lower and upper bounds is formulated as an l(1) minimization problem augmented with the addi...
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We describe a new finite continuation algorithm for linear programming. The dual of the linear programming problem with unit lower and upper bounds is formulated as an l(1) minimization problem augmented with the addition of a linear term. This nondifferentiable problem is approximated by a smooth problem. It is shown that the minimizers of the smooth problem define a family of piecewise-linear paths as a function of a smoothing parameter. Based on this property, a finite algorithm that, traces these paths to arrive at an optimal solution of the linear program is developed. The smooth problems are solved by a Newton-type algorithm. Preliminary numerical results indicate that the new algorithm is promising.
We consider a two-person, general-sum, rational-data, undiscounted stochastic game in which one player (player II) controls the transition probabilities. We show that the set of stationary equilibrium points is the un...
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We consider a two-person, general-sum, rational-data, undiscounted stochastic game in which one player (player II) controls the transition probabilities. We show that the set of stationary equilibrium points is the union of a finite number of sets such that, every element of each of these sets can be constructed from a finite number of extreme equilibrium strategies for player I and from a finite number of pseudo-extreme equilibrium strategies for player II. These extreme and pseudo-extreme strategies can themselves be constructed by finite (but inefficient) algorithms. Analogous results can also be established in the more straightforward case of discounted single-controller games.
We consider the basic covariate model with one treatment at I levels and an additional first-order regression on a cofactor with values in [−1,1] as well as the corresponding intra-class regression model where the slo...
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We consider the basic covariate model with one treatment at I levels and an additional first-order regression on a cofactor with values in [−1,1] as well as the corresponding intra-class regression model where the slope of the regression line depends on the treatment level. In applied design problems often upper and lower bounds on the number of replications for each treatment level are given reflecting the limited availability of some levels and the need for accurate estimators respectively. With respect to these constraints necessary and sufficient conditions for D-optimum designs for inference on (i) the treatment effects (ii) the regression parameters and (iii) all parameters are derived. Also these conditions give simple finite algorithms generating optimum designs.
We consider finite-dimensional minimax problems for two traditional models: firstly, with box constraints at variables and, secondly, taking into account a finite number of linear inequalities. We present finite exact...
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We consider finite-dimensional minimax problems for two traditional models: firstly, with box constraints at variables and, secondly, taking into account a finite number of linear inequalities. We present finite exact primal and dual methods. These methods are adapted to a great extent to the specific structure of the cost function which is formed by a finite number of linear functions. During the iterations of the primal method we make use of the information from the dual problem, thereby increasing effectiveness. To improve the dual method we use the "long dual step" rule (the principle of full relaxation). The results are illustrated by numerical experiments.
The idea of continuation applied to the proximal transform of a polyhedral convex function is used to develop a general procedure for the construction of descent algorithms which has the property that it specializes t...
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The idea of continuation applied to the proximal transform of a polyhedral convex function is used to develop a general procedure for the construction of descent algorithms which has the property that it specializes to variants of the projected gradient method when applied to such problems as l1
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