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Linear complementarity, linear and nonlinear programming
01 Jan 1988-
About: The article was published on 1988-01-01 and is currently open access. It has received 1012 citations till now. The article focuses on the topics: Mixed complementarity problem & Complementarity theory.
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TL;DR: In this paper, a generalized Newton method was proposed for solving the linear complementarity problem with the regular interval matrix based on the nonlinear penalized equation, where the penalty equation of LCP was transformed into the absolute value equation, and the existence of solutions for the penalty problem was proved by the regularity of the interval matrix.
Abstract: The penalty equation of LCP is transformed into the absolute value equation, and then the existence of solutions for the penalty equation is proved by the regularity of the interval matrix. We propose a generalized Newton method for solving the linear complementarity problem with the regular interval matrix based on the nonlinear penalized equation. Further, we prove that this method is convergent. Numerical experiments are presented to show that the generalized Newton method is effective.
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01 Sep 2019TL;DR: It is proposed to use a specially designed quadratic smoothing regularizing functional, which ensures the uniqueness of the solution of a linear programming problem with resource constraints in conditions of inaccurate information, which is solved by effective subgradient methods.
Abstract: The article considers a linear programming problem with resource constraints in conditions of inaccurate information, where the number of constraints is substantially less than the number of variables. Noisiness of the data and possible poor conditionality of the constraint matrix determine the ambiguity of the solution. Alongside with this, solutions found by linear programming methods may not satisfy the conditions of proportional resource allocation for similar objects. In this paper, it is proposed to use a specially designed quadratic smoothing regularizing functional, which ensures the uniqueness of the solution. A certain choice of quadratic function coefficients also makes it possible to control the proportions of the resource distribution and to observe the conditions of equivalence of this distribution for close enterprises. To reduce the dimension of the problem under consideration, transition to a dual problem was made in the work. This made it possible to obtain the problem of unconditional non-smooth minimization of a lower dimension. Tikhonov's quadratic regularization can be applied to ensure the uniqueness of the solution for the latter. The dual problem is solved by effective subgradient methods. The computational experiment confirms the efficiency and effectiveness of the proposed approach. The paper presents an example of solving an applied problem.
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TL;DR: In this article, a continuation method for linear complementarity problem (LCP) is proposed, which solves one system of linear equations and carries out only a one-line search at each iteration.
Abstract: In this article, we propose a new continuation method for solving the linear complementarity problem (LCP). The method solves one system of linear equations and carries out only a one-line search at each iteration. The continuation method is based on a modified smoothing function. The existence and continuity of a smooth path for solving the LCP with a P 0 matrix are discussed. We investigate the boundedness of the iteration sequence generated by our continuation method under the assumption that the solution set of the LCP is nonempty and bounded. It is shown to converge to an LCP solution globally linearly and locally superlinearly without the assumption of strict complementarity at the solution under suitable assumption. In addition, some numerical results are also reported in this article.
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Cites background from "Linear complementarity, linear and ..."
...This linear complementary problem is one for which Murty [18] has shown that principal pivot method I is known to run in a number of pivots exponential in the number of variables in the problem....
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...1 1 0 BBBBBB@ 1 CCCCCCA : This linear complementary problem is one for which Murty [18] has shown that principal pivot method I is known to run in a number of pivots exponential in the number of variables in the problem....
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TL;DR: A strongly polynomial reduction from Grid-LPs to linear programs over cubes (Cube-LLP) and generalized linear complementarity problems (GLCPs) with hidden K-matrices is obtained through a generalization of the reduction scheme for Grid- LPs.
Abstract: We investigate the duality relation between linear programs over grids (Grid-LPs) and generalized linear complementarity problems (GLCPs) with hidden K-matrices Moreover, the two problems share their combinatorial structure with discounted Markov decision processes (MDPs) Through proposing reduction schemes for the GLCP, we obtain a strongly polynomial reduction from Grid-LPs to linear programs over cubes (Cube-LPs) As an application, discounted MDPs admit formulations as Cube-LPs This result also suggests that Cube-LPs are the key problems with respect to solvability of linear programming in strongly polynomial time We then consider two-player stochastic games with perfect information as a natural generalization of discounted MDPs We identify the subclass of the GLCPs with P-matrices that corresponds to these games and also provide a characterization in terms of unique-sink orientations A strongly polynomial reduction from the games to their binary counterparts is obtained through a generalization of the reduction scheme for Grid-LPs Remark This is a report on ongoing research activities The content is subject to change
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TL;DR: In this article, a special class of fixed point problems for piecewise constant mappings of a simplex into itself is considered, which arise from the consideration of models with fixed budgets and possessing a property of monotonicity.
Abstract: We consider a special class of the fixed point problems for piecewise constant mappings of a simplex into itself. These are polyhedral complementarity problems arising in studying the classical exchange model and its variations. We study the problems that stem from the consideration of models with fixed budgets and possessing a certain property of monotonicity (logarithmic monotonicity). Our considerations are purely mathematical and not associated with the economic models that gave rise to these mathematical objects. The class of regular mappings is investigated, and their potentiality is proved.
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