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Study On Smoothing Algorithms For Weighted Complementarity Problem Over Nonnegative Quadrant And Second Order Cone

Posted on:2022-01-16Degree:MasterType:Thesis
Country:ChinaCandidate:W L LiuFull Text:PDF
GTID:2480306554472454Subject:Mathematics
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As an extension of the complementarity problem,the weighted complementarity problem(WCP)can be used in engineering,image processing and finance.For example,the Fisher market equilibrium problem can be modeled as the WCP.The WCP is to find a pair of vectors that belongs to the intersection of a manifold with a cone,so that their algebraic product is equal to a given weight vector.However,due to the nonzero weight vector in WCP,it is difficult to study the theory and the algorithms of WCP.Therefore,there are few researches on WCP.In this paper,we study the nonmonotone smoothing Newton algorithms to solve the WCP over nonnegative quadrant and linear weighted second-order cone(SOC)complementarity problem.The main contributions are described as follows:1.Constructing a new smoothing function and analyzing the properties of the function,such as continuous differentiability and strong semismoothness.Based on the new derivative-free nonmonotone line search,we present smoothing Newton algorithm for solving the WCP over nonnegative quadrant.By algebraic theory,the well-definedness and global convergence of the algorithm are proved.Numerical examples verify the stability and effectiveness of the algorithm.2.Based on the new smoothing function,the linear weighted SOC complementarity problem is transformed into a system of equations.Then the equations are solved by the inexact nonmonotone smoothing Newton method.With the assumption of the positive semidefinite matrix,we show that the algorithm is globally and locally superlinear convergence.The well performance of the method is illustrated by numerical examples.3.We propose a new weighted SOC complementarity function and its smoothing function.Then we derive the computable formula for Jacobian matrix and prove Jacobian consistency of a smoothing function for weighted SOC complementarity problem.In order to adjust a parameter appropriately in smoothing algorithms,we estimate the distance between the subgradient of the weighted SOC complementarity function and the gradient of its smoothing function.
Keywords/Search Tags:weighted complementarity problem, linear weighted second-order cone complementarity problem, derivative-free nonmonotone line search, global convergence, local superlinear convergence, Jacobian consistency
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