This paper is devoted to the study of the proximal point algorithm for solving monotone second-order cone complementarity problems. The proximal point algorithm is to generate a sequence by solving subproblems that ar...
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This paper is devoted to the study of the proximal point algorithm for solving monotone second-order cone complementarity problems. The proximal point algorithm is to generate a sequence by solving subproblems that are regularizations of the original problem. After given an appropriate criterion for approximate solutions of subproblems by adopting a merit function, the proximal point algorithm is verified to have global and superlinear convergence properties. For the purpose of solving the subproblems efficiently, we introduce a generalized Newton method and show that only one Newton step is eventually needed to obtain a desired approximate solution that approximately satisfies the appropriate criterion under mild conditions. Numerical comparisons are also made with the derivative-free descent method used by Pan and Chen (Optimization 59:1173-1197, 2010), which confirm the theoretical results and the effectiveness of the algorithm.
By using auxiliary principle technique, a new proximal point algorithm based on decomposition method is suggested for computing a weakly efficient solution of the constrained multiobjective optimization problem (MOP) ...
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By using auxiliary principle technique, a new proximal point algorithm based on decomposition method is suggested for computing a weakly efficient solution of the constrained multiobjective optimization problem (MOP) without assuming the nonemptiness of its solution set. The optimality conditions for (MOP) are derived by the Lagrangian function of its subproblem and corresponding mixed variational inequality. Some basic properties and convergence results of the proposed method are established under some mild assumptions. As an application, we employ the proposed method to solve a split feasibility problem. Finally, numerical results are also presented to illustrate the feasibility of the proposed algorithm.
We study the convergence of a diagonal process for minimizing a closed proper convex function f, in which a proximalpoint iteration is applied to a sequence of functions approximating f. We prove that, when the appro...
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We study the convergence of a diagonal process for minimizing a closed proper convex function f, in which a proximalpoint iteration is applied to a sequence of functions approximating f. We prove that, when the approximation is sufficiently fast, and also when it is sufficiently slow, the sequence generated by the method converges toward a minimizer of f. Comparison to previous work is provided through examples in penalty methods for linear programming and Tikhonov regularization.
In this paper we present an extension of the proximal point algorithm with Bregman distances to solve constrained minimization problems with quasiconvex and convex objective function on Hadamard manifolds. The propose...
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In this paper we present an extension of the proximal point algorithm with Bregman distances to solve constrained minimization problems with quasiconvex and convex objective function on Hadamard manifolds. The proposed algorithm is a modified and extended version of the one presented in Papa Quiroz and Oliveira (J Convex Anal 16(1): 49-69, 2009). An advantage of the proposed algorithm, for the nonconvex case, is that in each iteration the algorithm only needs to find a stationary point of the proximal function and not a global minimum. For that reason, from the computational point of view, the proposed algorithm is more practical than the earlier proximal method. Another advantage, for the convex case, is that using minimal condition on the problem data as well as on the proximal parameters we get the same convergence results of the Euclidean proximalalgorithm using Bregman distances.
Very recently, the author gave an upper bound on a decreasing positive sequence. And, he made use of it to improve a classical result of Br,zis and Lions concerning the proximal point algorithm for monotone inclusion ...
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Very recently, the author gave an upper bound on a decreasing positive sequence. And, he made use of it to improve a classical result of Br,zis and Lions concerning the proximal point algorithm for monotone inclusion in an infinite-dimensional Hilbert space. One assumption is the algorithm's strong convergence. In this paper, we derive a new upper bound on this decreasing positive sequence and thus achieve the same improvement without requiring this assumption.
In this paper, we give a sufficient condition which guarantees that the sequence generated by the proximal point algorithm terminates after a finite number of iterations.
In this paper, we give a sufficient condition which guarantees that the sequence generated by the proximal point algorithm terminates after a finite number of iterations.
This paper presents a proximal point algorithm for solving discrete e(infinity) approximation problems of the form minimize \\Ax - b\\(infinity). Let epsilon(infinity), be a preassigned positive constant and let epsil...
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This paper presents a proximal point algorithm for solving discrete e(infinity) approximation problems of the form minimize \\Ax - b\\(infinity). Let epsilon(infinity), be a preassigned positive constant and let epsilon(e), e = 0, 1,2,..., be a sequence of positive real numbers such that 0 < epsilon(e) < epsilon(infinity). Then, starting from an arbitrary point ao, the proposed method generates a sequence of points z(e), e = 0, 1,2,..., via the rule z(e+1) = arg min(z) {epsilon(e)1/2\\z -z(e)\\(2)(2) + 1/2\\Az - b\\(2)(infinity)}. One feature that characterizes this algorithm is its finite termination property. That is, a solution is reached within a finite number of iterations. The smaller are the numbers epsilon(e) the smaller is the number of iterations. In fact, if epsilon(0) is sufficiently small then z(1) solves the original minimax problem. The practical value of the proposed iteration depends on the availability of an efficient code for solving a regularized minimax problem of the form minimize P(x) = 1/2\\x\\(2)(2) + 1/2\\Ax - b\\(2)(infinity)/epsilon where epsilon is a given positive constant. It is shown that the dual of this problem has the form maximize D(y) = b(T)y - 1/2\\A(T)y\\(2)(2) - 1/2 epsilon\\y\\(2)(1), and if y solves the dual then x = A(T)y solves the primal. The simple structure of the dual enables us to apply a wide range of methods. In this paper we design and analyze a row relaxation method which is suitable for solving large sparse problems. Numerical experiments illustrate the feasibility of our ideas.
In this article we use techniques of proof mining to analyse a result, due to Yong=hong Yao and Muhammad Aslam Noor, concerning the strong convergence of a generalized proximal point algorithm which involves multiple ...
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In this article we use techniques of proof mining to analyse a result, due to Yong=hong Yao and Muhammad Aslam Noor, concerning the strong convergence of a generalized proximal point algorithm which involves multiple parameters. Yao and Noor's result ensures the strong convergence of the algorithm to the nearest projection point onto the set of zeros of the operator. Our quantitative analysis, guided by Fernando Ferreira and Paulo Oliva's bounded functional interpretation, provides a primitive recursive bound on the metastability for the convergence of the algorithm, in the sense of Terence Tao. Furthermore, we obtain quantitative information on the asymptotic regularity of the iteration. The results of this paper are made possible by an arithmetization of the lim sup.
In this paper, we consider a proximal point algorithm ( PPA) for solving monotone nonlinear complementarity problems (NCP). PPA generates a sequence by solving subproblems that are regularizations of the original prob...
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In this paper, we consider a proximal point algorithm ( PPA) for solving monotone nonlinear complementarity problems (NCP). PPA generates a sequence by solving subproblems that are regularizations of the original problem. It is known that PPA has global and superlinear convergence properties under appropriate criteria for approximate solutions of subproblems. However, it is not always easy to solve subproblems or to check those criteria. In this paper, we adopt the generalized Newton method proposed by De Luca, Facchinei, and Kanzow to solve subproblems and adopt some NCP functions to check the criteria. Then we show that the PPA converges globally provided that the solution set of the problem is nonempty. Moreover, without assuming the local uniqueness of the solution, we show that the rate of convergence is superlinear in a genuine sense, provided that the limit point satis es the strict complementarity condition.
In this paper, we consider the proximal point algorithm for the problem of finding zeros of any given maximal monotone operator in an infinite-dimensional Hilbert space. For the usual distance between the origin and t...
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In this paper, we consider the proximal point algorithm for the problem of finding zeros of any given maximal monotone operator in an infinite-dimensional Hilbert space. For the usual distance between the origin and the operator's value at each iterate, we put forth a new idea to achieve a new result on the speed at which the distance sequence tends to zero globally, provided that the problem's solution set is nonempty and the sequence of squares of the regularization parameters is nonsummable. We show that it is comparable to a classical result of Br,zis and Lions in general and becomes better whenever the proximal point algorithm does converge strongly. Furthermore, we also reveal its similarity to Guler's classical results in the context of convex minimization in the sense of strictly convex quadratic functions, and we discuss an application to an I mu-approximation solution of the problem above.
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