Above are papers related to the following paper: [Paper_ID] Paper #258259. 	[Title] ERROR STABILITY PROPERTIES OF GENERALIZED GRADIENT-TYPE ALGORITHMS [Abstract] We present a unified framework for convergence analysis of the generalized subgradient-type algorithms in the presence of perturbations. One of the principal novel features of our analysis is that perturbations need not tend to zero in the limit. It is established that the iterates of the algorithms are attracted, in a certain sense, to an "-stationary set of the problem, where " depends on the magnitude of perturbations. Characterization of those attraction sets is given in the general (nonsmooth and nonconvex) case. The results are further strengthened for convex, weakly sharp and strongly convex problems. Our analysis extends and unifies previously known results on convergence and stability properties of gradient and subgradient methods, including their incremental, parallel and "heavy ball" modifications. fl The first author is supported in part by CNPq grant 300734/95-6. Research of the second author was supported in part by the International Science Foundation Grant NBY000, the International Science Foundation and Russian Goverment Grant NBY300 and the Russian Foundation for Fundamental Research Grant N 95-01-01448. y Instituto de Matematica Pura e Aplicada, Estrada Dona Castorina 110, Jardim Bot^anico, Rio de Janeiro, RJ, CEP 22460-320, Brazil. Email : solodov@impa.br. z Operations Research Department, Faculty of Computational Mathematics and Cybernetics, Moscow State University, Moscow, Russia, 119899. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #168958. 	[Title] New Inexact Parallel Variable Distribution Algorithms  Editor: [Abstract] We consider the recently proposed parallel variable distribution (PVD) algorithm of Ferris and Mangasarian [4] for solving optimization problems in which the variables are distributed among p processors. Each processor has the primary responsibility for updating its block of variables while allowing the remaining "secondary" variables to change in a restricted fashion along some easily computable directions. We propose useful generalizations that consist, for the general unconstrained case, of replacing exact global solution of the subproblems by a certain natural sufficient descent condition, and, for the convex case, of inexact subproblem solution in the PVD algorithm. These modifications are the key features of the algorithm that has not been analyzed before. The proposed modified algorithms are more practical and make it easier to achieve good load balancing among the parallel processors. We present a general framework for the analysis of this class of algorithms and derive some new and improved linear convergence results for problems with weak sharp minima of order 2 and strongly convex problems. We also show that nonmonotone synchronization schemes are admissible, which further improves flexibility of PVD approach. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #35335. 	[Title] Parallel Gradient Distribution in Unconstrained Optimization [Abstract] A parallel version is proposed for a fundamental theorem of serial unconstrained optimization. The parallel theorem allows each of k parallel processors to use simultaneously a different algorithm, such as a descent, Newton, quasi-Newton or a conjugate gradient algorithm. Each processor can perform one or many steps of a serial algorithm on a portion of the gradient of the objective function assigned to it, independently of the other processors. Eventually a synchronization step is performed which, for differentiable convex functions, consists of taking a strong convex combination of the k points found by the k processors. For nonconvex, as well as convex, differentiable functions, the best point found by the k processors is taken, or any better point. The fundamental result that we establish is that any accumulation point of the parallel algorithm is stationary for the nonconvex case, and is a global solution for the convex case. Computational testing on the Thinking Machines CM-5 multiprocessor indicate a speedup of the order of the number of processors employed. Please summarise all the given information.

Paper #258259 has the title `ERROR STABILITY PROPERTIES OF GENERALIZED GRADIENT-TYPE ALGORITHMS  
` and the abstract `We present a unified framework for convergence analysis of the generalized subgradient-type algorithms in the presence of perturbations. One of the principal novel features of our analysis is that perturbations need not tend to zero in the limit. It is established that the iterates of the algorithms are attracted, in a certain sense, to an "-stationary set of the problem, where " depends on the magnitude of perturbations. Characterization of those attraction sets is given in the general (nonsmooth and nonconvex) case. The results are further strengthened for convex, weakly sharp and strongly convex problems. Our analysis extends and unifies previously known results on convergence and stability properties of gradient and subgradient methods, including their incremental, parallel and "heavy ball" modifications. fl The first author is supported in part by CNPq grant 300734/95-6. Research of the second author was supported in part by the International Science Foundation Grant NBY000, the International Science Foundation and Russian Goverment Grant NBY300 and the Russian Foundation for Fundamental Research Grant N 95-01-01448. y Instituto de Matematica Pura e Aplicada, Estrada Dona Castorina 110, Jardim Bot^anico, Rio de Janeiro, RJ, CEP 22460-320, Brazil. Email : solodov@impa.br. z Operations Research Department, Faculty of Computational Mathematics and Cybernetics, Moscow State University, Moscow, Russia, 119899. 
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Question: Which category should Paper #258259 be classified as ?
You can select one from ['Neural_Networks', 'Case_Based', 'Theory', 'Reinforcement_Learning', 'Probabilistic_Methods', 'Rule_Learning', 'Genetic_Algorithms'].


Paper #258259 has the title `ERROR STABILITY PROPERTIES OF GENERALIZED GRADIENT-TYPE ALGORITHMS  
` and the abstract `We present a unified framework for convergence analysis of the generalized subgradient-type algorithms in the presence of perturbations. One of the principal novel features of our analysis is that perturbations need not tend to zero in the limit. It is established that the iterates of the algorithms are attracted, in a certain sense, to an "-stationary set of the problem, where " depends on the magnitude of perturbations. Characterization of those attraction sets is given in the general (nonsmooth and nonconvex) case. The results are further strengthened for convex, weakly sharp and strongly convex problems. Our analysis extends and unifies previously known results on convergence and stability properties of gradient and subgradient methods, including their incremental, parallel and "heavy ball" modifications. fl The first author is supported in part by CNPq grant 300734/95-6. Research of the second author was supported in part by the International Science Foundation Grant NBY000, the International Science Foundation and Russian Goverment Grant NBY300 and the Russian Foundation for Fundamental Research Grant N 95-01-01448. y Instituto de Matematica Pura e Aplicada, Estrada Dona Castorina 110, Jardim Bot^anico, Rio de Janeiro, RJ, CEP 22460-320, Brazil. Email : solodov@impa.br. z Operations Research Department, Faculty of Computational Mathematics and Cybernetics, Moscow State University, Moscow, Russia, 119899. 
`.
Question: Which category should Paper #258259 be classified as ?
You can select one from ['Neural_Networks', 'Case_Based', 'Theory', 'Reinforcement_Learning', 'Probabilistic_Methods', 'Rule_Learning', 'Genetic_Algorithms'].

This paper should be classified into: Neural_Networks