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Neural network training with optimal bounded ellipsoid algorithm

NEURAL COMPUTING & APPLICATIONS 2009年第6期18卷 623-631页

作者： de Jesus Rubio, Jose Yu, Wen Ferreyra, Andres UAM Azcapotzalco Dept Elect Area Instrumentac Mexico City DF Mexico Inst Politecn Nacl ESIME Azcapotzalco Secc Estudios Posgrad & Invest Mexico City DF Mexico IPN CINVESTAV Dept Automat Control Mexico City 07360 DF Mexico

Compared to normal learning algorithms, for example backpropagation, the optimal bounded ellipsoid (OBE) algorithm has some better properties, such as faster convergence, since it has a similar structure as Kalman filter. OBE has some advantages over Kalman filter training, the noise is not required to be Guassian. In this paper OBE algorithm is applied in training the weights of the feedforward neural network for nonlinear system identification. Both hidden layers and output layers can be updated. From a dynamic system point of view, such training can be useful for all neural network applications requiring real-time updating of the weights. Two simulations give the effectiveness of the suggested algorithm.

关键词： Neural networks Identification ellipsoid algorithm Stability

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MODIFICATIONS AND IMPLEMENTATION OF THE ellipsoid algorithm FOR LINEAR-PROGRAMMING

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MATHEMATICAL PROGRAMMING 1982年第1期23卷 1-19页

作者： GOLDFARB, D TODD, MJ CORNELL UNIV ITHACANY 14853

We give some modifications of the ellipsoid algorithm for linear programming and describe a numerically stable implementation. We are concerned with practical problems where user-supplied bounds can usually be provided. Our implementation allows constraint dropping and updates bounds on the optimal value, and should be able to terminate with an indication of infeasibility or with a provably good feasible solution in a moderate number of iterations.

关键词： ellipsoid algorithm Linear Programming Polynomial Boundedness Khachian's Method Linear Inequalities

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An Oblivious ellipsoid algorithm for Solving a System of (In)Feasible Linear Inequalities

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MATHEMATICS OF OPERATIONS RESEARCH 2024年第1期49卷 1-651, C2页

作者： Lamperski, Jourdain Freund, Robert M. Todd, Michael J. Univ Pittsburgh Ind Engn Pittsburgh PA 15213 USA MIT Sloan Sch Management Cambridge MA 02139 USA Cornell Univ Sch Operat Res & Informat Engn Ithaca NY 14853 USA

The ellipsoid algorithm is a fundamental algorithm for computing a solution to the system of m linear inequalities in n variables (P) : A(T)x = 0. This paper develops an oblivious ellipsoid algorithm (OEA) that either... 详细信息

The ellipsoid algorithm is a fundamental algorithm for computing a solution to the system of m linear inequalities in n variables (P) : A(T)x <= u when its set of solutions has positive volume. However, when (P) is infeasible, the ellipsoid algorithm has no mechanism for proving that (P) is infeasible. This is in contrast to the other two fundamental algorithms for tackling (P), namely, the simplex and interior-point methods, each of which can be easily implemented in a way that either produces a solution of (P) or proves that (P) is infeasible by producing a solution to the alternative system (Alt) : A lambda = 0, u(T)lambda < 0, lambda >= 0. This paper develops an oblivious ellipsoid algorithm (OEA) that either produces a solution of (P) or produces a solution of (Alt). Depending on the dimensions and other condition measures, the computational complexity of the basic OEA may be worse than, the same as, or better than that of the standard ellipsoid algorithm. We also present two modified versions of OEA, whose computational complexity is superior to that of OEA when n MUCH LESS-THAN m. This is achieved in the first modified version by proving infeasibility without producing a solution of (Alt), and in the second version by using more memory.

关键词： ellipsoid algorithm certificates condition measures computational complexity linear inequalities

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An objective-function ellipsoid-algorithm for convex quadratical programming

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Optimization 1986年第3期17卷 333-347页

作者： Recht, P. Universität Karlsruhe Institut für Statistik und Mathematische Wirtsehaftstheorie Germany

An ellipsoid algorithm for convex, quadratical programming is given, based on the use of the objective function itself for ellipsoidal iterations. An implemented version is presented and numerical results are discusse... 详细信息

关键词： ellipsoid algorithm Quadratical programming

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ellipsoid algorithmS IN MATHEMATICAL-PROGRAMMING

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HUMAN SYSTEMS MANAGEMENT 1980年第2期1卷 173-178页

作者： ZELENY, M European Institute for Advanced Studies in Management Brussels Belgium

Newly proposed ellipsoid algorithms provide considerable hope for advancing the relevance of mathematical programming applications. They are extremely suitalbe for multiobjective programming, goal programming and interactive programming methodologies. More intimate cooperation between the decision maker and a model of his problem is likely to be enhanced. In this short article the basic ideas of ellipsoid algorithms and views of their potential applicability are communicated.

关键词： Khachiyan Shor ellipsoid algorithm polynomial time exponential time simplex method satisfactory solutions

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USING 2 SUCCESSIVE SUBGRADIENTS IN THE ellipsoid METHOD FOR NONLINEAR-PROGRAMMING

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JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS 1994年第3期82卷 543-554页

作者： KIM, S KIM, D CHANG, KN Department of Management Science Korea Advanced Institute of Science and Technology Taejon Korea

A variant of the ellipsoid method for nonlinear programming is introduced to enhance the speed of convergence. This variant is based on a new simple scheme to reduce the ellipsoid volume by using two center cuts generated in two consecutive iterations of the ellipsoid method. Computational tests show a significant improvement in computational efficiency. The tests show that the improvement is more significant for larger-size problems.

关键词： NONLINEAR PROGRAMMING ellipsoid algorithm ALTERNATIVE CUTS ellipsoidS OF SMALLER VOLUME

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AN ellipsoid TRUST REGION BUNDLE METHOD FOR NONSMOOTH CONVEX MINIMIZATION

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SIAM JOURNAL ON CONTROL AND OPTIMIZATION 1989年第4期27卷 737-757页

作者： KIWIEL, KC

This paper presents a bundle method of descent for minimizing a convex (possibly nonsmooth) function f of several variables. At each iteration the algorithm finds a trial point by minimizing a polyhedral model of f subject to an ellipsoid trust region constraint. The quadratic matrix of the constraint, which is updated as in the ellipsoid method, is intended to serve as a generalized “Hessian” to account for “second-order” effects, thus enabling faster convergence. The interpretation of generalized Hessians is largely heuristic, since so far this notion has been made precise by J. L. Goffin only in the solution of linear inequalities. Global convergence of the method is established and numerical results are given.

关键词： 65K05 90C25 nonsmooth optimization nondifferentiable programming convex programming descent methods ellipsoid algorithm

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Equivalence of Convex Problem Geometry and Computational Complexity in the Separation Oracle Model

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MATHEMATICS OF OPERATIONS RESEARCH 2009年第4期34卷 869-879页

作者： Freund, Robert M. Vera, Jorge R. MIT Alfred P Sloan Sch Management Cambridge MA 02142 USA Pontificia Univ Catolica Chile Dept Ingn Ind & Sistemas Fac Ingn Santiago 7820436 Chile

Consider the supposedly simple problem of computing a point in a convex set that is conveyed by a separation oracle with no further information (e. g., no domain ball containing or intersecting the set, etc.). The authors' interest in this problem stems from fundamental issues involving the interplay of (i) the computational complexity of computing a point in the set, (ii) the geometry of the set, and (iii) the stability or conditioning of the set under perturbation. Under suitable definitions of these terms, the authors show herein that problem instances with favorable geometry have favorable computational complexity, validating conventional wisdom. The authors also show a converse of this implication by showing that there exist problem instances that require more computational effort to solve in certain families characterized by unfavorable geometry. This in turn leads, under certain assumptions, to a form of equivalence among computational complexity, geometry, and the conditioning of the set. The authors' measures of the geometry, relative to a given reference point, are based on the radius of a certain domain ball whose intersection with the set contains a certain inscribed ball. The geometry of the set is then measured by the radius of the domain ball, the radius of the inscribed ball, and the ratio between these two radii, the latter of which is called the aspect ratio. The aspect ratio arises in the analysis of many algorithms for convex problems, and its importance in convex algorithm analysis has been well-known for several decades. However, the presence in our bound of terms involving only the radius of the domain ball and only the radius of the inscribed ball are a bit counterintuitive;nevertheless, we show that the computational complexity must involve these terms in addition to the aspect ratio, even when the aspect ratio itself is small. This lower-bound complexity analysis relies on simple features of the separation oracle model;if we instead assume

关键词： convex optimization ellipsoid algorithm computational complexity

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General models in min-max continuous location: Theory and solution techniques

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JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS 1996年第1期89卷 39-63页

作者： Frenk, JBG Gromicho, J Zhang, S UNIV LISBON FAC CIENCIAS DEIO LISBON PORTUGAL

In this paper, a class of min-max continuous location problems is discussed. After giving a complete characterization of the stationary points, we propose a simple central and deep-cut ellipsoid algorithm to solve these problems for the quasiconvex case. Moreover, an elementary convergence proof of this algorithm and some computational results are presented.

关键词： location theory min-max programming ellipsoid algorithm quasiconvexity

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On Vaidya's volumetric cutting plane method for convex programming

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MATHEMATICS OF OPERATIONS RESEARCH 1997年第1期22卷 63-89页

作者： Anstreicher, KM Department of Management Sciences University of Iowa Iowa City IA 52242 United States

We describe a simplified and strengthened version of Vaidya's volumetric cutting plane method for finding a point in a convex set C subset of R(n). At each step the algorithm has a system of linear inequality constraints which defines a polyhedron P superset of C, and an interior point x is an element of P. The algorithm then either drops one constraint, or calls an oracle to check if x is an element of C, and, if not, obtain a new constraint that separates x from C. Following the addition or deletion of a constraint, the algorithm takes a small number of Newton steps for the volumetric barrier V(.). Progress of the algorithm is measured in terms of changes in V(.). The algorithm is terminated when either it is discovered that x is an element of C, or V(.) becomes large enough to demonstrate that the volume of C must be below some prescribed amount. The complexity of the algorithm compares favorably with that of the ellipsoid method, especially in terms of the number of calls to the separation oracle. Compared to Vaidya's original analysis, we decrease the total number of Newton steps required for termination by a factor of about 1.3 million, white at the same time decreasing the maximum number of constraints used to define P from 10(7)n to 200n.

关键词： linear programming convex programming interior point algorithm volumetric barrier cutting plane method ellipsoid algorithm

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