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OPTIMAL-CONTROL COMPUTATION FOR PARABOLIC-SYSTEMS WITH BOUNDARY-CONDITIONS INVOLVING TIME DELAYS

JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS 1987年第3期53卷 475-507页

作者： WONG, KH 1.Department of Applied Mathematics University of the Witnwatersrand Johannesburg South Africa

In this paper, we consider a class of optimal control problems involving a second-order, linear parabolic partial differential equation with Neumann boundary conditions. The time-delayed arguments are assumed to appear in the boundary conditions. A necessary and sufficient condition for optimality is derived, and an iterative method for solving this optimal control problem is proposed. The convergence property of this iterative method is also investigated.

关键词： Parabolic differential equations second boundary-value problems time-delayed arguments distributed optimal control problems necessary and sufficient conditions existence theory algorithms convergence of algorithms Galerkin's scheme lumped-parameter systems

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Progressive regularization of variational inequalities and decomposition algorithms

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JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS 1998年第2期97卷 407-433页

作者： El Farouq, N Cohen, G Univ Clermont Ferrand 2 IUT Montlucon F-63006 Clermont Ferrand France INRIA Rocquencourt France Ecole Mines Ctr Automat & Syst Paris France

For nonsymmetric operators involved in variational inequalities, the strong monotonicity of their possibly multivalued inverse operators (referred to as the Dunn property) appears to be the weakest requirement to ensure convergence of most iterative algorithms of resolution proposed in the literature. This implies the Lipschitz property, and both properties are equivalent for symmetric operators. For Lipschitz operators, the Dunn property is weaker than strong monotonicity, but is stronger than simple monotonicity. Moreover, it is always enforced by the Moreau-Yosida regularization and it is satisfied by the resolvents of monotone operators. Therefore, algorithms should always be applied to this regularized version or they should use resolvents: in a sense, this is what is achieved in proximal and splitting methods among others. However, the operation of regularization itself or the computation of resolvents may be as complex as solving the original variational inequality. In this paper, the concept of progressive regularization is introduced and a convergent algorithm is proposed for solving variational inequalities involving nonsymmetric monotone operators. Essentially, the idea is to use the auxiliary problem principle to perform the regularization operation and, at the same time, to solve the variational inequality in its approximately regularized version;thus, two iteration processes are performed simultaneously, instead of being nested in each other, yielding a global explicit iterative scheme. Parallel and sequential versions of the algorithm are presented. A simple numerical example demonstrates the behavior of these two versions for the case where previously proposed algorithms fail to converge unless regularization or computation of a resolvent is performed at each iteration. Since the auxiliary problem principle is a general framework to obtain decomposition methods, the results presented here extend the class of problems for which decomposition methods c

关键词： variational equalities optimization problems monotonicity regularization Dunn property firm contraction convergence of algorithms

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An algebraic, analytic, and algorithmic investigation on the capacity and capacity-achieving input probability distributions of finite-input-finite-output discrete memoryless channels (vol 54, pg 1003, 2008)

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IEEE TRANSACTIONS ON INFORMATION THEORY 2008年第9期54卷 4395-4395页

作者： Liang, Xue-Bin Louisiana State Univ Dept Elect & Comp Engn Baton Rouge LA 70803 USA

In this paper, we investigate the capacity and capacity-achieving input probability distributions (IPDs) of finite-input--finite-output discrete memoryless channels (DMCs). In the general respect, we establish a novel and simple characterization for the capacity-achieving IPDs of a DMC, which is equivalent to the conventional Kuhn-Tucker conditions. We then prove a conjecture of Majani and Rumsey, which claims that every probability component of each capacity-achieving IPD of a DMC with positive capacity is less than 1 - e -1,where c 2.71828182... is the base of natural logarithms. It remains an open problem whether there exists an explicit closed-form solution for the capacity and capacity-achieving IPDs of a general finite-input-finite-output DMC, except for the two-input-two-output DMC. In the algebraic respect, we demonstrate that there does not, in general, exist an algebraic solution for the capacity-achieving IPDs of an m-input-n-output DMC for any m ≥ 2 and any n ≥ 3. In the analytic respect, how- ever, we can obtain an explicit closed-form analytic solution, represented as an infinite series, for the capacity-achieving IPD of a two-input-three-output DMC. We also provide a formula for the average capacity of weakly symmetric DMCs and show that the average capacity in nats per channel use of the n-input-n-output weakly symmetric DMCs increases for n ≥ 2 but has a finite limit of 1 - γ as n → ∞, where γ = 0.57721566... is Euler's constant. In the algorithmic respect, the convergence of the Arimoto--Blahut algorithm is proved in a direct and elementary way. A new and simple iterative algorithm for calculating a capacity-achieving IPD is then proposed, which is provably convergent for all DMCs with positive transition probabilities. Finally, the characterization and determination of the set of all capacity-achieving IPDs of a DMC are addressed. [ABSTRACT FROM AUTHOR]

关键词： Algebraic solutions Arimoto-Blahut algorithm average capacity capacity capacity-achieving probability distributions convergence of algorithms convex hull discrete memoryless channels (DMCs) extreme points infinite-series solutions Kuhn-Tucker conditions linear-algebraic method polyhedral sets projection equations trinomial equations

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Convergent algorithm based on progressive regularization for solving pseudomonotone variational inequalities

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JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS 2004年第3期120卷 455-485页

作者： El Farouq, N Univ Clermont Ferrand Toulouse France CNRS LAAS F-31077 Toulouse France

In this paper, we extend the Moreau-Yosida regularization of monotone variational inequalities to the case of weakly monotone and pseudomonotone operators. With these properties, the regularized operator satisfies the pseudo-Dunn property with respect to any solution of the variational inequality problem. As a consequence, the regularized version of the auxiliary problem algorithm converges. In this case, when the operator involved in the variational inequality problem is Lipschitz continuous (a property stronger than weak monotonicity) and pseudomonotone, we prove the convergence of the progressive regularization algorithm introduced in Refs. 1, 2.

关键词： variational inequalities generalized monotonicity pseudomonotonicity regularization convergence of algorithms decomposition

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Minimizing and stationary sequences of constrained optimization problems

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SIAM JOURNAL ON CONTROL AND OPTIMIZATION 1998年第6期36卷 1908-1936页

作者： Chou, CC Ng, KF Pang, JS Univ Perpignan Dept Math F-66025 Perpignan France Chinese Univ Hong Kong Dept Math Shatin New Territory Hong Kong Johns Hopkins Univ Dept Math Sci Baltimore MD 21218 USA

We study some fundamental asymptotic properties associated with the well-posedness of constrained optimization problems. Emphasis is placed on the relation between minimizing and stationary sequences and their characterizations in terms of a set of asymptotic Karush-Kuhn-Tucker (KKT) optimality conditions. Unlike the subdifferential approach used by Auslender, Cominetti, and Crouzeix, we use a residual function approach that is closely tied to the theory of error bounds;this approach handles constraints explicitly and allows the effective treatment of infeasible, stationary sequences. The asymptotic KKT conditions provide an asymptotic optimality certificate for inequality constrained programs. Specializations of the results to convex quadratically constrained convex quadratic spline minimization problems and convex programs with Holderian minima are discussed. An application of the results to the convergence of a family of Newton-type iterative descent methods involving singular quasi-Newton matrices for solving a constrained minimization problem is also presented.

关键词： well-posed optimization problems minimizing sequences stationary sequences approximate optimality conditions convex analysis residual functions error bounds weak sharp minima quadratic splines convergence of algorithms

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Error bounds in mathematical programming

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MATHEMATICAL PROGRAMMING 1997年第1-3期79卷 299-332页

作者： Pang, JS Department of Mathematical Sciences The Johns Hopkins University Baltimore USA

Originated from the practical implementation and numerical considerations of iterative methods for solving mathematical programs, the study of error bounds has grown and proliferated in many interesting areas within mathematical programming, This paper gives a comprehensive, state-of-the-art survey of the extensive theory and rich applications of error bounds for inequality and optimization systems and solution sets of equilibrium problems. (C) 1997 The Mathematical Programming Society, Inc, Published by Elsevier Science B.V.

关键词： error bounds inequality systems complementarity problems variational inequalities metric regularity penalty function convergence of algorithms

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DECOMPOSITION COORDINATION algorithms IN STOCHASTIC OPTIMIZATION

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SIAM JOURNAL ON CONTROL AND OPTIMIZATION 1990年第6期28卷 1372-1403页

作者： CULIOLI, JC COHEN, G INST NATL RECH INFORMAT & AUTOMAT F-78153 LE CHESNAYFRANCE

This paper considers an extension to the situation of stochastic programming of the Auxiliary Problem Principle formerly introduced in a deterministic setting to serve as a general framework for decomposition/coordination optimization algorithms. The idea is based upon that of the stochastic gradient, that is, independent noise realizations are considered successively along the iterations. As a consequence, deterministic subproblems are solved at each iteration whereas iterations fulfill the two tasks of coordination and stochastic approximation at the same time. Coupling cost function (expectation of some performance index) and deterministic coupling constraints are considered. Price (dual) decomposition (encompassing extensions of the Uzawa and Arrow–Hurwicz algorithms to this stochastic case) are studied as well as resource allocation (primal decomposition).

关键词： 49D27 49D29 62L20 65K10 stochastic optimization stochastic gradient decomposition coordination dual methods price decomposition resource allocation convergence of algorithms

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OPTIMIZATION WITH AN AUXILIARY CONSTRAINT AND DECOMPOSITION

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SIAM JOURNAL ON CONTROL AND OPTIMIZATION 1990年第1期28卷 137-157页

作者： COHEN, G MIARA, B INST NATL RECH INFORMAT & AUTOMAT F-78150 LE CHESNAYFRANCE ECOLE SUPER INGN ELECTROTECH & ELECTR F-93162 NOISY LE GRANDFRANCE

In the context of decomposition/coordination of a linear quadratic optimal control problem, Takahara’s algorithm was an earlier version of the so-called Interaction Prediction Principle that can be examined in the more general framework of infinite-dimensional constrained optimization problems. This principle is both a decomposition principle and a coordination strategy based on a fixed point scheme. It has been later revisited in the general theory of the Auxiliary Problem Principle and the convergence of corresponding iterative algorithms has been analyzed. In this paper, we keep the same decomposition principle but we propose an alternative coordination strategy. The improvement brought by this new strategy is proved theoretically and illustrated by a numerical example. All of this is based on some manipulation of constrained optimization problems that we call the Auxiliary Constraint Principle.

关键词： 49D27 65K10 93A15 optimization algorithms decomposition coordination interaction prediction principle convergence of algorithms

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The largest step path following algorithm for monotone linear complementarity problems

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MATHEMATICAL PROGRAMMING 1997年第2期76卷 309-332页

作者： Gonzaga, CC Department of Mathematics Federal University of Santa Catarina Florianopolis SC Brasil

Path-following algorithms take at each iteration a Newton step for approaching a point on the central path, in such a way that all the iterates remain in a given neighborhood of that path. This paper studies the case in which each iteration uses a pure Newton step with the largest possible reduction in complementarity measure (duality gap). This algorithm is known to converge superlinearly in objective values. We show that with the addition of a computationally trivial safeguard it achieves Q-quadratic convergence, and show that this behaviour cannot be proved by usual techniques for the original method.

关键词： linear complementarity problem primal-dual interior-point algorithm convergence of algorithms

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Getchell's method for conversion of Cartesian-geocentric to geodetic coordinates - Its properties and Newtonian alternative

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JOURNAL OF APPLIED GEODESY 2021年第1期15卷 47-60页

作者： Kadaj, Roman Rzeszow Univ Technol Dept Geodesy & Geotech 12 Powstancow Warszawy St PL-35959 Rzeszow Poland

The paper concentrates on the iterative Getchell's method (formulated in 1972) and its alternative Newtonian implementation for conversion of Cartesian geocentric coordinates into geodetic coordinates. The same basic equation formulated in the Getchell's method is used in both cases. The equation has a stable form in the whole range of argument (latitude) variation <-pi/2, pi/2>. The original Getchell's method (somehow "forgotten") has a simple geometric interpretation and its applications turn out to be particularly effective. Many studies on iterative algorithms usually omit theoretical proofs of convergence replacing them with conclusions based on numerical examples. This paper presents theoretical proofs of algorithms convergence both for the Getchell's method and the Newton procedure. The convergence parameter and numerical error of results were estimated in each case. Numerical tests were carried out for a set of points distributed on the Earth's space, also for extreme h values. For typical practical applications of the Getchell's method, sufficiently accurate results are obtained after 1-3 iterations, while in the Newton procedure already after one iteration, assuming the same numerical error and initial conditions. The accuracy of the geodetic coordinates determinations meets all practical requirements with some margin. For example an absolute numerical error for latitude is approx. 0.4 - 10(-13) [rad] i. e. about 0.00026 mm in the length of the meridian arc. The proposed methods were compared with other methods (algorithms), including in terms of stability and non-singularity in the entire usable space of the Earth, but excluding the near geocenter, which has no practical significance. Both the modification of the Getchell method and its Newtonian alternative are very good determined in this area (in the Earth's poles, the final solution is directly the starting value of iterative algorithms). The discussed algorithms were implemented in the form of proced

关键词： geocentric coordinates geodetic coordinates conversion iterative formula convergence of algorithms

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