Optimization and Operations Research: Proceedings of a Conference Held at Oberwolfach, July 27-August 2, 1975

ISBN-13: 9783540076162 / Angielski / Miękka / 1976 / 318 str.

W. Oettli; K. Ritter

Opis

The variable metric algorithm is widely recognised as one of the most efficient ways of solving the following problem: - Locate x* a local minimum point n ( 1) of f(x) x E R Considerable attention has been given to the study of the convergence prop- ties of this algorithm especially for the case where analytic expressions are avai- ble for the derivatives g. = af/ax. i 1 n (2) In particular we shall mention the results of Wolfe (1969) and Powell (1972), (1975). Wolfe established general conditions under which a descent algorithm will converge to a stationary point and Powell showed that two particular very efficient algorithms that cannot be shown to satisfy , olfe's conditions do in fact converge to the minimum of convex functions under certain conditions. These results will be st- ed more completely in Section 2. In most practical problems analytic expressions for the gradient vector g (Equ. 2) are not available and numerical derivatives are subject to truncation error. In Section 3 we shall consider the effects of these errors on Wolfe's convergent prop- ties and will discuss possible modifications of the algorithms to make them reliable in these circumstances. The effects of rounding error are considered in Section 4, whilst in Section 5 these thoughts are extended to include the case of on-line fu- tion minimisation where each function evaluation is subject to random noise."

Spis treści

Reduction and Decomposition of Modulo Optimization Problems.- Optimization of Elastic Structures by Mathematical Programming Techniques.- On Closed Sets Having a Least Element.- On the Convergence of the Variable Metric Method with Numerical Derivatives and the Effect of Noise in the Function Evaluation.- Parallel Path Strategy. Theory and Numerical Illustrations.- On Minimization under Linear Equality Constraints.- On Constrained Shortest-Route Problems.- Bang-Bang Solution of a Control Problem for the Heat Equation.- Generalized Stirling-Newton Methods.- On a Method for Computing Pseudoinverses.- Charakterisierung lokaler Pareto-Optima.- On the Exact Evaluation of Finite Activity Networks with Stochastic Durations of Activities.- Approximation of a Parabolic Boundary Control Problem by the Line Method.- Regularisation of Optimization Problems and Operator Equations.- Some Extensions of Linearly Constrained Nonlinear Programming.- Boundary Control of the Higher-Dimensional Wave Equation.- Nondifferentiable Optimisation. Subgradient and ? — Subgradient Methods.- Approximations to Stochastic Optimization Problems.- Models in Resource Allocation — Trees of Queues.- Dual Methods in Convex Control Problems.- A Subgradient Algorithm for Solving K-convex Inequalities.- Computation of Bang-Bang-Controls and Lower Bounds for a Parabolic Boundary-Value Control Problem.- Numerical Solution of Systems of Nonlinear Inequalities.- Lower Bounds and Inclusion Balls for the Solution of Locally Uniformly Convex Optimization Problems.- Über Vektormaximierung und Analyse der Gewichtung von Subzielen.- Preference Optimality.- Decomposition Procedures for Convex Programs.