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.

Inhalt

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.