W. Oettli – författare

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.

This volume constitutes the proceedings of the workshop "Optimierung und Operations Research", held at the Elly Holterhoff Backing Stift (Bad Honnef) of the University of Bonn, October 2-8, 1977. This conference was devoted to recent advances in the field of mathe- matical programming, optimization techniques, and operations research. It was attended by about 50 invited participants. Furthermore many scholars in these areas showed a great interest in this workshop, despite several other conferences and activities on similar topics in the year 1977. The organizers regret that considerations of avail- able space for conference activities limited the number of participants. This widespread interest, the high quality of the lectures presented and the active and stimulating discussions at the conference mani- fested the breadth of the activity ongoing in the field covered by this workshop and the necessity that this field be cultivated to a greater extent by the scientific community.The workshop was organized by the Institute of Operations Research (Sonderforschungsbereich 21), University of Bonn and was generously sponsored by the Gesellschaft der Freunde und Forderer der Rheinischen Friedrich-Wilhelms-Universitat and by IBM Germany. Only through this invaluable support was this workshop possible; for this the editors wish to express their sincere thanks and appreciation. Bonn, December 1977 R. Henn B. Korte W. Oettli III TABLE OF CONTENTS A. Bachem The theorem of Minkowski for polyhedral monoids and aggregated linear diophantine systems * * * . * * * * * * * S. Baum and L.E. Trotter, Jr.

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.