Location of the Maximum on Unimodal Surfaces

ABSTRACT

Pinning down the maximum of a function in one or more variable is a basic computing problem. Without further a priori information regarding the nature of the function, of course, the problem is not feasible for computation. Thus if the function is permitted to oscillate infinitely often then no number of evaluations can give information regarding its maximum. J. Kiefer, in his paper, treats the one-dimensional problem and shows that the correct a priori assumption regarding the function is that of “unimodality.” He then gives the complete and exact optimal procedure within this framework. In this paper is given what the author believes is the correct a priori background for functions of several variables. (This is analogous to unimodality in one dimension.) However, there is not obtained the exactness of Kiefer's result, but rather the determination of the correct procedures to within “order of magnitude.”

REFERENCES

• 1: KIEFER, J. Sequential minimax search for a maximum. Proc. Amer. Math. Soc. 4 (1953), 502-506.
• 2: GOODMAN, R. Machine Program 600-239. Sylvania Electronics, Needham, Mass.
• 3: HAMMER, P.C. The centroid of a convexbody. Proc. Amer. Math. Soc. 2 (1951), 522-525.

INDEX TERMS

Primary Classification:
  G. Mathematics of Computing
  G.2 DISCRETE MATHEMATICS

Additional Classification:
  G. Mathematics of Computing
  G.1 NUMERICAL ANALYSIS
      G.1.2 Approximation
          Subjects: Minimax approximation and algorithms

General Terms:
Theory, Verification