Abstract
We discuss the average case complexity of global optimization problems. By the average complexity, we roughly mean the amount of work needed to solve the problem with the expected error not exceeding a preassigned error demand. The expectation is taken with respect to a probability measure on a classF of objective functions.
Since the distribution of the maximum, max x f (x), is known only for a few nontrivial probability measures, the average case complexity of optimization is still unknown. Although only preliminary results are available, they indicate that on the average, optimization is not as hard as in the worst case setting. In particular, there are instances, where global optimization is intractable in the worst case, whereas it is tractable on the average.
We stress, that the power of the average case approach is proven by exhibiting upper bounds on the average complexity, since the actual complexity is not known even for relatively simple instances of global optimization problems. Thus, we do not know how much easier global optimization becomes when the average case approach is utilized.
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References
R.J. Adler,The Geometry of Random Fields (Wiley, New York, 1981).
J.E. Dennis and R.B. Schnabel,Numerical Methods for Unconstrained Optimization and Nonlinear Equations (Prentice-Hall, Englewood Cliffs, NJ, 1983).
M.R. Leadbetter, G. Lindgren and H. Rootzén,Extremes and Related Properties of Random Sesquences and Processes (Springer, New York, 1983).
D. Lee and G.W. Wasilkowski, “Approximation of linear functionals on a Banach space with a Gaussian measure,”Journal of Complexity 2 (1986) 12–43.
J. Mockus,Bayesian Approach to Glogal Optimization (Kluwer Academic Publishers, Dordrecht, 1989).
A.S. Nemirovsky and D.B. Yudin,Problem Complexity and Method Efficiency in Optimization (Wiley, New York, 1983).
E. Novak,Deterministic and Stochastic Error Bounds in Numerical Analysis. Lecture Notes in Mathematics No. 1349 (Springer, Berlin, 1988).
K.R. Parthasarathy,Probability Measures on Metric Spaces (Academic Press, New York, 1967).
K. Ritter, “Approximation and optimization on the Wiener space,”Journal of Complexity 6 (1990) 337–364.
J. Sacks and D. Ylvisaker, “Statistical design and integral approximation,” in:Proceeding of the 12th Biennial Seminary of the Canadian Mathematical Congress 1970, pp. 115–136.
P. Speckman, “L p approximation of autoregressive Gaussian processes,” Technical Report, Department of Statistics, University of Oregon (Eugene, OR, 1979).
A.G. Sukharev,Minimax Algorithms in Problems of Numerical Analysis (Nauka, Moscow, 1989).
J.F. Traub and H. Woźniakowski, “Complexity of linear programming,”Operations Research Letters 1 (1982) 59–62.
J.F. Traub, G.W. Wasilkowski and H. Woźniakowski,Information-based Complexity (Academic Press, New York, 1988).
S.A. Vavasis, “Black-box complexity of local minimization,” Technical Report, Department of Computer Science, Cornell University (Ithaca, NY, 1990).
G. Wahba,Spline Models for Observational Data. SIAM-NSF Regional Conference Series in Applied Mathematics 59 (1990).
G.W. Wasilkowski, “Information of varying cardinality,”Journal of Complexity 2 (1985) 204–228.
G.W. Wasilkowski and F. Gao, “On the power of adaptive information for functions with singularities,”Mathematics of Computation 58 (1992) 285–304.
H. Woźnakowski, “Average case complexity of multivariate integration,”Bulletin of the AMS 24 (1991) 185–194.
D. Ylvisaker, “Designs on random fields,” in: J.N. Srivastava, ed.,A Survey of Statistical Design and Linear Models (North-Holland, New York, 1975) pp. 593–607.
A.G. Zhilinskas, “Single-step Bayesian search method for an extremum of functions of a single variable,”Cybernetics 11 (1975) 160–166.
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Research partially supported by the National Science Foundation under Grant CCR-89-0537.
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Wasilkowski, G.W. On average complexity of global optimization problems. Mathematical Programming 57, 313–324 (1992). https://doi.org/10.1007/BF01581086
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DOI: https://doi.org/10.1007/BF01581086