Abstract
Stochastic algorithms for optimization problems, where function evaluations are done by Monte Carlo simulations, are presented. At each iteratex i, they draw a predetermined numbern(i) of sample points from an underlying probability space; based on these sample points, they compute a feasible-descent direction, an Armijo stepsize, and the next iteratex i+1. For an appropriate optimality function σ, corresponding to an optimality condition, it is shown that, ifn(i) → ∞, then σ(x i) → 0, whereJ is a set of integers whose upper density is zero. First, convergence is shown for a general algorithm prototype: then, a steepest-descent algorithm for unconstrained problems and a feasible-direction algorithm for problems with inequality constraints are developed. A numerical example is supplied.
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Communicated by E. Polak
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Wardi, Y. Stochastic algorithms with armijo stepsizes for minimization of functions. J Optim Theory Appl 64, 399–417 (1990). https://doi.org/10.1007/BF00939456
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DOI: https://doi.org/10.1007/BF00939456