Probabilistic version of the method of feasible directions. (Q1855739)

This paper is a sequel of the results obtained by one of the authors [see \textit{J. Korychki} and \textit{M. Kostreva}, J. Optimization Theory 92, 311--330 (1994; Zbl 0886.90128) and 91, 389--418 (1996; Zbl 0883.90101)]. It is devoted to the discussion of random procedure that implements the solution of the non linear programming inequality-constrained problem \(\operatorname{Argmin} \{f_{0}(x)\mid f_{j}(x) \leq 0, j=1,2,\dots, m\}\) where the \(f_{j}\)'s are smooth and concave functions. Two lemmata establish that the solutions of the direction finding subproblems are unique. Its dual is used for performing the numerical calculations. A description of a line search problem is developed and its closureness is established. The global convergence of the proposed algorithm and the convergence of the random multidirectional algorithm are proved. An example illustrates graphically the behavior of the method. A set of the problems proposed by \textit{W. Hock} and \textit{K. Schittkowski} [see Test examples of nonlinear programming codes, Lecture Notes in Economics and Math. Systems, Vol. 187, Springer Verlag (1981; Zbl 0452.90038)], is used for testing the behavior of their proposal. The numerical results suggest that the procedure performs similarly to its deterministic counterpart.