Stochastic global optimization using tangent minorants for Lipschitz functions

This paper deals with global stochastic optimization where the decision variable belongs to a compact subset X of R. The objective function is the mathematical expectation of a partial bivariate Lipschitz function fx,Θ depending on a decision variable x and a random variableΘ, whose probability distribution depends on x. In the first part of the present paper, we propose a branch and bound algorithm based on tangent minorants that provides a global minimum. In the second part, we consider the case where the function f is discontinuous. We show the manner we correct that discontinuity without modifying the global minimum of Efx,Θ. We also illustrate how to extend this framework to multidimensional stochastic optimization problems by using the Alienor method. Then we validate the proposed method by applying it to some test functions and compare it to known algorithms.