On the Use of Augmented Lagrangians in the Solution of Generalized Semi-Infinite Min-Max Problems
Computational Optimization and Applications
An aggregate deformation homotopy method for min-max-min problems with max-min constraints
Computational Optimization and Applications
On sample size control in sample average approximations for solving smooth stochastic programs
Computational Optimization and Applications
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We develop an implementable algorithm for the solution of a class of generalized semi-infinite min-max problems. To this end, first we use exact penalties to convert a generalized semi-infinite min-max problem into a finite family of semi-infinite min-max-min problems. Second, the inner min-function is smoothed and the semi-infinite max part is approximated, using discretization, to obtain a three-parameter family of finite min-max problems. Under a calmness assumption, we show that when the penalty is sufficiently large the semi-infinite min-max-min problems have the same solutions as the original problem, and that when the smoothing and discretization parameters go to infinity the solutions of the finite min-max problems converge to solutions of the original problem, provided the penalty parameter is sufficiently large.Our algorithm combines tests for adjusting the penalty, the smoothing, and the discretization parameters and makes use of a min-max algorithm as a subroutine. In effect, the min-max algorithm is applied to a sequence of gradually better-approximating min-max problems, with the penalty parameter eventually stopping to increase, but with the smoothing and discretization parameters driven to infinity. A numerical example demonstrates the viability of the algorithm.