## Adaptive Approximations and Exact Penalization for
the Solution of Generalized Semi-Infinite Min-Max Problems

Johannes O. Royset^{1} and Armen Der Kiureghian^{2}

(Professor Elijah Polak)

Norwegian Research Council, (NSF) ECS-9900985, and Space Sciences Laboratory-Lockheed Martin Mini-grant Program

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 ceasing to increase, but the smoothing and
discretization parameters driven to infinity. A numerical example
demonstrates the viability of the algorithm.

^{1}Graduate Student (non-EECS)

^{2}Outside Adviser (non-EECS)

Send mail to the author : (joroyset@ce.berkeley.edu)

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