Approximation to Optimization Problems: An Elementary Review
@article{Kall1986ApproximationTO,
title={Approximation to Optimization Problems: An Elementary Review},
author={Peter Kall},
journal={Math. Oper. Res.},
year={1986},
volume={11},
pages={9-18}
}During the last two decades the concept of epi-convergence was introduced and then was used in various investigations in optimization and related areas. The aim of this review is to show in an elementary way how closely the arguments in the epi-convergence approach are related to those of the classical theory of convergence of functions.
73 Citations
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On consistency of bounding operations in deterministic global optimization
- Mathematics
- 1989
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- 2004
Upper and lower bounds for the minimum VaR are developed and it is shown how the combined bounding procedures can be used to compute the latter value to global optimality.
References
SHOWING 1-10 OF 12 REFERENCES
Designing approximation schemes for stochastic optimization problems, in particular for stochastic programs with recourse
- Computer Science
- 1986
Various approximation schemes for stochastic optimization problems involving either approximates of the probability measures and/or approximates of the objective functional, are investigated. We…
On the convergence of sequences of convex sets in finite dimensions
- Mathematics
- 1979
Four types of convergence for sequences of convex sets are investigated. Their interrelationships are explored.