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Approximations and Solution Estimates in Optimization

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Technical Report

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Naval Postgraduate School Monterey United States

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Approximation is central to many optimization problems and the supporting theory provides insight as well as foundation for algorithms. In this paper, we lay out a broad framework for quantifying approximations by viewing finite- and infinite-dimensional constrained minimization problems as instances of extended real-valued lower semicontinuous functions defined on a general metric space. Since the Attouch-Wets distance between such functions quantifies epi-convergence, we are ableto obtain estimates of optimal solutions and optimal values through estimates of that distance. In particular, we show that near-optimal and near-feasible solutions are effectively Lipschitz continuous with modulus one in this distance. We construct a general class of approximations of extended real-valued lower semicontinuous functions that can be made arbitrarily accurate and that involve only a finite number of parameters under additional assumptions on the underlying metric space.

Subject Categories:

  • Numerical Mathematics
  • Operations Research
  • Numerical Mathematics
  • Operations Research

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