Abstract
This paper gives several sets of sufficient conditions that alocal solutionx k exists of the problem\(\min _{R^k } f^k (x)\),k=1, 2,..., such that {x k} has cluster points that arelocal solutions of a problem of the form min R f(x). The results are based on a well-known concept of topological, orpoint-wise convergence of the sets {R k} toR. Such results have been used to construct and validate large classes of mathematical programming methods based on successive approximations of the problem functions. They are also directly applicable to parametric and sensitivity analysis and provide additional characterizations of optimality for large classes of nonlinear programming problems.
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Communicated by G. B. Dantzig
Results closely related to those obtained here were presented inSome Properties of Convergent Sequences of Mathematical Programs by P. D. Robers and the author to the 36th National Meeting of the Operations Research Society of America, Miami, Florida, 1969. The author deeply acknowledges the contributions made by Dr. Roberts in developing these earlier results.
This work was supported in part by the US Army Research Office, Durham, North Carolina, Contract No. DAHC19-69-C-0017, and in part by the Office of Naval Research, Contract No. N00014-67-0214, Task 0001, Project NR-347-020.
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Fiacco, A.V. Convergence properties of local solutions of sequences of mathematical programming problems in general spaces. J Optim Theory Appl 13, 1–12 (1974). https://doi.org/10.1007/BF00935606
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DOI: https://doi.org/10.1007/BF00935606