Abstract
In this paper we introduce the concept of solving strategy for a linear semi-infinite programming problem, whose index set is arbitrary and whose coefficient functions have no special property at all. In particular, we consider two strategies which either approximately solve or exactly solve the approximating problems, respectively. Our principal aim is to establish a global framework to cope with different concepts of well-posedness spread out in the literature. Any concept of well-posedness should entail different properties of these strategies, even in the case that we are not assuming the boundedness of the optimal set. In the paper we consider three desirable properties, leading to an exhaustive study of them in relation to both strategies. The more significant results are summarized in a table, which allows us to show the double goal of the paper. On the one hand, we characterize the main features of each strategy, in terms of certain stability properties (lower and upper semicontinuity) of the feasible set mapping, optimal value function and optimal set mapping. On the other hand, and associated with some cells of the table, we recognize different notions of Hadamard well-posedness. We also provide an application to the analysis of the Hadamard well-posedness for a linear semi-infinite formulation of the Lagrangian dual of a nonlinear programming problem.
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References
M.S. Bazaraa, H.D. Sherali and C.M. Shetty, Nonlinear Programming: Theory and Algorithms (Wiley, New York, 1993).
B. Brosowski, Parametric semi–infinite linear programming I. Continuity of the feasible set and of the optimal value, Math. Programming Study 21 (1984) 18–42.
M.J. Cánovas, M.A. López, J. Parra and M.I. Todorov, Stability and well–posedness in linear semiinfinite programming, SIAM J. Optim. 10 (1999) 82–98.
A.L. Dontchev and T. Zolezzi, Well–Posed Optimization Problems (Springer, Berlin, 1993).
T. Fischer, Contributions to semi–infinite linear optimization, in: Approximation and Optimization in Mathematical Physics, eds. B. Brosowski and E. Martensen (Peter Lang, Frankfurt–am–Main, 1983) pp. 175–199.
M.A. Goberna and M.A. López, Linear Semi–Infinite Optimization (Wiley, Chichester, UK, 1998).
M.A. Goberna, M.A. López and M.I. Todorov, Stability theory for linear inequality systems, SIAM J. Matrix Anal. Appl. 17 (1996) 730–743.
M.A. Goberna, M.A. López and M.I. Todorov, Stability theory for linear inequality systems II: upper semicontinuity of the solution set mapping, SIAM J. Optim. 7 (1997) 1138–1151.
R.T. Rockafellar, Convex Analysis (Princeton University Press, Princeton, NJ, 1970).
M.I. Todorov, Generic existence and uniqueness of the solution set to linear semi–infinite optimization problems, Numer. Funct. Anal. Optim. 8 (1985–86) 27–39.
Y.J. Zhu, Generalizations of some fundamental theorems on linear inequalities, Acta Math. Sinica 16 (1966) 25–40.
S. Zlobec, R. Gardner and A. Ben–Israel, Regions of stability for arbitrarily perturbed convex programs, in: Mathematical Programming with Data Perturbations I, ed. A.V. Fiacco (Dekker, New York, 1982) pp. 69–89.
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Cánovas, M., López, M., Parra, J. et al. Solving Strategies and Well-Posedness in Linear Semi-Infinite Programming. Annals of Operations Research 101, 171–190 (2001). https://doi.org/10.1023/A:1010920507183
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DOI: https://doi.org/10.1023/A:1010920507183