×

On the arg min multifunction for lower semicontinuous functions. (English) Zbl 0731.49009

Let L(X) denote the set of lower semicontinuous functions on a Hausdorff space X and for each \(f\in L(X)\) let arg min f\(=\{x\in X:\) \(f(x)=\inf f(X)\}.\)
The authors show that with respect to the epi-topology on L(X), the graph of arg min is a closed subset of L(X)\(\times X\) if and only if X is locally compact. Moreover, if X is locally compact, then the epi-topology is the weakest topology on L(X) for which the graph of arg min is closed, and the operators \(f\to f\vee g\) and \(f\to f\wedge g\) are continuous for each continuous real function g on X.

MSC:

49J45 Methods involving semicontinuity and convergence; relaxation
26A15 Continuity and related questions (modulus of continuity, semicontinuity, discontinuities, etc.) for real functions in one variable
54C60 Set-valued maps in general topology
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] H. Attouch, Variational convergence for functions and operators, Applicable Mathematics Series, Pitman (Advanced Publishing Program), Boston, MA, 1984. · Zbl 0561.49012
[2] Hédy Attouch and Roger J.-B. Wets, Approximation and convergence in nonlinear optimization, Nonlinear programming, 4 (Madison, Wis., 1980) Academic Press, New York-London, 1981, pp. 367 – 394.
[3] Dino Dal Maso, Questions of topology associated with \Gamma -convergence, Ricerche Mat. 32 (1983), no. 1, 135 – 162 (Italian). · Zbl 0539.49005
[4] Ennio De Giorgi, Sulla convergenza di alcune successioni d’integrali del tipo dell’area, Rend. Mat. (6) 8 (1975), 277 – 294 (Italian, with English summary). Collection of articles dedicated to Mauro Picone on the occasion of his ninetieth birthday. · Zbl 0316.35036
[5] Szymon Dolecki, Gabriella Salinetti, and Roger J.-B. Wets, Convergence of functions: equi-semicontinuity, Trans. Amer. Math. Soc. 276 (1983), no. 1, 409 – 430. · Zbl 0504.49006
[6] James Dugundji, Topology, Allyn and Bacon, Inc., Boston, Mass., 1966. · Zbl 0144.21501
[7] Edward G. Effros, Convergence of closed subsets in a topological space, Proc. Amer. Math. Soc. 16 (1965), 929 – 931. · Zbl 0139.40403
[8] J. M. G. Fell, A Hausdorff topology for the closed subsets of a locally compact non-Hausdorff space, Proc. Amer. Math. Soc. 13 (1962), 472 – 476. · Zbl 0106.15801
[9] John L. Kelley, General topology, D. Van Nostrand Company, Inc., Toronto-New York-London, 1955. · Zbl 0066.16604
[10] Erwin Klein and Anthony C. Thompson, Theory of correspondences, Canadian Mathematical Society Series of Monographs and Advanced Texts, John Wiley & Sons, Inc., New York, 1984. Including applications to mathematical economics; A Wiley-Interscience Publication. · Zbl 0556.28012
[11] R. T. Rockafellar and Roger J.-B. Wets, Variational systems, an introduction, Multifunctions and integrands (Catania, 1983) Lecture Notes in Math., vol. 1091, Springer, Berlin, 1984, pp. 1 – 54. · doi:10.1007/BFb0098800
[12] H. L. Royden, Real analysis, The Macmillan Co., New York; Collier-Macmillan Ltd., London, 1963. · Zbl 0121.05501
[13] Bernard Van Cutsem, Problems of convergence in stochastic linear programming, Techniques of optimization (Fourth IFIP Colloq., Los Angeles, Calif., 1971), Academic Press, New York, 1972, pp. 445 – 454.
[14] Wim Vervaat, Une compactification des espaces fonctionnels \? et \?; une alternative pour la démonstration de théorèmes limites fonctionnels, C. R. Acad. Sci. Paris Sér. I Math. 292 (1981), no. 8, 441 – 444 (French, with English summary). · Zbl 0468.60008
[15] Wim Vervaat, Stationary self-similar extremal processes and random semicontinuous functions, Dependence in probability and statistics (Oberwolfach, 1985) Progr. Probab. Statist., vol. 11, Birkhäuser Boston, Boston, MA, 1986, pp. 457 – 473. · Zbl 0606.60049
[16] R. J.-B. Wets, Convergence of convex functions, variational inequalities and convex optimization problems, Variational inequalities and complementarity problems (Proc. Internat. School, Erice, 1978) Wiley, Chichester, 1980, pp. 375 – 403.
[17] R. A. Wijsman, Convergence of sequences of convex sets, cones and functions. II, Trans. Amer. Math. Soc. 123 (1966), 32 – 45. · Zbl 0146.18204
[18] T. Zolezzi, Characterizations of some variational perturbations of the abstract linear-quadratic problem, SIAM J. Control Optimization 16 (1978), no. 1, 106 – 121. · Zbl 0391.49019 · doi:10.1137/0316008
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.