Characterizing the Single-Valuedness of Multifunctions
We characterize the local single-valuedness and continuity of multifunctions (set-valued mappings) in terms of their ‘premonotonicity’ and lower semicontinuity. This result completes the well-known fact that lower semicontinuous, monotone multifunctions are single-valued and continuous. We also show that a multifunction is actually a Lipschitz single-valued mapping if and only if it is premonotone and has a generalized Lipschitz property called ‘Aubin continuity’. The possible single-valuedness and continuity of multifunctions is at the heart of some of the most fundamental issues in variational analysis and its application to optimization. We investigate the impact of our characterizations on several of these issues; discovering exactly when certain generalized subderivatives can be identified with classical derivatives, and determining precisely when solutions to generalized variational inequalities are locally unique and Lipschitz continuous. As an application of our results involving generalized variational inequalities, we characterize when the Karush–Kuhn–Tucker pairs associated with a parameterized optimization problem are locally unique and Lipschitz continuous.
- Aubin, J. P.: Lipschitz behavior of solutions to convex minimization problems, Math. Oper. Res. 9 (1984), 87–111.
- Clarke, F. H.: Optimization and Nonsmooth Analysis, Wiley, New York, 1983.
- Dontchev, A. L. and Rockafellar, R. T.: Characterizations of strong regularity for variational inequalities over polyhedral convex sets, SIAM J. Optim. 6 (1995), 1087–1105.
- Kenderov, P.: Semi-continuity of set-valued mappings, Fund. Math. 88 (1975), 61–69.
- Kenderov, P.: Dense strong continuity of pointwise continuous mappings, Pacific J. Math. 89 (1980), 111–130.
- Levy, A. B. and Rockafellar, R. T.: Variational conditions and the proto-differentiation of partial subgradient mappings, Nonlinear Anal. 26 (1995), 1951–1964.
- Mordukhovich, B.: Complete characterization of openness, metric regularity, and Lipschitzian properties of multifunctions, Trans. Amer. Math. Soc. 340 (1993), 1–35.
- Poliquin, R. A.: Integration of subdifferentials of nonconvex functions, Nonlinear Anal. 17 (1991), 385–398.
- Poliquin, R. A.: An extension of Attouch's theorem and its application to second-order epidifferentiation of convexly composite functions, Trans. Amer. Math. Soc. 332 (1992), 861–874.
- Poliquin, R. A. and Rockafellar, R. T.: Prox-regular functions in variational analysis, Trans. Amer. Math. Soc. 348 (1996), 1805–1838.
- Rockafellar, R. T.: Favorable classes of Lipschitz-continuous functions in subgradient optimization, in: E. Nurminski (ed.), Progress in Nondifferentiable Optimization, Institute for Applied Systems, 1982, pp. 125–143.
- Characterizing the Single-Valuedness of Multifunctions
Volume 5, Issue 4 , pp 351-364
- Cover Date
- Print ISSN
- Online ISSN
- Kluwer Academic Publishers
- Additional Links
- Aubin continuity
- proximal subdifferential
- sensitivity analysis