Abstract
Second-order derivatives of Chaney’s type are introduced for an arbitrary subdifferential. Their uses for necessary optimality conditions and sufficient optimality conditions are put in light when the subdifferential satisfies a weak sum rule.
References
Auslender, A.: Stability in mathematical programming with nondifferentiable data. SIAM J. Control Opt. 22, 239–254 (1984)
Aussel, D., Corvellec, J.-N., Lassonde, M.: Mean value property, and subdifferential criteria for lower semicontinuous functions. Trans. Am. Math. Soc. 347, 4147–4161 (1995)
Chaney, R.W.: On second derivatives for nonsmooth functions. Nonlinear Anal. Th. Methods Appl. 9, 1189–1209 (1985)
Chaney, R.W.: Second-order directional derivatives for nonsmooth functions. J. Math. Anal. Appl. 128, 495–511 (1987)
Chaney, R.W.: Second-order necessary conditions in constrained semismooth optimization. SIAM J. Control Optim. 25, 1072–1081 (1987)
Chaney, R.W.: Second-order sufficient conditions in nonsmooth optimization. Math. Oper. Res. 13, 660–673 (1988)
Chou, C.C., Li, X., Ng, K.F., Shi, S.: Optimality conditions for lower semi-continuous functions. J. Math. Anal. Appl. 202(2), 686–701 (1996)
Clarke, F.H.: Optimization and Nonsmooth Analysis. Wiley, New York (1983)
Correa, R., Jofre, A., Thibault, L.: Subdifferential monotonicity as characterization of convex functions. Numer. Funct. Anal. Optim. 15(5–6), 531–535 (1994)
Correa, R., Jofre, A., Thibault, L.: Subdifferential characterization of convexity. In: Du, D.Z., Qi, L., Womersley, R.S. (eds.) Recent Advances in Nonsmooth Optimization, pp. 18–23 World Scientific, Singapore (1995)
Huang, L.R., Ng, K.F.: Second-order necessary and sufficient conditions in nonsmooth optimization. Math. Program. 66, 379–402 (1994)
Huang, L.R., Ng, K.F.: On second-order directional derivatives in nonsmooth optimization. In: Du, D.Z., Qi, L., Womersley, R.S. (eds.) Recent Advances in Nonsmooth Optimization, pp. 159–171. World Scientific, Singapore (1995)
Huang, L.R., Ng, K.F.: On some relations between Chaney’s generalized second order directional derivative and that of Ben-Tal and Zowe. SIAM J. Control Optim. 34(4), 1220–1235 (1996)
Huang, L.R., Ng, K.F., Penot, J.-P.: On minimizing and critical sequences in nonsmooth optimization. SIAM J. Optim. 10(4), 999–1019 (2000)
Ioffe, A.D.: On subdifferentiability spaces. Ann. N Y Acad. Sci. 410, 107–119 (1983)
Ioffe, A.D.: Subdifferentiability spaces and nonsmooth analysis. Bull. Am. Math. Soc. 10, 87–89 (1984)
Ioffe, A.D.: Calculus of Dini subdifferentials of functions and contingent coderivatives of set-valued maps. Nonlinear Anal. Theory, Methods Appl. 8, 517–539 (1984)
Ioffe, A.D.: Codirectional compactness, metric regularity and subdifferential calculus. Can. Math. Soc. 27, 123–163 (2000)
Ioffe, A.D.: Theory of subdifferentials. Adv. Nonlinear Anal. 1, 47–120 (2012)
Ioffe, A.D., Penot, J.-P.: Subdifferentials of performance functions and calculus of coderivatives of set-valued mappings. Serdica Math. J. 22, 359–384 (1996)
Ph Loewen, A.: Mean value theorem for Fréchet subgradients. Nonlinear Anal. Theory Methods Appl. 23, 1365–1381 (1994)
Penot, J.-P.: Local minimality versus spongious minimality. Optimization 19(3), 301–306 (1988)
Penot, J.-P.: Second-order generalized derivatives: comparisons of two types of epi-derivatives, Advances in optimization, Proceedings, Lambrecht, FRG, 1991. In: Oettli, W., Pallaschke, D. (eds.) Lecture Notes in Economics and Math. Systems, vol. 382, pp. 52–76. Springer, Berlin (1992)
Penot, J.-P.: A mean value theorem with small subdifferentials. J. Optim. Theory Appl. 94(1), 209–221 (1997)
Penot, J.-P.: Miscellaneous incidences of convergence theories in optimization and nonlinear analysis, part II: applications in nonsmooth analysis. In: Du, D.Z., Qi, L., Womersley, R.S. (eds.) Recent Advances in Nonsmooth Optimization, pp. 289–321. World Scientific, Singapore (1995)
Penot, J.-P.: Compactness properties, openness criteria and coderivatives. Set-Valued Anal. 6, 363–380 (1998)
Penot, J.-P.: Subdifferential calculus and subdifferential compactness. In: Sofonea, M., Corvellec, J.-N. (eds.) Proceedings of the Second Catalan Days on Applied Mathematics, pp. 209–226. Presses universitaires de Perpignan, Perpignan (1995)
Penot, J.-P.: Are generalized derivatives useful for generalized convexity? In: Crouzeix J.-P., et al. (eds.) Proceedings of the Symposium on Generalized Convexity, pp. 1–60. Kluwer, Dordrecht (1998)
Penot, J.-P.: Well-behavior, well-posedness and nonsmooth analysis. Pliska Stud. Math. Bulgar. 12, 141–190 (1998)
Penot J.-P.: A variational subdifferential for quasiconvex functions. J. Optim. Th. Appl. 111, 165–171 (2001)
Penot J.-P.: Towards a new era in subdifferential analysis? Contemporary Math. (2013)
Rockafellar, R.T.: Directionally Lipschitzian functions and subdifferential calculus. Proc. London Math. Soc. 39, 145–154 (1979)
Rockafellar, R.T.: First- and second-order epi-differentiability in nonlinear programming. Trans. Am. Math. Soc. 307, 75–107 (1988)
Rockafellar, R.T.: Second-order optimality conditions in nonlinear programming obtained by the way of pseudo-derivatives. Math. Oper. Res. 14, 462–484 (1989)
Rockafellar, R. T., Wets, R. J.-B.: Variational Analysis. Grundlehren der Math. Wissenschaften, vol. 317. Springer, Berlin (1998)
Treiman, J.S.: Clarke’s gradients and \(\varepsilon \)-subgradients in Banach spaces. Trans. Am. Math. Soc. 294, 65–78 (1986)
Treiman, J.S.: Shrinking generalized gradients. Nonlinear Anal. Theory Methods Appl. 12, 1429–1450 (1988)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Penot, JP. Directionally limiting subdifferentials and second-order optimality conditions. Optim Lett 8, 1191–1200 (2014). https://doi.org/10.1007/s11590-013-0663-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11590-013-0663-0