Abstract
In this paper we demonstrate that second order subdifferentials constructed via the accumulation of local Hessian information provided by an integral convolution approximation of the function, provide useful information only for a limited class of nonsmooth functions. When local finiteness of associated second order directional derivative is demanded this forces the first order subdifferential to possess a local Lipschitz property. To enable the study of a broader classes of nonsmooth functions we show that a combination of the infimal and integral convolutions needs to be used when constructing approximating smooth functions.
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References
H. Attouch (1984) Variational Convergence for Functions and Operators, Pitman Adv. Publ. Prog. Boston-London-Melbourne.
G. Beer (1993) Topologies on Closed and Convex Sets, mathematics and its applications Vol. 268, Kluwer Academic Publishers.
B. Craven (1986) Non-Differential Optimization by Smooth Approximations, Optimization, Vol. 17 no. 1, pp. 3–17.
B. Craven (1986) A Note on Non-Differentiable Symmetric Duality, Journal of Australian Mathematical society Series B, Vol. 28 no. 1, 30–35.
R. Cominetti and R. Correa (1990) A Generalized Second Order Derivative in Nonsmooth Optimization, SIAM J. Control and Optimization, Vol. 28, pp. 789–809.
G. Crespi, D. La Torre and M. Rocca, Mollified Derivative and Second-order Optimality Conditions, Preprint communicated from the authors.
R. M. Dudley (1977) On Second Derivatives of Convex Functions, Math. Scand., Vol. 41, pp. 159–174.
A. Eberhard, M. Nyblom and D. Ralph (1998) Applying Generalised Convexity Notions to Jets, J.P Crouzeix et al. (eds), it Generalized Convexity, Generalized Monotonicity: Recent Results, Kluwer Academic Pub., pp. 111–157.
A. Eberhard and M. Nyblom (1998) Jets Generalized Convexity, Proximal Normality and Differences of Functions, Non-Linear Analysis Vol. 34, pp. 319–360.
A. Eberhard (2000) Prox-Regularity and Subjets, Optimization and Related Topics, Ed. A. Rubinov, Applied Optimization Volumes, Kluwer Academic Pub., pp. 237–313.
Y.M. Ermoliev, V.I. Norkin, R. J-B. Wets (1995) The Minimization of Semicontinuous Functions: Mollifier Subgradients, SIAM J. Control and Optimization, Vol. 33. No. 1, pp. 149–167.
R. F. Gariepy and W. P. Zoemer (1995) Modern Real Analysis, PWS Publishing Company, PWS Publishing Company, Boston Massachusetts.
H. Halkin (1976) Interior Mapping Theorem with Set-Valued Derivatives, J. d’Analyse Mathèmatique, Vol. 30, pp 200–207.
H. Halkin (1976) Mathematical Programming without Differentiability, Calculus of Variations and Control Theory, ed D. L. Russell, Academic Press, NY.
A. D. Ioffe (1989), On some Recent Developments in the Theory of Second Order Optimality Conditions, Optimization-fifth French German Conference, Castel Novel 1988, Lecture Notes in Mathematics, Vol. 405, Springer Verlag, pp. 55–68.
A.D. Ioffe and J-P. Penot (1987) Limiting Subhessians, Limiting Subjets and their Calculus, Transactions of the American Mathematics Society, Vol. 349, no. 2, pp 789–807.
F. Mignot (1976) Contrôle dans Inéquations Variationelles Elliptiques, J. of Functional Analysis, No. 22, pp. 130–185.
M. Nyblom (1998) Smooth Approximation and Generalized Convexity in Nonsmooth Analysis and Optimization, PhD thesis RMIT University.
Z. Páles and V. Zeidan (1996) Generalized Hessians for Functions in Infinite-Dimensional Normed Spaces, Mathematical Programming, Vol. 74, pp. 59–78.
J.-P. Penot (1994) Sub-Hessians, Super-Hessians and Conjugation, Nonlinear Analysis, Theory Methods and Applications, Vol. 23, no. 6, pp. 689–702.
D. Ralph (1990) Rank-1 Support Functional and the Rank-1 Generalised Jacobian, Piecewise Linear Homeomorphisms, Ph.D. Thesis, Computer Science Technical Reports #938, University of Wisconsin Madison.
S. M. Robinson, Local Epi-Continuity and Local Optimization, Mathematical Programming, Vol. 37, pp. 208–222.
R. T. Rockafellar and R. J-B. Wets (1998) Variational Analysis, Volume 317, A series of Comprehensive Studies in Mathematics, Springer.
R. Sivakumaran (2003) A Study of the Viscosity and Weak Solutions to a Class of Boundary Valued Problems, PhD Thesis, RMIT University.
J. Warga (1975) Necessary Conditions without Differentiability Assumptions in Optimal Control, J. of Diff. Equ., Vol. 15, pp. 41–61.
J. Warga (1976) Derivative Containers, Inverse Functions and Controllability, Calculus of Variations and Control Theory, ed D. L. Russell, Academic Press, NY.
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Eberhard, A., Nyblom, M., Sivakumaran, R. (2005). Second Order Subdifferentials Constructed Using Integral Convolutions Smoothing. In: Eberhard, A., Hadjisavvas, N., Luc, D.T. (eds) Generalized Convexity, Generalized Monotonicity and Applications. Nonconvex Optimization and Its Applications, vol 77. Springer, Boston, MA. https://doi.org/10.1007/0-387-23639-2_14
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DOI: https://doi.org/10.1007/0-387-23639-2_14
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