Hostname: page-component-cd9895bd7-jn8rn Total loading time: 0 Render date: 2024-12-24T00:48:00.286Z Has data issue: false hasContentIssue false
  • English
  • Français

A Calculus of EPI-Derivatives Applicable to Optimization

Published online by Cambridge University Press:  20 November 2018

R. A. Poliquin  and
R. T. Rockafellar
R. A. Poliquin
Affiliation:
Department of Mathematics, University of Alberta, Edmonton, Alberta, T6G 2G1
R. T. Rockafellar
Affiliation:
Department of Mathematics, University of Washington, Seattle, Washington 98195, U.S.A.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

When an optimization problem is represented by its essential objective function, which incorporates constraints through infinite penalties, first- and secondorder conditions for optimality can be stated in terms of the first- and second-order epi-derivatives of that function. Such derivatives also are the key to the formulation of subproblems determining the response of a problem's solution when the data values on which the problem depends are perturbed. It is vital for such reasons to have available a calculus of epi-derivatives. This paper builds on a central case already understood, where the essential objective function is the composite of a convex function and a smooth mapping with certain qualifications, in order to develop differentiation rules covering operations such as addition of functions and a more general form of composition. Classes of "amenable" functions are introduced to mark out territory in which this sharper form of nonsmooth analysis can be carried out.

Keywords

90C4849A5258C2090C30epi-derivativesgeneralized second derivativesamenable sets and functionsnonsmooth analysiscomposite optimizationparametric optimizationsensitivity analysis
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 45 , Issue 4 , 01 August 1993 , pp. 879 - 896
Copyright
Copyright © Canadian Mathematical Society 1993

References

1. Clarke, F. H., Generalized gradients and applications, Transactions Amer. Math. Soc. 205(1975), 247–262.Google Scholar
2. Clarke, F. H., Optimization and Nonsmooth Analysis, Wiley, 1983.Google Scholar
3. Clarke, F. H., Methods of Dynamic and Nonsmooth Optimization, CBMS-NSF Regional Conference Series in Applied Mathematics, 57, SIAM Publications, Philadelphia, PA, 1989.Google Scholar
4. Cominetti, R., On Pseudo-differentiability, Transactions Amer. Math. Soc, 324(1991), 843–865.Google Scholar
5. Cominetti, R. and Correa, R., A generalized second order derivative in nonsmooth optimization, SIAM J. Control Opt. 28(1990), 789–809.Google Scholar
6. Do, C. N., Generalized second derivatives of convex functions in reflexive Banach spaces, Transactions Amer. Math. Soc, to appear.Google Scholar
7. Ioffe, A. D., Variational analysis of a composite function: a formula for the lower second order directional derivative, J. Math. Anal. Appl., 160(1991), 379–405.Google Scholar
8. Ioffe, A. D., Variational analysis of a composite function: perturbations, value function and sensitivity. Google Scholar
9. McLinden, L. and R. C. Bergstrom, Preservation of convergence of convex sets and functions, Transactions Amer. Math. Soc. 268(1981), 127–142.Google Scholar
10. Poliquin, R. A., Proto-differentiation of subgradient set-valued mappings, Canadian J. Math 42( 1990), 520- 532. 11 , An extension ofAttouch 's Theorem and its application to second-order epi-differentiation of convex composite functions, Trans. Amer. Math. Soc. 322(1992), 861–874.Google Scholar
13. Robinson, S. M., Some continuity properties of polyhedral multifunctions, Math. Programming Study 14 (1981), 206–214.Google Scholar
12. Poliquin, R. A. and Rockafellar, R. T., Differentiability of solution mappings and sensitivity analysis in parametric optimization, preprint.Google Scholar
14. Rockafellar, R. T., Convex Analysis, Princeton University Press, Princeton, NJ, 1970.Google Scholar
15. Rockafellar, R. T., Directionally Lipschitzian functions and subdifferential calculus, Proc. London Math. Soc. 39 (1979), 331–355.Google Scholar
16. Rockafellar, R. T., First- and second-order epi-differentiability in nonlinear programming, Trans. Amer. Math. Soc. 307(1988),75–107.Google Scholar
17. Rockafellar, R. T., Proto-differentiability of set-valued mappings and its applications in optimization. In: Analyse Non Linéaire, (editor H. Attouch et al.), Gauthier-Villars, Paris (1989), 449-482.Google Scholar
18. Rockafellar, R. T., Second-order optimality conditions in nonlinear programming obtained by way ofepi-derivatives, Math, of Open Research 14(1989), 462–484.Google Scholar
19. Rockafellar, R. T., Generalized second derivatives of convex functions and saddle functions, Transactions Amer. Math. Soc. 320(1990), 810–822.Google Scholar
20. Rockafellar, R. T., Nonsmooth analysis and parametric optimization. In: Methods of Nonconvex Analysis., (editor A. Cellina), Springer-Verlag Lecture Notes in Math. (1446), (1990), 137-151.Google Scholar
21. Rockafellar, R. T., Dualization of subgradient conditions for optimality. Google Scholar
22. Sun, J., On Monotropic Piecewise quadratic Programming, dissertation, Department of Applied Math., University of Washington, August, 1986.Google Scholar
23. Hiriart-Urruty, J.-B., Calculus rules on the approximate second order directional derivative of a convex Junction, SIAM J. Control Opt. 22 (1984), 381–404.Google Scholar
24. Hiriart-Urruty, J.-B. and Seeger, A., Calculus rules on a new set-valued second order derivative for convex functions, Nonlinear Analysis Th. Meth. Appl. 13(1989), 721–738.Google Scholar
25. Ward, D. E. and Borwein, J. M., Nonsmooth calculus infinite dimensions, SIAM J. Control Opt. 25(1987), 1312–1340 Google Scholar