Abstract
This paper introduces examples of convex functions in a general class of functions having what we call primal-dual gradient structure. The class contains finitely defined maximum value functions and maximum eigenvalue functions as well as other infinitely determined max-functions. For such a function there is a space decomposition that allows us to identify a subspace u on which the function appears smooth. Moreover, the special structure combined with sufficiency conditions implies the existence of smooth trajectories on which the function has a certain second order expansion. This results in an explicit expression for the Hessian of a related u-Lagrangian.
Research supported by the National Science Foundation under Grant No. DMS-9703952 and by FAPERJ (Brazil) under Grant No. E-26/171.393/97.
Research supported by FAPERJ (Brazil) under Grant No.E26/150.205/98.
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References
A. Ben-Tal and M. Bendsoe. A new method for optimal truss topology design. SIAM Journal on Optimization, 3: 322 - 358, 1993.
A. Ben-Tal and J. Zowe. Necessary and sufficient optimality conditions for a class of nonsmooth minimization problems. Mathematical Programming, 24 (1): 70 - 91, 1982.
R. Cominetti and R. Correa. A generalized second-order derivative in nonsmooth optimization. SIAM Journal on Control and Optimization, 28 (4): 789 - 809, 1990.
F.H. Clarke. Optimization and Nonsmooth Analysis. Wiley, 1983. Reprinted by Siam, 1990.
R., Cominetti. On pseudo-differentiability. Trans. Amer. Math. Soc., 2324: 843 - 865, 1991.
J.-B. Hiriart-Urruty. Refinements of necessary optimality conditions in non-differentiable programming I. Applied Mathematics and Optimization, 5: 6382, 1979.
J.-B. Hiriart-Urruty. Refinements of necessary optimality conditions in non-differentiable programming II. Mathematical Programming Study, 19: 120139, 1982.
A.D. Ioffe. Variational analysis of a composite function: a formula for thelower second order epi-derivative. Journal of Mathematical Analysis and Applications, pages 379 - 405, 1991.
C. Lemaréchal and R. Mifflin. Global and superlinear convergence of an algorithm for one-dimensional minimization of convex functions. Mathematical Programming, 24: 241 - 256, 1982.
C. Lemaréchal, F. Oustry, and C. Sagastizâbal. The U-Lagrangian of a convex function. Transactions of the AMS, 1997. Accepted for publication.
R. Mifflin and C. Sagastizâbal. On VU-theory for functions with primal-dual gradient structure. Submitted, 1999.
R. Mifflin and C. Sagastizâbal. VU-decomposition derivatives for convex max-functions. In R. Tichatschke and M. Théra, editors, Ill-posed problems and variational inequalities, Lecture Notes. Springer, 1999. Accepted for publication.
F. Oustry. A second-order bundle method to minimize the maximum eigen-value function. Submitted, 1997.
Ous98] F. Oustry. The U-Lagrangian of the maximum eigenvalue function. SIAM Journal on Optimization. To appear.
M. L. Overton and R.S. Womersley. Second derivatives for optimizing eigen-values of symmetric matrices. Journal of Mathematical Analysis and Applications, 3: 667 - 718, July 1995.
J.-P. Penot. Subhessians, superhessians and conjugation. Nonlinear Analysis: Theory, Methods and Applications, 23 (6): 689 - 702, 1994.
R.A. Poliquin. Proto-differentiation of subgradient set-valued mappings. Canadian Journal of Mathematics, 42 (3): 520 - 532, 1990.
S.M. Robinson. Local structure of feasible sets in nonlinear programming, part III: stability and sensitivity. Mathematical Programming Study, 30: 4566, 1987.
R.T. Rockafellar. First and second-order epi-differentiability in nonlinear programming. Trans. Amer. Math. Soc., 307: 75 - 107, 1988.
R.T. Rockafellar. Proto-differentiability of set-valued mappings and its applications in optimization. Annales de Institut Henri Poincaré, Analyse non linéaire, 6: 449 - 482, 1989.
R.T. Rockafellar. Second-order optimality conditions in nonlinear programming obtained by way of epi-derivatives. Mathematics of Operations Research, 14: 462 - 484, 1989.
R.T. Rockafellar and R.J.-B. Wets. Variational Analysis. Number 317 in Grund. der math. Wiss. Springer-Verlag, 1998.
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Mifflin, R., Sagastizábal, C. (2000). Functions with Primal-Dual Gradient Structure and u-Hessians. In: Pillo, G.D., Giannessi, F. (eds) Nonlinear Optimization and Related Topics. Applied Optimization, vol 36. Springer, Boston, MA. https://doi.org/10.1007/978-1-4757-3226-9_12
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DOI: https://doi.org/10.1007/978-1-4757-3226-9_12
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