The Mordukhovich Subdifferentials and Directions of Descent
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The problem of finding minima of weakly sequentially lower semicontinuous functions on reflexive Banach spaces is studied by means of convex and nonconvex subdifferentials. Finding a descent direction for a non-stationary point is a question of importance for many optimization algorithms. The existence or non-existence of such a direction is clarified through several theorems and a series of selective examples. For the general problem, a notion called radius of descent is proposed and shown to be useful for the analysis related to descent directions.
KeywordsWeakly sequentially lower semicontinuous function Minimization Subdifferential Descent direction Radius of descent
Mathematics Subject Classification49J52 49J53 90C26 49J45
The researches of Khanh, Yao, Yen were funded, respectively, by the Vietnam Institute for Advanced Study in Mathematics (VIASM), the Grant MOST 102-2115-M-039-003-MY3, and the Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 101.01-2014.37. We would like to thank Professor Nguyen Nang Tam for a useful discussion on generalized Weierstrass Theorems and the referees for their constructive comments.
- 3.Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation. Vol. I: Basic Theory, Vol. II: Applications. Springer, Berlin (2006)Google Scholar
- 5.Kiwiel, K.C.: Methods of Descent for Nondifferentiable Optimization. Lecture Notes in Mathematics, vol. 1133. Springer, Berlin (1985)Google Scholar
- 9.Polyak, B.T.: Introduction to Optimization. Translated from the Russian. Optimization Software, Inc., New York (1987)Google Scholar
- 11.Vasilev, F.P.: Numerical Methods for Solving Extremal Problems (in Russian), 2nd edn. Nauka, Moscow (1988)Google Scholar