Abstract
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.
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Acknowledgments
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.
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Khanh, P.D., Yao, JC. & Yen, N.D. The Mordukhovich Subdifferentials and Directions of Descent. J Optim Theory Appl 172, 518–534 (2017). https://doi.org/10.1007/s10957-015-0774-0
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DOI: https://doi.org/10.1007/s10957-015-0774-0
Keywords
- Weakly sequentially lower semicontinuous function
- Minimization
- Subdifferential
- Descent direction
- Radius of descent