Journal of Optimization Theory and Applications

, Volume 172, Issue 2, pp 518–534 | Cite as

The Mordukhovich Subdifferentials and Directions of Descent

  • P. D. Khanh
  • J.-C. Yao
  • N. D. YenEmail author


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.


Weakly sequentially lower semicontinuous function Minimization Subdifferential Descent direction Radius of descent 

Mathematics Subject Classification

49J52 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.


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Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of Pedagogy of Ho Chi Minh CityHo Chi Minh CityVietnam
  2. 2.Department of Applied MathematicsNational Sun Yat-Sen UniversityKaohsiungTaiwan
  3. 3.Center for General EducationChina Medical UniversityTaichungTaiwan
  4. 4.Department of MathematicsKing Abdulaziz UniversityJeddahSaudi Arabia
  5. 5.Institute of MathematicsVietnam Academy of Science and TechnologyHanoiVietnam

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