Sigma-subdifferential and its application to minimization problem
In this paper, we study \(\sigma \)-subdifferentials of \(\sigma \)-convex functions. Two equivalent conditions for \(\sigma \)-convexity are given. The formula for the \(\sigma \)-subdifferential of a sum of two functions is established. In terms of \(\sigma \)-subdifferential and Clarke’s normal cone, some Fermat’s rules for minimization problems are obtained.
Keywords\(\sigma \)-Convex function \(\sigma \)-Subdifferential Fermat’s rule Banach space
Mathematics Subject Classification49J52 90C26
The authors are greatly indebted to the reviewers and the Editor for their valuable comments. Huang’s research was supported by the National Natural Science Foundation of China (Grant No. 11461080). Sun’s research was supported by the National Natural Science Foundation of China (Grant Nos. 11522109 and 11871169).
- 10.Luc, D.T., Ngai, H.V., Thera, M.: On \(\epsilon \)-monotonicity and \(\epsilon \)-convexity. In: Ioffe, A. (ed.) Calculus of Variations and Differential Equations (Haifa, 1998). Research Notes in Mathematics Series, vol. 410, pp. 82–100. Chapman & Hall, Boca Raton (1999)Google Scholar
- 14.Rockafellar, R.T.: Convex functions and dual extremum problems. Ph.D. Thesis, Hardvard University, Cambridge (1963)Google Scholar