Sigma-subdifferential and its application to minimization problem
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Positivity
Sigma-subdifferential and its application to minimization problem Hui Huang1
· Chunyou Sun2
Received: 18 February 2019 / Accepted: 8 July 2019 © Springer Nature Switzerland AG 2019
Abstract In this paper, we study σ -subdifferentials of σ -convex functions. Two equivalent conditions for σ -convexity are given. The formula for the σ -subdifferential of a sum of two functions is established. In terms of σ -subdifferential and Clarke’s normal cone, some Fermat’s rules for minimization problems are obtained. Keywords σ -Convex function · σ -Subdifferential · Fermat’s rule · Banach space Mathematics Subject Classification 49J52 · 90C26
1 Introduction Let (X , ·) be a Banach space and X ∗ be the dual space of X . Let f : X → R∪{+∞} be an extended real-valued function. The domain of f is defined by dom( f ) = {x ∈ X | f (x) < +∞}. We say that f is proper if dom( f ) = ∅. The epigraph of f is defined by epi( f ) = {(x, t) ∈ X × R | f (x) ≤ t}. Let ε > 0. Jofre, Luc and Thera [9] introduced the notion of ε-convex function as follows. A function f is said to be ε-convex if for all x, y ∈ X and t ∈ (0, 1), f (t x + (1 − t)y) ≤ t f (x) + (1 − t) f (y) + εt(1 − t)x − y.
B
Hui Huang [email protected]
1
Department of Mathematics, Yunnan University, Kunming 650091, People’s Republic of China
2
School of Mathematics and Statistics, Lanzhou University, Lanzhou 730000, People’s Republic of China
H. Huang, C. Sun
However, in many functions, ε should be a variable depending on x and y. For this reason, recently, Alizadeh and Roohi [1] introduced the notion of σ -convex function as follows. Let σ : X → R+ ∪ {+∞} be a function with dom( f ) ⊆ dom(σ ); f is said to be σ -convex if for all x, y ∈ X and t ∈ (0, 1), f (t x + (1 − t)y) ≤ t f (x) + (1 − t) f (y) + t(1 − t) min{σ (x), σ (y)}x − y. Clearly, dom( f ) is a convex set if f is a σ -convex function. In the special case of min{σ (x), σ (y)} = ε, σ -convex function reduces to ε-convex function. But the converse is not true; see Example 3 in [3]. Moreover, ε-convex function reduces to convex function if ε = 0. But the converse is not true. For example, the function f (x) = ||x| − 1| for all x ∈ R is a 4-convex function but not a convex function; see [15, named as paraconvex function]. In a word, {convex function} ⊂ {ε-convex function} ⊂ {σ -convex function}. In [9,10], the relationship between ε-convexity and ε-monotonicity was studied. In [1], the relationship between σ -convexity and σ -monotonicity was also studied. For more details about monotonicity, σ -monotonicity and maximal σ -monotonicity, we refer the readers to the papers [2,4,8]. In [13], some related issues on generalized convexity are also studied. Subdifferentials of convex functions were initiated by Rockafellar [14], and have been developed into a rich theory which is known as convex analysis. Since subdifferentials play important roles in optimization and variational inequality (for example, see the references [6,7]), various kinds of subdifferentials such as Fr´echet subdifferential,
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