On a Nonsmooth Vector Optimization Problem with Generalized Cone Invexity

and Applied Analysis 3 Definition 2.2 see 16 . Let ψ : R → R be a locally Lipschitz function, then ψ◦ u;v denotes Clarke’s generalized directional derivative of ψ at u ∈ R in the direction v and is defined as ψ◦ u;v lim sup y→u t→ 0 ψ ( y tv ) − ψ(y) t . 2.4 Clarke’s generalized gradient of ψ at u is denoted by ∂ψ u and is defined as ∂ψ u { ξ ∈ R | ψ◦ u;v ≥ 〈ξ, v〉, ∀v ∈ Rn}. 2.5 Let f : R → R be a vector-valued function given by f f1, f2, . . . , fm , where fi : R → R, i 1, 2, . . . , m. Then f is said to be locally Lipschitz on R if each fi is locally Lipschitz on R. The generalized directional derivative of a locally Lipschitz function f : R → R at u ∈ R in the direction v is given by f◦ u;v { f◦ 1 u;v , f ◦ 2 u;v , . . . , f ◦ m u;v } . 2.6 The generalized gradient of f at u is the set ∂f u ∂f1 u × ∂f2 u × · · · × ∂fm u , 2.7 where ∂fi u i 1, 2, . . . , m is the generalized gradient of fi at u. Every A a1, a2, . . . , am ∈ ∂f u is a continuous linear operator from R to R and Au 〈a1, u〉, 〈a2, u〉, . . . , 〈am, u〉 ∈ R, ∀u ∈ R. 2.8 Lemma 2.3 see 16 . (a) If fi : R → R is locally Lipschitz then, for each u ∈ R, f◦ i u;v max {〈ξ, v〉 | ξ ∈ ∂fi u } , ∀v ∈ R, i 1, 2, . . . , m. 2.9 (b) Let fi i 1, 2, . . . , m be a finite family of locally Lipschitz functions on R, then ∑m i 1 fi is also locally Lipschitz and