Generalized Second-Order Mixed Symmetric Duality in Nondifferentiable Mathematical Programming

and Applied Analysis 3 It can be easily seen that for a compact convex set C, y is in NC x if and only if S y | C xy, or equivalently, x is in ∂S y | C . Definition 2.2. A functional F : X × X × R → R where X ⊆ R is sublinear with respect to the third variable if for all x, u ∈ X ×X, i F x, u; a1 a2 ≤ F x, u; a1 F x, u; a2 for all a1, a2 ∈ R, ii F x, u;αa αF x, u; a , for all α ∈ R and for all a ∈ R. Let ψ : X → R be a real-valued twice differentiable function. Definition 2.3. ψ is said to be second-order F-convex at u ∈ X with respect to q ∈ R, if for all x ∈ X, ψ x − ψ u 1 2 q∇xxψ u q ≥ F ( x, u;∇xψ u ∇xxψ u q ) . 2.5 Definition 2.4. ψ is said to be second-order F-pseudoconvex at u ∈ X with respect to q ∈ R, if for all x ∈ X, F ( x, u;∇xψ u ∇xxψ u q ) ≥ 0 ⇒ ψ x ≥ ψ u − 1 2 q∇xxψ u q. 2.6 ψ is second-order F-concave/pseudoconcave at u ∈ X with respect to q ∈ R if −ψ is secondorder F-convex/pseudoconvex at u ∈ X with respect to q ∈ R. 3. Second-Order Mixed Nondifferentiable Symmetric Dual Programs ForN {1, 2, . . . , n} andM {1, 2, . . . , m}, let J1 ⊆ N,K1 ⊆ M, J2 N \J1, andK2 M \K1. Let |J1| denote the number of elements in J1. The other symbols |J2|, |K1| and |K2| are defined similarly. Let x1 ∈ R|J1|, x2 ∈ R|J2|. Then, any x ∈ R can be written as x1, x2 . Similarly, for y1 ∈ R|K1|, y2 ∈ R|K2|, y ∈ R can be written as y1, y2 . It may be noted here that if J1 ∅, then |J1| 0, J2 N, and therefore |J2| n. In this case, R|J1|, R|J2| and R|J1| ×R|K1| will be zerodimensional, n-dimensional and |K1|-dimensional Euclidean spaces, respectively. The other situations are J2 ∅,K1 ∅ or K2 ∅. Now we formulate the following pair of mixed nondifferentiable second-order symmetric dual programs and discuss their duality results. 4 Abstract and Applied Analysis Primal problem (SMNP) minimize G ( x1, y1, x2, y2, z2, p, r ) f ( x1, y1 ) S ( x1 | C1 ) g ( x2, y2 ) S ( x2 | C2 ) − ( y2 )T z2 − ( y1 T∇y1f ( x1, y1 ) ∇y1y1f ( x1, y1 ) p ] − 1 2 p∇y1y1f ( x1, y1 ) p − 1 2 r∇y2y2g ( x2, y2 ) r, 3.1