By Shashi Kant Mishra, Giorgio Giorgi (auth.)
Invexity and Optimization offers effects on invex functionality and their houses in tender and nonsmooth situations, pseudolinearity and eta-pseudolinearity. effects on optimality and duality for a nonlinear scalar programming challenge are provided, moment and better order duality effects are given for a nonlinear scalar programming challenge, and saddle element effects also are awarded. Invexity in multiobjective programming difficulties and Kuhn-Tucker optimality stipulations are given for a multiobjecive programming challenge, Wolfe and Mond-Weir style twin types are given for a multiobjective programming challenge and ordinary duality effects are awarded in presence of invex features. Continuous-time multiobjective difficulties also are mentioned. Quadratic and fractional programming difficulties are given for invex capabilities. Symmetric duality effects also are given for scalar and vector cases.
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Extra info for Invexity and Optimization
We want to show that limn→∞ η(xn , y n ) = η(x, y). Three separate cases must be considered: (a) From the deﬁnition of η, we have η(x, y) = (f (x) − f (y))∇f (y) . ∇f (y)T ∇f (y) 36 2 Invex Functions (The Smooth Case) By continuity of f , there exists an N ∈ ℵ such that ∀n ∈ N, f (xn ) < f (y n ). Therefore, for n ≥ N, η(xn , y n ) = (f (xn ) − f (y n ))∇f (y n ) . ∇f (y n )T ∇f (y n ) by continuity of ∇f, limn→∞ η(xn , y n ) = η(x, y). (b) By hypothesis, η(x, y) = 0. Again, by continuity of f there exists an N ∈ ℵ such that ∀n ∈ N, f (xn ) > f (y n ), and thus η(xn , y n ) = 0.
Let x ≥ 0, y ≥ 0, in this case we have, for the preinvexity of f : −(y + λ(x − y)) ≤ λ(−x) + (1 − λ)(−y), relation which is obviously true for any λ ∈ [0, 1]. The same result holds, if x ≤ 0, y ≤ 0. Let now be x < 0 and y > 0 : we must have −|y + λ(y − x)| = −y − λ(y − x) ≤ λx − (1 − λ)y, ∀λ ∈ [0, 1]. 1) holds for all λ ∈ [0, 1]. , x > 0, y < 0. , λ(2y) ≤ 0. 1) holds for all λ ∈ [0, 1]. The function is therefore preinvex on R, but obviously it is not convex. Similarly to convex functions, it is possible to characterize preinvex functions in terms of invexity of their epigraph, however not with reference to the same function η (ﬁrst of all, one is an n-vector, the other is an (n + 1)-vector).
There is a sizable literature on pseudolinear functions; see for example Chew and Choo , Bianchi and Schaible , Bianchi et al. , Jeyakumar and Yang , Kaul et al. , Komlosi , Kortanek and Evans , Kruk and Wolkowicz , Martos , Mishra , Rapcsak . η-Pseudolinear functions have been introduced by Rueda  and studied in a more detailed way by Ansari et al. . 1. A diﬀerentiable function f deﬁned on an open set X ⊆ Rn is called η-pseudolinear if f and −f are pseudo-invex with respect to the same η.
Invexity and Optimization by Shashi Kant Mishra, Giorgio Giorgi (auth.)