By Shashi Kant Mishra, Giorgio Giorgi (auth.)
Invexity and Optimization provides effects on invex functionality and their homes in tender and nonsmooth instances, pseudolinearity and eta-pseudolinearity. effects on optimality and duality for a nonlinear scalar programming challenge are offered, moment and better order duality effects are given for a nonlinear scalar programming challenge, and saddle aspect effects also are provided. Invexity in multiobjective programming difficulties and Kuhn-Tucker optimality stipulations are given for a multiobjecive programming challenge, Wolfe and Mond-Weir kind twin versions are given for a multiobjective programming challenge and traditional duality effects are provided in presence of invex capabilities. Continuous-time multiobjective difficulties also are mentioned. Quadratic and fractional programming difficulties are given for invex features. Symmetric duality effects also are given for scalar and vector cases.
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Extra resources for Invexity and Optimization
This last inequality together with quasi-invexity of f imply η(y, x)T ∇f (x) ≤ 0. As η is skew symmetric we have also η(x, y)T ∇f (x) ≥ 0. Hence ∇f is quasi-invex monotone. The converse of the above result is not true, as can be seen by the following example. Consider f : R → R deﬁned by f (x) = x3 . Hence ∇f (x) is quasiinvex monotone with respect to η(x, y) = sin x−sin y. , x = π/3, y = π/6). Indeed f is not invex for η. The necessary conditions for generalized monotonicity given by RuizGarzon et al.
F (y + λη(x, y)) ≤ λf (x) + (1 − λ)f (y), ∀x, y ∈ X, ∀λ ∈ [0, 1].
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. Obviously every pseudolinear function is η-pseudolinear with η(x, y) = (x − y) but the converse is not true (see a counter example in ). The next two deﬁnitions will be considered again in the next chapter. 2 (Mohan and Neogy ). For a given η : K × K → Rn a non-empty set K ⊆ Rn is called η-convex or simply invex, if for each x, y ∈ K, 0 ≤ λ ≤ 1, y + λη(x, y) ∈ K.