Invexity and Optimization by Shashi Kant Mishra, Giorgio Giorgi (auth.)

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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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 [3]). The next two definitions will be considered again in the next chapter. 2 (Mohan and Neogy [165]). 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.

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