By Mishra Shashi Kant Et.Al
This quantity summarizes and synthesizes a facet of study paintings that has been performed within the region of Generalized Convexity over the last few a long time. particularly, the publication specializes in V-invex features in vector optimization that experience grown out of the paintings of Jeyakumar and Mond within the 1990’s. The authors combine comparable learn into the e-book and show the vast context from which the world has grown and maintains to develop.
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Extra info for V-Invex Functions and Vector Optimization (Springer Optimization and Its Applications)
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M , x ∈ X . The following lemma from Singh and Hanson (1991) connects the conditionally properly efficient solutions of (FP) and (FP )V ' . 1 [Singh and Hanson (1991)]: Let x * be conditionally properly efficient solution of (FP). Then there exists v * ∈ R+p such that x * is conditionally properly efficient solution of (FP )V * . Conversely, if x * is conditionally properly efficient solution of (FP )V * where vi* = ( ) ( ) f i x* , i = 1, 2 , ... , p , gi x* then x * is conditionally properly efficient solution for (FP).
35), we get ∑τ f (u )η (x , u ) ≥ 0 . p 0 i i =1 Since τ ≥ 0 , p ∑τ i =1 i 0 0 i = 1, we have ∑τ (∇ f (u ) + B z )η (x , u ) ≥ 0 . 40) 32 Chapter 2: V-Invexity in Nonlinear Multiobjective Programming ∑ α i (x0 , u )τ i ( f i (x0 ) + xoT Bi z i ) ≥ ∑ α i (x0 , u )τ i ( f i (u ) + u T Bi z i ). p p i =1 i =1 Since α i ( x0 , u ) > 0 , ∀ i = 1, ... , p , we have ∑τ ( f (x ) + x p i =1 i i 0 T o ) p ( ) Bi z i ≥ ∑τ i f i (u ) + u T Bi z i . i =1 Since the objective values of (NDVP) and (NDVD) are equal, we have ⎛ τ i ⎜⎜ f i (x0 ) + xoT Bi x0 ∑ i =1 ⎝ ( p ) 1 2 ⎞ p ⎛ ⎟⎟ ≥ ∑τ i ⎜⎜ f i ( x ) + x T Bi x ⎠ i =1 ⎝ ( ) 1 2 ⎞ ⎟⎟ .
7) p m i =1 j =1 *T ( ) ( ) τ , v , w ≥ 0, * * * p ∑τ i =1 * i = 1. 8) The following necessary optimality criteria for a feasible point x * of (MFP) to be conditionally properly efficient can be proved on similar lines as that of Theorem 2 of Weir (1988). 1: Let x* be a conditionally properly efficient solution for (MFP). Assume that there exists x ∈ X such that h j ( x ) < 0, ( ) { } for j = 1,… , m and for j ∈ I x * = j : h j ( x ) = 0 any one of the following conditions holds (i) h j is V − invex (ii) h j is V − pseudo-invex on X with respect to η and α i > 0 , i = 1, ...