Dr. Shashi Kant Mishra, Prof. Shou-Yang Wang, Prof. Kin's Generalized Convexity and Vector Optimization PDF

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If G is finite-dimensional, then h is strongly compact Lipschitian at x¯ if and only if it is locally Lipschitz near x. ¯ If h is strongly compact Lipschitzian, then for all u∗ ∈ G∗ , (u∗ ◦ f )(x) =< u∗ , h(x) > is locally Lipschitz. From now on, let Q ⊂ F and K ⊂ G denote pointed closed convex cones with nonempty interior; let Q∗ , K ∗ be their respective dual cones. The cone Q induces a partial order ≤ on F given by z ≤z z ≤z if z–z ∈ Q if z–z ∈ int Q z ≥ z is the negation of the first one of the above relations and z > z is the negation of the second one of the above relations.

Let S be a nonempty set in Rn and ψ : S → R p be a preinvex function on S. Then either ψ (x) < 0 has a solution x ∈ S, or λ T ψ (x) 0 for all x ∈ S, or some λ ∈ Rm + , but both alternatives are never true. 3. (Fritz John necessary optimality condition). Let x¯ be a weak Pareto efficient solution for (P). Suppose that gj is continuous for j ∈ J˜(x), ¯ f and g are ¯ η (x, x)), ¯ and gJ(x) (x, ¯ η (x, x)) ¯ pre-invex directionally differentiable at x¯ with f (x, k m ¯ ¯ ¯ ¯ ξ , μ¯ satisfies functions of x on X.

3. We say the problem (VP) is of weak quasistrictly pseudo type I univex b0 , b1 , φ0 , φ1 and η at a ∈ X0 if there exist a real-valued function b0 , b1 , φ0 , φ1 and η such that b0 (x, a) φ0 [ f (x) − f (a)] ≤ 0 ⇒ (∇ f (a)) η (x, a) − b1 (x, a) φ1 [g (a)] 0, 0 ⇒ (∇g (a)) η (x, a) ≤ 0, for every x ∈ X0 and for all i = 1, . . , p, and j = 1, . . , m. If (VP) is of weak quasistrictly pseudo type I univex at each a ∈ X , we say (VP) is of weak quasistrictly pseudo type I univex on X. 3. The function f : R2 → R2 and g : R2 → R defined by f (x) = x31 x21 + 1 , x22 (x2 − 1)3 and g (x) = (2x1 − 4) e−x2 , (x1 + x2 − 2) x21 + 2x1 + 4 are weak quasistrictly pseudo type I univex with respect to b0 = 1 = b1 , φ0 , and φ1 are identity function on R and η (x, a) = (x1 , x2 (1 − x2)) at a = (0, 0), but ( f , g) are not type I univex with respect to same b0 , b1 , φ0 , φ1 and η as can be seen by taking x = (1, 0).

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