## On generalised convex mathematical programming.(English)Zbl 0773.90061

In this paper, $$V$$-minimization means vector minimization and the term “weak minimum” is used in the sense introduced by T. Weir, the second author and B. D. Craven [Optimization 17, 711-721 (1986; Zbl 0606.49011)]. The authors consider the problem (VP) of $$V$$-minimizing $$(f_ 1(x),\dots,f_ p(x))$$ subject to $$g(x)\leq 0$$ where $$f_ i: X_ 0\to R$$, $$g: X_ 0\to R^ m$$ are differentiable functions, $$X_ 0$$ is an open set in $$R^ n$$, and their goal is to find the set of weak minimum points. A vector function $$f: X_ 0\to R^ n$$ is called $$V$$- invex if there exist functions $$\mu: X_ 0\times X_ 0\to R^ b$$ and $$\alpha_ i: X_ 0\times X_ 0\to R_ +\backslash\{0\}$$ such that for each $$x$$, $$a\in X_ 0$$, and for $$i=1,2,\dots,p$$, $$f_ i(x)f_ i(a)- \alpha_ i(x,a)f_ i(a)\eta(x,a)\geq 0$$. The problem (VP) is called $$V$$- invex if the vector function $$(f_ i,\dots,f_ p, g_ 1,\dots,g_ m)$$ is $$V$$-invex. The concept of $$V$$-invexity is then generalized to yield the concepts of $$V$$-pseudo-invexity and $$V$$-quasi-invexity. For the problem (VP), sufficient conditions for weak global optimality are then obtained, the dual problem (VD) is introduced, and a weak and a strong duality theorems are proved under the assumptions involving $$V$$-pseudo- invexity for the vector objective function and $$V$$-quasi-invexity of a function of the form $$(\lambda_ 1 g_ 1,\dots,\lambda_ m g_ m)$$ (where all $$\lambda_ i\geq 0)$$. One section deals with nonconvex multi- objective fractional programming. The last section (conclusion) summarizes the results, points some possible extensions and announces further results to be discussed in another paper.
Reviewer: J.Abrham (Toronto)

### MSC:

 90C29 Multi-objective and goal programming 90C25 Convex programming 90C32 Fractional programming 90C26 Nonconvex programming, global optimization

Zbl 0606.49011
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