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)


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


Zbl 0606.49011
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