Efficiency and duality for nonlinear multiobjective programs involving \(n\)-set functions (Q1323895)

The paper considers some duality relationships between the following multiobjective nonlinear programming problem (MOP) and Wolfe-type vector dual multiobjective programming problem (MOD): \[ \text{(MOP)}\quad\text{Minimize}\quad F(S_ 1,\dots,S_ n)= [F_ 1(S_ 1,\dots, S_ n),\dots, F_ k(S_ 1,\dots, S_ n)] \] \[ \text{subject to}\quad G_ j(S_ 1,\dots, S_ n)\leq 0\;(j= 1,\dots,m)\;(S_ 1,\dots, S_ n)\in A^ n, \] where \(A^ n\) is the \(n\)-fold product of \(\sigma\)-algebra \(A\) of subset of a given set \(X\), \(F_ i\), \(i= 1,\dots,k\), and \(G_ j\), \(j= 1,\dots,m\) are real valued functions defined on \(A^ n\); \[ \begin{multlined}\text{(MOD)}\quad\text{Maximize}\quad \phi[(T_ 1,\dots, T_ n), u,\tau]=\\=\left[F_ 1(T_ 1,\dots, T_ n)+ \sum^ m_{j= 1} u_ j G_ j(T_ 1,\dots, T_ n),\dots, F_ k(T_ 1,\dots, T_ n)+ \sum^ m_{j=1} u_ j G_ j (T_ 1,\dots, T_ n)\right]\end{multlined} \] subject to \(\left\langle \sum^ k_{r=1} \tau_ r D_ i F_{r T_ 1\cdots T_ n}+\sum^ m_{j=1} u_ j D_ i G_{j T_ 1\cdots T_ n}, I_{S_ i}- I_{T_ i}\right\rangle\geq 0\) for all \(S_ i\in A\), \(i= 1,\dots, n\), \((T_ 1,\dots, T_ n)\in A^ n\), \(u\in \mathbb{R}^ m_ +\), \(\tau_ r\geq 0\) \((r= 1,\dots, k)\), \(\sum^ k_{r=1} \tau_ r= 1\). Moreover, the paper considers some duality relationships between the problem (MOP) and a general Mond-Weir-type vector dual problem. Also, weak duality theorems and strong duality theorems are obtained for convex functions, \(\rho\)- convex functions, and \(\rho\)-pseudoconvex functions.