Duality for nondifferentiable static multiobjective variational problems involving generalized \((F,\rho)\)-convex functions
Consider the following multiobjective continuous programming problem: \[ \begin{multlined}\text{Minimize}\quad \Biggl(\int^b_a f^1(t, x(t),\dot x(t))+ (x(t)^T B_1(t)x(t))^{1/2}dt,\dots,\\ \int^b_a f^p(t, x(t),\dot x(t))+ (x(t)^T B_p(t)x(t))^{1/2}dt\Biggr),\end{multlined} \] \[ \text{Subject to }x(a)=\alpha,\quad x(b)=\beta,\quad g(t,x,\dot x)\geq 0, \] where \(f^i: I\times \mathbb{R}^n\times \mathbb{R}\), \(i= 1,2,\dots,p\), \(g: I\times \mathbb{R}^n\times \mathbb{R}^n\mapsto \mathbb{R}^m\) are assumed to be continuously differentiable functions, and for each \(t\in I\), \(i\in P= \{1,\dots, p\}\), \(B_i(t)\) is as \(n\times n\) positive semidefinite (symmetric) matrix, with \(B(\cdot)\) continuous on \(I\). Wolfe and Mond-Weir type dual problems are formulated and weak duality theorems established under generalized \((F,\rho)\) convexity conditions. (The function \(\Phi\) is said to be \((F,\rho)\)-convex at \(x^0\in X\) if for all \(x\in X\), \[ \begin{multlined} \int^b_a \{\Phi(t,x,\dot x)- \Phi(t,x^0,\dot x^0)\}dt\geq \int^b_a F\Biggl(t, x,\dot x, x^0,\dot x^0; \Phi_x(t, x^0,\dot x^0)-\\ {d\over dt} (\Phi_{\dot x}(t, x^0,\dot x^0))\Biggr) dt+ \rho \int^b_a d^2(t, x,x^0)dt.\end{multlined} \] The results are then extended to a multiobjective variational nondifferentiable fractional programming problem. The use of the word `static' in the title, abstract and introduction seems inappropriate.
multiobjective variational nondifferentiable fractional programming
scientific article; zbMATH DE number 915157