Optimality conditions for vector mathematical programming via a theorem of the alternative
The paper is concerned with the optimality conditions for the following nonsmooth and nonconvex vector mathematical programming: \[ \begin{matrix} &\qquad &\qquad {\l}\\ (P) & \text{Minimize} & f(x)=(f_1(x), \ldots, f_q(x)) \\ &\text{subject to} & g_{i}(x)\leq{0}, \quad i=1, \ldots, m\\ & & x\in{\Omega}, \end{matrix} \] where \(f:R^{N}\rightarrow{R^q}\), \(g:R^{N}\rightarrow{R^m}\) with \(g=(g_1, \ldots, g_m)\) and \(\Omega\) is a nonempty subset of \(R^N\). By using a generalization of the Gordan alternative theorem given by \textit{V. Jeyakumar} [J. Optim. Theory Appl. 48, 525-533 (1986; Zbl 0578.49008)], the author proves the Lagrange multipliers type of optimal condition for problem~\((P)\) under the assumption of the existence and cone-subconvexlikeness of the Hadamard directional derivatives of objective and constraint functions. Based on convexity on \(f_i\), \(g_j\), and \(\Omega\), the sufficient optimality conditions of problem~\((P)\) is established, too. Moreover, the paper gives an application to the following vector fractional mathematical programming problem: \[ \begin{matrix} &\qquad &\qquad {\l}\\ (FP) & \text{Minimize} & \left(f_1(x)/g_1(x), \ldots, f_q(x)/g_q(x)\right) \\ &\text{subject to} & x\in{\Omega}, \quad h_{i}(x)\leq{0}, \quad i=1, \ldots, m, \end{matrix} \] where \(f_i:R^{N}\rightarrow{R}\) (\(i=1, \ldots, q\)), \(h_j:R^{N}\rightarrow{R}\) (\(j=1, \ldots, m\)) are continuous, convex, and finite at \(\bar{x}\in{S}\), and \(g_i:R^{N}\rightarrow{R}\) (\(i=1, \ldots, q\)) are continuous, concave, finite at \(\bar{x}\), and positive on \(S\).
nonconvex vector mathematical programming
vector fractional mathematical programming
scientific article; zbMATH DE number 1289958