A Newton-like method for variable order vector optimization problems (Q1637359)

The convexity and strong convexity with respect to a mapping are introduced in the class of vector functions defined on an \(n\)-dimensional non-empty open and convex set. The relation between the critical points of a convex, respectively strongly convex vector function and weakly efficient points, respectively efficient points is studied. Newton's method is extended for minimizing a strongly convex vector function with respect to a variable order, using a quadratic approximation of the scalarized convex function. The existence and the uniqueness of the solution is proved and properties of Newton's direction are obtained: the characterization of efficiency, the continuity of the direction, upper bounds and the descent of the objective functions. The algorithm converges. The convergence rate of the sequence generated by Newton's method for a variable order vector optimization is analyzed.