Convergence Analysis of the Inexact Nonmonotone Projected Gradient Method for Vector Optimization Problems

In this paper, we put forward the inexact nonmonotone projected gradient algorithm for solving the smooth and convex vector optimization on the constrained set with regard to a fixed ordered cone. Under general assumptions, we demonstrate that the sequence produced by the algorithm has cluster points and this cluster point is a stationary point. If we consider that the objective function has <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$K$</tex> -convexity, then this cluster point is also a weakly efficient point. In addition, we use the property of quasi-Fejer convergence to prove that the full sequence produced by the method in our paper converges to a weakly efficient point.