Nonmonotone trust region algorithm for solving the unconstrained multiobjective optimization problems

In this work an iterative method to solve the nonlinear multiobjective problem is presented. The goal is to find locally optimal points for the problem, that is, points that cannot simultaneously improve all functions when we compare the value at the point with those in their neighborhood. The algorithm uses a strategy developed in previous works by several authors but globalization is obtained through a nonmonotone technique. The construction of a new ratio between the actual descent and predicted descent plays a key role for selecting the new point and updating the trust region radius. On the other hand, we introduce a modification in the quadratic model used to determine if the point is accepted or not, which is fundamental for the convergence of the method. The combination of this strategy with a Newton-type method leads to an algorithm whose convergence properties are proved. The numerical experimentation is performed using a known set of test problems. Preliminary numerical results show that the nonmonotone method can be more efficient when it is compared to another algorithm that use the classic trust region approach.