Linear convergence of a nonmonotone projected gradient method for multiobjective optimization

A nonmonotone gradient method for constrained multiobjective optimization problems

“…Recently, we also used this assumption to show the convergence of another nonmonotone projected gradient method [11]. Next, by applying the assumption (A1) again, we study the convergence of Algorithm 3.1 considered in the present paper.…”

“…is the average of all the previous function values. Thus, similar to the selection of the parameters involved in other nonmonotone linear search rules ( [11,33]), the choice of η k controls the degree of the non-monotonicity of the line search (3.2) (the line search closely approximates the monotone one as η k approaches 0, and the scheme becomes more nonmonotone as η k approaches 1). Now, we illustrate the validity of the stopping criteria in Algorithm 3.1.…”

“…In view of its extensive applications, the research on the numerical algorithms for solving multiobjective optimization problems has received a lot of attention and many iterative methods have been proposed, including projected gradient method [9,10,11], steepest descent method [12,13], proximal point method [14,15,16], conjugate gradient method [17], Newton method [18,19], trust-region method [20,21] and so on.…”

“…Recently, nonmonotone line search techniques have also been applied to multiobjective optimization [11,30,31,32]. Particularly, Fazzio and Schuverdt [33] extended the nonmonotone line search rule in [29] for scalar optimization to the multiobjective optimization and changed the Armijo search rule (1.2) as follows: given γ ∈ (0, 1),…”

“…They showed the stationarity of the accumulation points of the sequences generated by the proposed algorithm, and then established the convergence to weak Pareto points when the function F is convex. Very recently in [11], by adopting the nonmonotone line search rule (1.4), we considered a projected gradient method with exogenously given square summable sequence in the computation of the search direction for the same multiobjective optimization problem when F is convex. We proved the convergence of the full sequence generated by the algorithm to a weak Pareto optimal point.…”