| Multiobjective optimization refers to the process of optimizing several objective functions simultaneously in the feasible region.Usually,for nontrivial problems,it is almost impossible to find a point to optimize all given objective functions at the same time.Therefore,the concept of optimality is replaced by the concept of Pareto optimality(or Pareto efficiency).Nowadays,multiobjective optimization problems have been widely used in many fields,such as economy,industry,agriculture and so on,specifically including engineering,statistics,management science,environmental analysis,design and others.Due to its wide application,the research of multiobjective optimization problems has attracted extensive attention of scholars,and has made rich research results in the existence of solutions,the relationship between solutions under different definitions,Farkas-type results,optimality conditions and numerical optimization algorithms.In this paper,we mainly study the numerical optimization algorithm for constrained multiobjective optimization problems.In this paper,we propose a projected gradient method for constrained multiobjective optimization problem.This method calculates the search direction by using a variable steplength instead of a fixed parameter.Further,we give the convergence analysis of the algorithm in the case when the multiobjective function is nonconvex,respectively,convex.Specifically,when the multiobjective function is nonconvex,we show that the accumulation point of the sequence generated by the algorithm is a Pareto stationary point of the problem.When the multiobjective function is quasiconvex,we prove that the sequence generated by this method converges to a Pareto stationary point of the problem.The convergence of the sequence generated by this method to a weak Pareto optimal point of the problem is obtained in the case when the multiobjective function is pseudoconvex.Finally,when the multiobjective function is convex,by imposing some approximate conditions on the gradients of the objective functions and the search direction,we obtain the linear convergence result of the algorithm. |
