Augmented Lagrangian cone method for multiobjective optimization problems with an application to an optimal control problem

This paper proposes an augmented Lagrangian method to compute Pareto optimal sets of multiobjective optimization problems. The method neither requires a prior information about the locations of the Pareto surface nor the convexity of the objective and constraint functions. To generate Pareto optimal points, we convert a multiobjective optimization problem into a set of direction-based parametric scalar optimization problems by using the cone method. Subsequently, we apply the augmented Lagrangian method to the direction-based parametric problems to transform them into unconstrained problems. Transformed augmented Lagrangian subproblems are then solved by the steepest descent method with a max-type nonmonotone line search method. A step-wise algorithmic implementation of the proposed method is provided. We discuss the convergence property of the proposed algorithm with regard to a feasibility measure and the global Pareto optimality. Under a few common assumptions, we prove that any subsequential limit of the sequence generated by the proposed algorithm is the global minimizer of an infeasibility measure corresponding to each direction. In addition, the obtained limit is found to be a global minimizer when the feasible region of the given multiobjective optimization problem is nonempty. It is observed that the solution of the proposed method is not affected by variable scaling. The efficiency of the proposed algorithm is shown by solving standard test problems. As a realistic application, we employ the proposed method on a deterministic unemployment optimal control model with the implementation of government policies to create employment and vacancies as their controls.