We propose an extension of Newton's method for unconstrained multiobjective optimization (multicriteria optimization). This method does not use a priori chosen weighting factors or any other form of a priori ranking or ordering information for the different objective functions. Newton's direction at each iterate is obtained by minimizing the max-ordering scalarization of the variations on the quadratic approximations of the objective functions. The objective functions are assumed to be twice continuously differentiable and locally strongly convex. Under these hypotheses, the method, as in the classical case, is locally superlinear convergent to optimal points. Again as in the scalar case, if the second derivatives are Lipschitz continuous, the rate of convergence is quadratic. Our convergence analysis uses a Kantorovich-like technique. As a byproduct, existence of optima is obtained under semilocal assumptions.

A number of first-order methods have been proposed for smooth multiobjective optimization for which some form of convergence to first-order criticality has been proved. Such convergence is global in the sense of being independent of the starting point. In this paper, we analyse the rate of convergence of gradient descent for smooth unconstrained multiobjective optimization, and we do it for non-convex, convex, and strongly convex vector functions. These global rates are shown to be the same as for gradient descent in single-objective optimization and correspond to appropriate worst-case complexity bounds. In the convex cases, the rates are given for implicit scalarizations of the problem vector function.

We propose a method for constrained and unconstrained nonlinear multiobjective optimization problems that is based on an SQP-type approach. The proposed algorithm maintains a list of nondominated p...