Stochastic method for the solution of unconstrained vector optimization problems (Q700771)

This paper presents a new stochastic algorithm for the solution of unconstrained vector optimization problems based on a special class of stochastic differential equations. The numerical treatment of these stochastic differential equations leads to algorithm for the computation of Pareto-optimal solutions of the problem \[ (\text{VOP})\qquad \min_{x\in\mathbb{R}^m} (f(x)), \] where \(f: \mathbb{R}^n\to \mathbb{R}^m\) is a twice-continuously differentiable vector valued function. Here a partial order is defined on \(\mathbb{R}^m\) as the following. A vector \(u= (u_i)\) is called less or equal to \(v= (v_i)\) if \(u_i\leq v_i\) for all \(i\), and a point \(\widehat x\) is called a Pareto optimal solution if there exists no \(w\) with \(f(w)\neq f(\widehat x)\) and \(f(w)\leq f(\widehat x)\).