A Projected Gradient Method for Vector Optimization Problems

Hybrid Approximate Proximal Method with Auxiliary Variational Inequality for Vector Optimization

“…Meantime, they admitted the possibility that F : X ---t Y takes the value ooc (this is made precise in Section 2), where X is a Hilbert space and C is a closed, convex, and pointed cone in Y with intC "/= 0, where intC denotes the interior of the set C. Such extensions can be traced back to the fashion of extension which existed in a finite-dimensional setting. For example, in nn, see the steepest descent method for multiobjective optimization [2], the same method for general finite-dimensional vector optimization [3], and the projected gradient method for convexly constrained vector optimization [4].…”

“…As pointed out in [4], the set argmin{ (G(x), z)lx E S} in Theorem 2.1 may be empty for some z E c+ \ {0}.…”

“…Because of its relaxed requirement, the inexact proximal point algorithm is more practical than the exact one. Thus, it has been studied widely and various forms of the method have been developed; see, e.g., [4,[7][8][9][10][11][12]. In most of these papers, the conditions ensuring that the error term being summable is an essential condition for the convergence of the method.…”

“…The set of weakly efficient solutions of the VOP is denoted by VO(Sl, F). Let Pn be the metric projection from X onto n. As noted in [4], any constrained vector optimization problem (CVOP):…”

Generalized viscosity approximation methods in multiobjective optimization problems

“…Because of these applications, a lot of literature have been published to study optimality conditions, duality theories and topological properties of solutions of multiobjective optimization problems (see, e.g., [9,18,20,25] and the references therein). Recently some numerical methods for solving convex multiobjective optimization problems have been proposed in following papers: the steepest descent method for multiobjective optimization was dealt with in [14], the extensions of the projective gradient method to the case of convex constrained vector optimization can be found in [16,17]. Bonnel et al [5] constructed a vector-valued proximal point algorithm to investigate convex vector optimization problem in Hilbert space, they generalized the famous Rockafellar's results [23] from scalar case to vector case.…”