G. C. Bento

Unconstrained Steepest Descent Method for Multicriteria Optimization on Riemannian Manifolds

In this paper, we present a steepest descent method with Armijo’s rule for multicriteria optimization in the Riemannian context. The sequence generated by the method is guaranteed to be well defined. Under mild assumptions on the multicriteria function, we prove that each accumulation point (if any) satisfies first-order necessary conditions for Pareto optimality. Moreover, assuming quasiconvexity of the multicriteria function and nonnegative curvature of the Riemannian manifold, we prove full convergence of the sequence to a critical Pareto point.

Proximal point method for a special class of nonconvex functions on Hadamard manifolds

In this article, we present the proximal point method for finding minima of a special class of nonconvex function on a Hadamard manifold. The well definedness of the sequence generated by the proximal point method is established. Moreover, it is proved that each accumulation point of this sequence satisfies the necessary optimality conditions and, under additional assumptions, its convergence for a minima is obtained.

Subgradient Method for Convex Feasibility on Riemannian Manifolds

In this paper, a subgradient type algorithm for solving convex feasibility problem on Riemannian manifold is proposed and analysed. The sequence generated by the algorithm converges to a solution of the problem, provided the sectional curvature of the manifold is non-negative. Moreover, assuming a Slater type qualification condition, we analyse a variant of the first algorithm, which generates a sequence with finite convergence property, i.e., a feasible point is obtained after a finite number of iterations. Some examples motivating the application of the algorithm for feasibility problems, nonconvex in the usual sense, are considered.

Finite termination of the proximal point method for convex functions on Hadamard manifolds

In this article, we proved that the sequence generated by the proximal point method, associated to a unconstrained optimization problem in the Riemannian context, has finite termination when the objective function has a weak sharp minima on the solution set of the problem.

Generalized Inexact Proximal Algorithms: Routine's Formation with Resistance to Change, Following Worthwhile Changes

This paper shows how, in a quasi-metric space, an inexact proximal algorithm with a generalized perturbation term appears to be a nice tool for Behavioral Sciences (Psychology, Economics, Management, Game theory,...). More precisely, the new perturbation term represents an index of resistance to change, defined as a “curved enough” function of the quasi-distance between two successive iterates. Using this behavioral point of view, the present paper shows how such a generalized inexact proximal algorithm can modelize the formation of habits and routines in a striking way. This idea comes from a recent “variational rationality approach” of human behavior which links a lot of different theories of stability (habits, routines, equilibrium, traps,...) and changes (creations, innovations, learning and destructions,...) in Behavioral Sciences and a lot of concepts and algorithms in variational analysis.

Generalized Proximal Distances for Bilevel Equilibrium Problems

We consider a bilevel problem involving two monotone equilibrium bifunctions and we show that this problem can be solved by a proximal point method with generalized proximal distances. We propose a framework for the convergence analysis of the sequence generated by the algorithm. This class of problems is very interesting because it covers mathematical programs and optimization problems under equilibrium constraints. As an application, we consider the problem of the stability and change dynamics of a leader-follower relationship in a hierarchical organization.

An Existence Result for the Generalized Vector Equilibrium Problem on Hadamard Manifolds

We present a sufficient condition for the existence of a solution to the generalized vector equilibrium problem on Hadamard manifolds using a version of the Knaster–Kuratowski–Mazurkiewicz Lemma. In particular, the existence of solutions to optimization, vector optimization, Nash equilibria, complementarity, and variational inequality problems is a special case of the existence result for the generalized vector equilibrium problem.

A New Approach to the Proximal Point Method: Convergence on General Riemannian Manifolds

In this paper, we present a new approach to the proximal point method in the Riemannian context. In particular, without requiring any restrictive assumptions about the sign of the sectional curvature of the manifold, we obtain full convergence for any bounded sequence generated by the proximal point method, in the case that the objective function satisfies the Kurdyka–Lojasiewicz inequality. In our approach, we extend the applicability of the proximal point method to be able to solve any problem that can be formulated as the minimizing of a definable function, such as one that is analytic, restricted to a compact manifold, on which the sign of the sectional curvature is not necessarily constant.

Iteration-Complexity of Gradient, Subgradient and Proximal Point Methods on Riemannian Manifolds

This paper considers optimization problems on Riemannian manifolds and analyzes the iteration-complexity for gradient and subgradient methods on manifolds with nonnegative curvatures. By using tools from Riemannian convex analysis and directly exploring the tangent space of the manifold, we obtain different iteration-complexity bounds for the aforementioned methods, thereby complementing and improving related results. Moreover, we also establish an iteration-complexity bound for the proximal point method on Hadamard manifolds.

Proximal point method for a special class of nonconvex multiobjective optimization functions

The proximal point method for a special class of nonconvex multiobjective functions is studied in this paper. We show that the method is well defined and that the accumulation points of any generated sequence, if any, are Pareto–Clarke critical points. Moreover, under additional assumptions, we show the full convergence of the generated sequence.