We show the validity of select existence results for a vector optimization problem, and a variational inequality. More generally, we consider generalized vector quasi-variational inequalities, as well as, fixed point problems on genuine Hadamard manifolds. Article Download View Existence Results for Particular Instances of the Vector Quasi-Equilibrium Problem on Hadamard Manifolds
We present a preconditioning of a generalized forward-backward splitting algorithm for finding a zero of a sum of maximally monotone operators \sum_{i=1}^n A_i + B with B cocoercive, involving only the computation of B and of the resolvent of each A_i separately. This allows in particular to minimize functionals of the form \sum_{i=1}^n g_i + … Read more
In this paper we present two inexact proximal point algorithms to solve minimization problems for quasiconvex objective functions on Hadamard manifolds. We prove that under natural assumptions the sequence generated by the algorithms are well defined and converge to critical points of the problem. We also present an application of the method to demand theory … Read more
We describe a Jordan-algebraic version of E. Lieb convexity inequalities. A joint convexity of Jordan-algebraic version of quantum entropy is proven. SA spectral theory on semi-simple complex Jordan algebras is used as atool to prove the convexity results. Possible applications to optimization and statistics are indicated Citation Preprint, University of Notre Dame, August 2014 Article … Read more
This article proposes a new algorithm for solving a class of composite convex-concave saddle-point problems. The new algorithm is a special instance of the hybrid proximal extragradient framework in which a Nesterov’s accelerated variant is used to approximately solve the prox subproblems. One of the advantages of the new method is that it works for … Read more
In this article we provide an erratum to Proposition 3.4 of E.A. Papa Quiroz and P.R. Oliveira. Proximal Point Methods for Quasiconvex and Convex Functions with Bregman Distances on Hadamard Manifolds, Journal of Convex Analysis 16 (2009), 49-69. More specifically, we prove that the function defined by the product of a fixed vector in the … Read more
In this paper we propose a scalarization proximal point method to solve multiobjective unconstrained minimization problems with locally Lipschitz and quasiconvex vector functions. We prove, under natural assumptions, that the sequence generated by the method is well defined and converges globally to a Pareto-Clarke critical point. Our method may be seen as an extension, for … Read more
Within a nonzero, real Banach space we study the problem of characterising a maximal extension of a monotone operator in terms of minimality properties of representative functions that are bounded by the Penot and Fitzpatrick functions. We single out a property of this space of representative functions that enable a very compact treatment of maximality … Read more
Vector optimization problems are a generalization of multiobjective optimization in which the preference order is related to an arbitrary closed and convex cone, rather than the nonnegative octant. Due to its real life applications, it is important to have practical solution approaches for computing. In this work, we consider the inexact projected gradient-like method for … Read more
This article considers the two-player composite Nash equilibrium (CNE) problem with a separable non-smooth part, which is known to include the composite saddle-point (CSP) problem as a special case. Due to its two-block structure, this problem can be solved by any algorithm belonging to the block-decomposition hybrid proximal-extragradient (BD-HPE) framework. The framework consists of a … Read more