Generalized Convexity/Monoticity

Relationships between constrained and unconstrained multi-objective optimization and application in location theory

  • Christian Günther
  • Christiane Tammer
    • Categories Constrained Nonlinear Optimization, Generalized Convexity/Monoticity, Multi-Criteria Optimization

    This article deals with constrained multi-objective optimization problems. The main purpose of the article is to investigate relationships between constrained and unconstrained multi-objective optimization problems. Under suitable assumptions (e.g., generalized convexity assumptions) we derive a characterization of the set of (strictly, weakly) efficient solutions of a constrained multi-objective optimization problem using characterizations of the sets … Read more

    How to Reach his Desires: Variational Rationality and the Equilibrium Problem on Hadamard Manifolds

  • Glaydston Bento
  • João Xavier da Cruz Neto
  • Antoine Soubeyran
  • Pedro A. Soares Jr.
    • Categories Generalized Convexity/Monoticity Tags , , , , ,

    In this paper we present a sufficient condition for the existence of a solution for an \mbox{equilibrium} problem on an Hadamard manifold and under suitable assumptions on the sectional curvature, we \mbox{propose} a framework for the convergence analysis of a proximal point algorithm to solve this equilibrium \mbox{problem}. Finally, we offer an application to the … Read more

    Existence Results for Particular Instances of the Vector Quasi-Equilibrium Problem on Hadamard Manifolds

  • Glaydston Bento
  • João Xavier da Cruz Neto
    • Categories Generalized Convexity/Monoticity

    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. ArticleDownload View PDF

    Preconditioning of a Generalized Forward-Backward Splitting and Application to Optimization on Graphs

  • Loïc Landrieu
  • Hugo Raguet
    • Categories Convex and Nonsmooth Optimization, Generalized Convexity/Monoticity

    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

    Inexact Proximal Point Methods for Quasiconvex Minimization on Hadamard Manifolds

  • Nancy Baygorrea
  • Nelson Maculan
  • Erik Alex Papa Quiroz
    • Categories Generalized Convexity/Monoticity, Global Optimization, Nonsmooth Optimization Tags , , , ,

    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

    E. Lieb convexity inequalities and noncommutative Bernstein inequality in Jordan-algebraic setting

  • Leonid Faybusovich
    • Categories Generalized Convexity/Monoticity, Other Topics Tags , ,

    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 CitationPreprint, University of Notre Dame, August 2014ArticleDownload View PDF

    An accelerated HPE-type algorithm for a class of composite convex-concave saddle-point problems

  • Yunlong He
  • Renato D.C. Monteiro
    • Categories Convex Optimization, Generalized Convexity/Monoticity, Nonsmooth Optimization Tags , , , , , , ,

    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

    About the Convexity of a Special Function on Hadamard Manifolds.

  • João Xavier da Cruz Neto
  • I.D. Melo
  • J.P. Silva
  • P.A. Sousa
    • Categories Generalized Convexity/Monoticity

    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

    A Scalarization Proximal Point Method for Quasiconvex Multiobjective Minimization

  • H. C. F. Apolinário
  • Paulo Roberto Oliveira
  • Erik Alex Papa Quiroz
    • Categories Generalized Convexity/Monoticity, Multi-Criteria Optimization, Nonsmooth Optimization Tags , , , , ,

    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

    On the Maximal Extensions of Monotone Operators and Criteria for Maximality

  • Andrew C. Eberhard
  • R Wenczel
    • Categories Convex and Nonsmooth Optimization, Generalized Convexity/Monoticity Tags ,

    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