Generalized Convexity/Monoticity

A fundamental proof to convergence analysis of alternating direction method of multipliers for weakly convex optimization

  • Tao Zhang
  • Shen Zhengwei
    • Categories Generalized Convexity/Monoticity, Global Optimization

    The convergence analysis of the alternating direction method of multipliers (ADMM) methods to convex/nonconvex combinational optimization have been well established in literature. Consider the extensive applications of weakly convex function in signal processing and machine learning(e.g. \textit{Special issue: DC-Theory, Algorithms and Applications, Mathematical Programming, 169(1):1-336,2018}), in this paper, we investigate the convergence analysis of ADMM … Read more

    Pareto efficient solutions in multi-objective optimization involving forbidden regions

  • Christian Günther
    • Categories Facility Planning and Design, Generalized Convexity/Monoticity, Multi-Criteria Optimization

    In this paper, the aim is to compute Pareto efficient solutions of multi-objective optimization problems involving forbidden regions. More precisely, we assume that the vector-valued objective function is componentwise generalized-convex and acts between a real topological linear pre-image space and a finite-dimensional image space, while the feasible set is given by the whole pre-image space … Read more

    Inner Conditions for Error Bounds and Metric Subregulerity of Multifunctions

  • Dominique Azé
    • Categories Convex and Nonsmooth Optimization, Generalized Convexity/Monoticity, Nonlinear Optimization Tags , , , , ,

    We introduce a new class of sets, functions and multifunctions which is shown to be large and to enjoy some nice common properties with the convex setting. Error bounds for objects attached to this class are characterized in terms of inner conditions of Abadie’s type, that is conditions bearing on normal cones and coderivatives at … Read more

    A Note on the Forward-Douglas–Rachford Splitting for Monotone Inclusion and Convex Optimization

  • Hugo Raguet
    • Categories Biomedical Applications, Convex Optimization, Generalized Convexity/Monoticity

    We shed light on the structure of the “three-operator” version of the forward-Douglas–Rachford splitting algorithm for finding a zero of a sum of maximally monotone operators $A + B + C$, where $B$ is cocoercive, involving only the computation of $B$ and of the resolvent of $A$ and of $C$, separately. We show that it … Read more

    A Hausdorff-type distance, a directional derivative of a set-valued map and applications in set optimization

  • Xuan Duc Ha Truong
    • Categories Generalized Convexity/Monoticity, Multi-Criteria Optimization Tags , , , ,

    In this paper, we follow Kuroiwa’s set approach in set optimization, which proposes to compare values of a set-valued objective map $F$ respect to various set order relations. We introduce a Hausdorff-type distance relative to an ordering cone between two sets in a Banach space and use it to define a directional derivative for $F$. … Read more

    On generalized-convex constrained multi-objective optimization

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

    In this paper, we consider multi-objective optimization problems involving not necessarily convex constraints and componentwise generalized-convex (e.g., semi-strictly quasi-convex, quasi-convex, or explicitly quasi-convex) vector-valued objective functions that are acting between a real linear topological pre-image space and a finite dimensional image space. For these multi-objective optimization problems, we show that the set of (strictly, weakly) … Read more

    An Inexact Proximal Method with Proximal Distances for Quasimonotone Equilibrium Problems

  • Lennin Mallma Ramirez
  • Paulo Roberto Oliveira
  • Erik Alex Papa Quiroz
    • Categories Generalized Convexity/Monoticity

    In this paper we propose an inexact proximal point method to solve equilibrium problem using proximal distances and the diagonal subdi erential. Under some natural assumptions on the problem and the quasimonotonicity condition on the bifunction, we prove that the sequence generated for the method converges to a solution point of the problem. Citation Report01-2016-PESC-COPPE-UFRJ Article … Read more

    Acceleration of the PDHGM on strongly convex subspaces

  • Thomas Pock
  • Tuomo Valkonen
    • Categories Applications - Science and Engineering, Convex Optimization, Generalized Convexity/Monoticity Tags , , ,

    We propose several variants of the primal-dual method due to Chambolle and Pock. Without requiring full strong convexity of the objective functions, our methods are accelerated on subspaces with strong convexity. This yields mixed rates, $O(1/N^2)$ with respect to initialisation and $O(1/N)$ with respect to the dual sequence, and the residual part of the primal … Read more

    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