6 research outputs found
Well-posedness in vector optimization and scalarization results
In this paper, we give a survey on well-posedness notions of Tykhonov's type for vector optimization problems and the links between them with respect to the classification proposed by Miglierina, Molho and Rocca in [33]. We consider also the notions of extended well-posedness introduced by X.X. Huang ([19],[20]) in the nonparametric case to complete the hierchical structure characterizing these concepts. Finally we propose a review of some theoretical results in vector optimization mainly related to different notions of scalarizing functions, linear and nonlinear, introduced in the last decades, to simplify the study of various well-posedness properties.
Pointwise well-posedness in vector optimization and variational inequalities
In this note we consider some notions of well-posedness for scalar
and vector variational inequalities and we recall their connections
with optimization problems. Subsequently, we investigate similar
connections between well-posedness of a vector optimization
problem and a related variational inequality problem and we
present an result obtained with scalar characterizations of vector
optimality concepts
α
The concepts of α-well-posedness, α-well-posedness in the
generalized sense, L-α-well-posedness and L-α-well-posedness in the generalized sense for
mixed quasi variational-like inequality problems are investigated. We present some metric
characterizations for these well-posednesses
Advances in Optimization and Nonlinear Analysis
The present book focuses on that part of calculus of variations, optimization, nonlinear analysis and related applications which combines tools and methods from partial differential equations with geometrical techniques. More precisely, this work is devoted to nonlinear problems coming from different areas, with particular reference to those introducing new techniques capable of solving a wide range of problems. The book is a valuable guide for researchers, engineers and students in the field of mathematics, operations research, optimal control science, artificial intelligence, management science and economics
Nonsmooth Convex Variational Approaches to Image Analysis
Variational models constitute a foundation for the formulation and understanding of models in many areas of image processing and analysis. In this work, we consider a generic variational framework for convex relaxations of multiclass labeling problems, formulated on continuous domains. We propose several relaxations for length-based regularizers, with varying expressiveness and computational cost. In contrast to graph-based, combinatorial approaches, we rely on a geometric measure theory-based formulation, which avoids artifacts caused by an early discretization in theory as well as in practice. We investigate and compare numerical first-order approaches for solving the associated nonsmooth discretized problem, based on controlled smoothing and operator splitting. In order to obtain integral solutions, we propose a randomized rounding technique formulated in the spatially continuous setting, and prove that it allows to obtain solutions with an a priori optimality bound. Furthermore, we present a method for introducing more advanced prior shape knowledge into labeling problems, based on the sparse representation framework