Stable strong Fenchel and Lagrange duality for evenly convex optimization problems

By means of a conjugation scheme based on generalized convex conjugation theory instead of Fenchel conjugation, we build an alternative dual problem, using the perturbational approach, for a general optimization one defined on a separated locally convex topological space. Conditions guaranteeing strong duality for primal problems which are perturbed by continuous linear functionals and their respective dual problems, which is named stable strong duality, are established. In these conditions, the fact that the perturbation function is evenly convex will play a fundamental role. Stable strong duality will also be studied in particular for Fenchel and Lagrange primal–dual problems, obtaining a characterization for Fenchel case.This research was partially supported by MINECO of Spain and FEDER of EU, [grant numberMIM2014-59179-C2-1-P]; Conselleria de la Educacion de la Generalitat Valenciana, Spain, Pre-doc Program Vali+d, DOCV 6791/07.06.2012, [grant number ACIF-2013-156]

New duality results for evenly convex optimization problems

We present new results on optimization problems where the involved functions are evenly convex. By means of a generalized conjugation scheme and the perturbation theory introduced by Rockafellar, we propose an alternative dual problem for a general optimization one defined on a separated locally convex topological space. Sufficient conditions for converse and total duality involving the even convexity of the perturbation function and c-subdifferentials are given. Formulae for the c-subdifferential and biconjugate of the objective function of a general optimization problem are provided, too. We also characterize the total duality by means of the saddle-point theory for a notion of Lagrangian adapted to the considered framework.Research partially supported by MINECO of Spain and ERDF of EU, Grant MTM2014-59179-C2-1-P, Austrian Science Fund (FWF), Project M-2045, and German Research Foundation (DFG), Project GR3367/4-1

E′-Convex Sets and Functions: Properties and Characterizations

The main properties of evenly convex sets and functions have been deeply studied by different authors, and a duality theory for evenly convex optimization problems has been well developed as well. In this theory, the notion of e′-convexity appears as a necessary requirement for obtaining important results in strong and stable strong duality. This fact has motivated the authors to study possible properties of this kind of convexity in sets and functions, which is closely connected to even convexity

Set-Valued Evenly Convex Functions: Characterizations and C-Conjugacy

In this work we deal with set-valued functions with values in the power set of a separated locally convex space where a nontrivial pointed convex cone induces a partial order relation. A set-valued function is evenly convex if its epigraph is an evenly convex set, i.e., it is the intersection of an arbitrary family of open half-spaces. In this paper we characterize evenly convex set-valued functions as the pointwise supremum of its set-valued e-affine minorants. Moreover, a suitable conjugation pattern will be developed for these functions, as well as the counterpart of the biconjugation Fenchel-Moreau theorem.Open Access funding provided thanks to the CRUE-CSIC agreement with Springer Nature. Research partially supported by MINECO of Spain and ERDF of EU, Grant MTM2014-59179-C2-1-P

Scaling Algorithms for Unbalanced Transport Problems

This article introduces a new class of fast algorithms to approximate variational problems involving unbalanced optimal transport. While classical optimal transport considers only normalized probability distributions, it is important for many applications to be able to compute some sort of relaxed transportation between arbitrary positive measures. A generic class of such "unbalanced" optimal transport problems has been recently proposed by several authors. In this paper, we show how to extend the, now classical, entropic regularization scheme to these unbalanced problems. This gives rise to fast, highly parallelizable algorithms that operate by performing only diagonal scaling (i.e. pointwise multiplications) of the transportation couplings. They are generalizations of the celebrated Sinkhorn algorithm. We show how these methods can be used to solve unbalanced transport, unbalanced gradient flows, and to compute unbalanced barycenters. We showcase applications to 2-D shape modification, color transfer, and growth models

Playing with Duality: An Overview of Recent Primal-Dual Approaches for Solving Large-Scale Optimization Problems

Optimization methods are at the core of many problems in signal/image processing, computer vision, and machine learning. For a long time, it has been recognized that looking at the dual of an optimization problem may drastically simplify its solution. Deriving efficient strategies which jointly brings into play the primal and the dual problems is however a more recent idea which has generated many important new contributions in the last years. These novel developments are grounded on recent advances in convex analysis, discrete optimization, parallel processing, and non-smooth optimization with emphasis on sparsity issues. In this paper, we aim at presenting the principles of primal-dual approaches, while giving an overview of numerical methods which have been proposed in different contexts. We show the benefits which can be drawn from primal-dual algorithms both for solving large-scale convex optimization problems and discrete ones, and we provide various application examples to illustrate their usefulness

Compressed Motion Sensing and Dynamic Tomography

Compressed sensing is a new sampling paradigm of mathematical signal processing which, under certain assumptions, allows signal recovery from highly undersampled measurements. The extension of the mathematical theory and the analysis and development of new applications in many fields are the subject of numerous international research activities. In this thesis an industrial problem from experimental fluid dynamics is consider, exemplarily. The current state of the art methodology solves the problem in two independent stages: First it recovers particle images by nonstandard tomography, and secondly it estimates the motion between two given time points. This motivates the problem of joint signal and motion estimation while raising theoretical questions in compressed sensing related to the recovery of sparse time-varying signals. In particular, two different approaches are presented for recovering a time-varying signal and its motion from undersampled linear measurements taken at two different points in time. The first approach formulates a problem at hand as optimal transport between two indirectly observed densities with a physical constraint. Several methods are proposed to integrate the projection constraints into the convex optimization framework of Benamou and Brenier. In the second approach, the signal is modeled as if observed by the real sensor specified by the experimental setup and an additional virtual sensor due to motion. The combination of these two sensors is called compressed motion sensor and its properties are examined from the viewpoint of compressed sensing. It is shown that in compressed motion sensing (CMS), besides sparsity, a sufficient change of signal leads to recovery guarantees and it is demonstrated that the compressed motion sensor at least doubles the performance of the real sensor. Moreover, for certain sparsity levels the signal motion can be established, too

Mathematical Methods in Quantum Chemistry

The field of quantum chemistry is concerned with the modelling and simulation of the behaviour of molecular systems on the basis of the fundamental equations of quantum mechanics. Since these equations exhibit an extreme case of the curse of dimensionality (the Schrödinger equation for N electrons being a partial differential equation on R3N ), the quantum-chemical simulation of even moderate-size molecules already requires highly sophisticated model-reduction, approximation, and simulation techniques. The workshop brought together selected quantum chemists and physicists, and the growing community of mathematicians working in the area, to report and discuss recent advances on topics such as coupled-cluster theory, direct approximation schemes in full configuration-interaction (FCI) theory, interacting Green’s functions, foundations and computational aspects of densityfunctional theory (DFT), low-rank tensor methods, quantum chemistry in the presence of a strong magnetic field, and multiscale coupling of quantum simulations