Local linear convergence analysis of Primal–Dual splitting methods

In this paper, we study the local linear convergence properties of a versatile class of Primal–Dual splitting methods for minimizing composite non-smooth convex optimization problems. Under the assumption that the non-smooth components of the problem are partly smooth relative to smooth manifolds, we present a unified local convergence analysis framework for these methods. More precisely, in our framework, we first show that (i) the sequences generated by Primal–Dual splitting methods identify a pair of primal and dual smooth manifolds in a finite number of iterations, and then (ii) enter a local linear convergence regime, which is characterized based on the structure of the underlying active smooth manifolds. We also show how our results for Primal–Dual splitting can be specialized to cover existing ones on Forward–Backward splitting and Douglas–Rachford splitting/ADMM (alternating direction methods of multipliers). Moreover, based on these obtained local convergence analysis result, several practical acceleration techniques are discussed. To exemplify the usefulness of the obtained result, we consider several concrete numerical experiments arising from fields including signal/image processing, inverse problems and machine learning. The demonstration not only verifies the local linear convergence behaviour of Primal–Dual splitting methods, but also the insights on how to accelerate them in practice.     

Beta Variables as Times Spent in [0, {infty}[ by Certain Perturbed Brownian Motions

The paper shows that the times spent in [0, +) by certain processesY which are defined by perturbations of Brownian motion involvingreflection at maxima and minima are beta distributed. This resultrelies heavily on Ray–Knight theorems for such perturbedBrownian motions.     

Microstructure and dynamics near an attractive colloidal glass transition

The microstructure and dynamics of a colloidal system interacting via short-ranged interparticle potential is studied by ultra-small-angle x-ray scattering and x-ray photon correlation spectroscopy. A colloidal gas-liquid type transition is induced when the short-ranged attractive interactions attain sufficient magnitude. The development of liquidlike structure is preceded by a systematic transition in the particle dynamics from diffusive to constrained motion and then completely frozen behavior. This demonstrates the existence of a jamming transition induced by strong short-ranged attractive interactions even at low packing fractions.     

Solving mixed-integer robust optimization problems with interval uncertainty using Benders decomposition

Uncertainty and integer variables often exist together in economics and engineering design problems. The goal of robust optimization problems is to find an optimal solution that has acceptable sensitivity with respect to uncertain factors. Including integer variables with or without uncertainty can lead to formulations that are computationally expensive to solve. Previous approaches for robust optimization problems under interval uncertainty involve nested optimization or are not applicable to mixed-integer problems where the objective or constraint functions are neither quadratic, nor linear. The overall objective in this paper is to present an efficient robust optimization method that does not contain nested optimization and is applicable to mixed-integer problems with quasiconvex constraints (? type) and convex objective funtion. The proposed method is applied to a variety of numerical examples to test its applicability and numerical evidence is provided for convergence in general as well as some theoretical results for problems with linear constraints.     

Implicit depot assignments and rotations in vehicle routing heuristics

Vehicle routing variants with multiple depots and mixed fleet present intricate combinatorial aspects related to sequencing choices, vehicle type choices, depot choices, and depots positioning. This paper introduces a dynamic programming methodology for efficiently evaluating compound neighborhoods combining sequence-based moves with an optimal choice of vehicle and depot, and an optimal determination of the first customer to be visited in the route, called rotation. The assignment choices, making the richness of the problem, are thus no more addressed in the solution structure, but implicitly determined during each move evaluation. Two meta-heuristics relying on these concepts, an iterated local search and a hybrid genetic algorithm, are presented. Extensive computational experiments demonstrate the remarkable performance of these methods on classic benchmark instances for multi-depot vehicle routing problems with and without fleet mix, as well as the notable contribution of the implicit depot choice and positioning methods to the search performance. New state-of-the-art results are obtained for multi-depot vehicle routing problems (MDVRP), and multi-depot vehicle fleet mix problems (MDVFMP) with unconstrained fleet size. The proposed concepts are fairly general, and widely applicable to many other vehicle routing variants.     

Primal-Dual Relationship Between Levenberg–Marquardt and Central Trajectories for Linearly Constrained Convex Optimization

We consider the minimization of a convex function on a bounded polyhedron (polytope) represented by linear equality constraints and non-negative variables. We define the Levenberg–Marquardt and central trajectories starting at the analytic center using the same parameter, and show that they satisfy a primal-dual relationship, being close to each other for large values of the parameter. Based on this, we develop an algorithm that starts computing primal-dual feasible points on the Levenberg–Marquardt trajectory and eventually moves to the central path. Our main theorem is particularly relevant in quadratic programming, where points on the primal-dual Levenberg–Marquardt trajectory can be calculated by means of a system of linear equations. We present some computational tests related to box constrained trust region subproblems.     

Asymptotic distributions of robust shape matrices and scales

It has been frequently observed in the literature that many multivariate statistical methods require the covariance or dispersion matrix Σ of an elliptical distribution only up to some scaling constant. If the topic of interest is not the scale but only the shape of the elliptical distribution, it is not meaningful to focus on the asymptotic distribution of an estimator for Σ or another matrix Γ∝Σ. In the present work, robust estimators for the shape matrix and the associated scale are investigated. Explicit expressions for their joint asymptotic distributions are derived. It turns out that if the joint asymptotic distribution is normal, the estimators presented are asymptotically independent for one and only one specific choice of the scale function. If it is non-normal (this holds for example if the estimators for the shape matrix and scale are based on the minimum volume ellipsoid estimator) only the scale function presented leads to asymptotically uncorrelated estimators. This is a generalization of a result obtained by Paindaveine [D. Paindaveine, A canonical definition of shape, Statistics and Probability Letters 78 (2008) 2240-2247] in the context of local asymptotic normality theory.     

The cut‐tree of large trees with small heights

We destroy a finite tree of size n by cutting its edges one after the other and in uniform random order. Informally, the associated cut‐tree describes the genealogy of the connected components created by this destruction process. We provide a general criterion for the convergence of the rescaled cut‐tree in the Gromov‐Prohorov topology to an interval endowed with the Euclidean distance and a certain probability measure, when the underlying tree has branching points close to the root and height of order . In particular, we consider uniform random recursive trees, binary search trees, scale‐free random trees and a mixture of regular trees. This yields extensions of a result in Bertoin (Probab Stat 5 (2015), 478–488) for the cut‐tree of uniform random recursive trees and also allows us to generalize some results of Kuba and Panholzer (Online J Anal Combin (2014), 26) on the multiple isolation of vertices. The approach relies in the close relationship between the destruction process and Bernoulli bond percolation, which may be useful for studying the cut‐tree of other classes of trees. © 2017 Wiley Periodicals, Inc. Random Struct. Alg., 51, 404–427, 2017     

Determination of cadmium in coal fly ash, soil and sediment samples by GFAAS with evaluation of different matrix modifiers

The determination of cadmium in different samples such as coal fly ashes, soils and sediments by Graphite Furnace Atomic Absorption Spectrometry (GFAAS) was studied. The best analytical conditions and instrumental parameters were deduced. Different types of matrix modifiers were tested and an optimization of several conditions (pyrolysis and atomization temperatures, heating rate, use of L’vov platform, etc.) was carried out. A mixture of 2% NH₄H₂PO₄ + 0.4% Mg(NO₃)₂ in 0.5 mol L^–1 HNO₃ as matrix modifier provided the highest sensitivity and an efficient behaviour for the three types of samples. A detection limit of 26 ng g^–1 was achieved and other figures of merit are shown. Analytical results obtained by direct calibration using standard samples prepared in the laboratory and by standard addition method were comparable.     

Nucleation and Growth Synthesis of Siloxane Gels to Form Functional,Monodisperse, and Acoustically Programmable Particles

Nucleation and growth methods offer scalable means of synthesizing colloidal particles with precisely specified size for applications in chemical research, industry, and medicine. These methods have been used to prepare a class of silicone gel particles that display a range of programmable properties and narrow size distributions. The acoustic contrast factor of these particles in water is estimated and can be tuned such that the particles undergo acoustophoresis to either the pressure nodes or antinodes of acoustic standing waves. These particles can be synthesized to display surface functional groups that can be covalently modified for a range of bioanalytical and acoustophoretic sorting applications.