Convex Extension Research Articles

Two new feather mite species of the family Analgidae (Acariformes: Analgoidea) from the Rufous-collared Sparrow Zonotrichia capensis (Müller, 1776 (Passeriformes: Passerellidae).

Two new feather mite species of the family Analgidae are described from the Rufous-collared Sparrow, Zonotrichia capensis (Müller, 1776) (Passeriformes: Passerellidae), from Brazil: Analges ticotico sp. nov. (Analginae) and Strelkoviacarus brasiliensis sp. nov. (Anomalginae). Analges ticotico sp. nov. is characterized by the heteromorphic males having three short and rounded spines on the inner margin of femur III, the anterior margin of adanal shield convex, the hysteronotal shield with the anterior margin sinuous, and the terminal lamella rectangular. Strelkoviacarus brasiliensis sp. nov. is characterized by the anterior ends of the adanal shields being convergent and encompassing the bases of setae ps3 in males, and by the strongly convex median extension of the prodorsal shield and short dorsal setae vi, c2, d2 and e2 in females. These species represent the first records of corresponding feather mite genera in Brazil.

Explicit formulas for 𝐶^{1,1} Glaeser-Whitney extensions of 1-Taylor fields in Hilbert spaces

We give a simple alternative proof for the C 1 , 1 C^{1,1} –convex extension problem which has been introduced and studied by D. Azagra and C. Mudarra (2017). As an application, we obtain an easy constructive proof for the Glaeser-Whitney problem of C 1 , 1 C^{1,1} extensions on a Hilbert space. In both cases we provide explicit formulae for the extensions. For the Glaeser-Whitney problem the obtained extension is almost minimal, that is, minimal up to a multiplicative factor in the sense of Le Gruyer (2009).

Properties of Combinatorial Optimization Problems Over Polyhedral-Spherical Sets

The class of combinatorial optimization problems over polyhedral-spherical sets is considered. The results of theory of convex extensions are generalized to certain classes of functions defined on sphere-located and vertex-located sets. The original problem is equivalently formulated as a mathematical programming problem with convex both objective function and functional constraints. A numerical illustration and possible applications of the results to solving combinatorial optimization problems are given.

Теория выпуклых продолжений в задачах комбинаторной оптимизации

Теория выпуклых продолжений в задачах комбинаторной оптимизации

Random aggregate model for mesoscopic structures and mechanical analysis of fully-graded concrete

The realization of approximate practical aggregate models is a crucial link to the mesoscopic mechanical analysis for fully-graded concrete. This study systematically presents the establishment of mesostructure models of three-phase fully-graded concrete composed of anisotropic high-contented aggregates, cement paste, and interfaces bonding between them. A convex extension method is demonstrated in details to construct convex polygons and polyhedrons represented as crushed coarse aggregates in two- and three-dimensional spaces. The Walraven formula combined with the definition of an equivalent diameter is adopted to ensure the equivalence between the gradation and content of circle and spherical aggregates and that of the actual anisotropic geometric aggregates. We also propose a so-called “occupation and removal method (ORM)” to improve the efficiency of generating mesostructure models with a high aggregate content. Combining the ORM, the inter-particle overlap detection, and the random sequential packing scheme, random aggregate models (RAMs) of polygonal and polyhedral particles are realized to characterize the mesostructure models of concrete. Moreover, by the present RAMs applied, we implement a mesoscopic finite element simulation to investigate the flexural failure process of fully-graded concrete. The numerical results for dynamical flexural strength and dynamic increasing factors show a good in agreement with the experimental data reported in the literature. The proposed RAMs can simulate and explain qualitatively the effects of the mesoscopic structural properties on the macroscopic dynamic characteristics of fully-graded concrete.

An Extension Theorem for convex functions of class C1,1 on Hilbert spaces

Let H be a Hilbert space, E⊂H be an arbitrary subset and f:E→R, G:E→H be two functions. We give a necessary and sufficient condition on the pair (f,G) for the existence of a convex function F∈C1,1(H) such that F=f and ∇F=G on E. We also show that, if this condition is met, F can be taken so that Lip(∇F)=Lip(G). We give a geometrical application of this result, concerning interpolation of sets by boundaries of C1,1 convex bodies in H. Finally, we give a counterexample to a related question concerning smooth convex extensions of smooth convex functions with derivatives which are not uniformly continuous.

Researches on convex extension problems of fuzzy valued functions

Researches on convex extension problems of fuzzy valued functions

Convex extensions and continuous functional representations in optimization, with their applications

Convex extensions and continuous functional representations in optimization, with their applications

Exact Penalty Functions and Convex Extensions of Functions in Schemes of Decomposition in Variables*

Using exact penalty functions in schemes of decomposition in variables for nonlinear optimization make it possible to overcome problems related to implicit description of the feasible region of master problem. The paper deals with determining the values of penalty coefficients in such an approach. In the case where the functions of the original problem are not defined on the whole space of variables, the author proposes to use convex extension of functions.

A Schur complement based semi-proximal ADMM for convex quadratic conic programming and extensions

This paper is devoted to the design of an efficient and convergent {semi-proximal} alternating direction method of multipliers (ADMM) for finding a solution of low to medium accuracy to convex quadratic conic programming and related problems. For this class of problems, the convergent two block semi-proximal ADMM can be employed to solve their primal form in a straightforward way. However, it is known that it is more efficient to apply the directly extended multi-block semi-proximal ADMM, though its convergence is not guaranteed, to the dual form of these problems. Naturally, one may ask the following question: can one construct a convergent multi-block semi-proximal ADMM that is more efficient than the directly extended semi-proximal ADMM? Indeed, for linear conic programming with 4-block constraints this has been shown to be achievable in a recent paper by Sun, Toh and Yang [arXiv preprint arXiv:1404.5378, (2014)]. Inspired by the aforementioned work and with the convex quadratic conic programming in mind, we propose a Schur complement based convergent semi-proximal ADMM for solving convex programming problems, with a coupling linear equality constraint, whose objective function is the sum of two proper closed convex functions plus an arbitrary number of convex quadratic or linear functions. Our convergent semi-proximal ADMM is particularly suitable for solving convex quadratic semidefinite programming (QSDP) with constraints consisting of linear equalities, a positive semidefinite cone and a simple convex polyhedral set. The efficiency of our proposed algorithm is demonstrated by numerical experiments on various examples including QSDP.

Energy-minimizing nematic elastomers

We prove weak lower semi-continuity and existence of energy-minimizers for a free energy describing stable deformations and the corresponding directorconfiguration of an incompressible nematic liquid-crystal elastomer subject to physically realistic boundary conditions. The energy is a sum ofthe trace formula developed by Warner, Terentjev and Bladon (coupling the deformation gradient and the director field) and the Landau-de Gennesenergy in terms of the gradient of the director field and the bulk term for the director with coefficients depending on temperature. A key step in our analysis is to prove that the energy density has a convex extension to non-unit length director fields.Our results apply to the setting of physical experimentsin which a thin incompressible elastomer in $\mathbb{R}^3$ is clamped on its sidesandstretched perpendicular to its initial director field, resulting inshape-changes and director re-orientation.

Diagonal-norm summation by parts operators for finite difference approximations of third and fourth derivatives

High-order accurate finite difference operators for third and fourth derivatives are derived. The closures are based on the summation-by-parts (SBP) framework, thereby guaranteeing linear stability. Stability for nonlinear equations that support a convex extension can be achieved if the SBP operators are based on a diagonal norm. The boundary conditions are imposed using a penalty technique. The accuracy and stability properties of the newly derived SBP operators are demonstrated for both linear and nonlinear dispersive wave propagation problems.

Generalized relativistic hydrodynamics with a convex extension

We propose a new model which describes relativistic hydrodynamics and generalizes the standard Euler system of isentropic perfect fluids. Remarkably, our system admits a convex extension which allows us to transform it to a symmetric hyperbolic form. This result sheds new light even on the relativistic Euler system.

Low-Rank Bilinear Classification: Efficient Convex Optimization and Extensions

In pattern classification, it is needed to efficiently treat not only feature vectors but also feature matrices defined as two-way data, while preserving the two-way structure such as spatio-temporal relationships. The classifier for the feature matrix is generally formulated in a bilinear form composed of row and column weights which jointly result in a matrix weight. The rank of the matrix should be low from the viewpoint of generalization performance and computational cost. For that purpose, we propose a low-rank bilinear classifier based on the efficient convex optimization. In the proposed method, the classifier is optimized by minimizing the trace norm of the classifier (matrix) to reduce the rank without any hard constraint on it. We formulate the optimization problem in a tractable convex form and provide the procedure to solve it efficiently with the global optimum. In addition, we propose two novel extensions of the bilinear classifier in terms of multiple kernel learning and cross-modal learning. Through kernelizing the bilinear method, we naturally induce a novel multiple kernel learning. The method integrates both the inter kernels between heterogeneous reproducing kernel Hilbert spaces (RKHSs) and the ordinary kernels within respective RKHSs into a new discriminative kernel in a unified manner using the bilinear model. Besides, for cross-modal learning, we consider to map into the common space the multi-modal features which are subsequently classified in that space. We show that the projection and the classification are jointly represented by the bilinear model, and then propose the method to optimize both of them simultaneously in the bilinear framework. In the experiments on various visual classification tasks, the proposed methods exhibit favorable performances compared to the other methods.

On removable sets for convex functions

In the present article we provide a sufficient condition for a closed set F∈Rd to have the following property which we call c-removability: Whenever a continuous function f:Rd→R is locally convex on the complement of F, it is convex on the whole Rd. We also prove that no generalized rectangle of positive Lebesgue measure in R2 is c-removable. Our results also answer the following question asked in an article by Jacek Tabor and Józef Tabor (2010) [5]: Assume the closed set F⊂Rd is such that any locally convex function defined on Rd∖F has a unique convex extension on Rd. Is F necessarily intervally thin (a notion of smallness of sets defined by their “essential transparency” in every direction)? We prove the answer is negative by finding a counterexample in R2.

Optimal diagonal-norm SBP operators

Optimal boundary closures are derived for first derivative, finite difference operators of order 2, 4, 6 and 8. The closures are based on a diagonal-norm summation-by-parts (SBP) framework, thereby guaranteeing linear stability on piecewise curvilinear multi-block grids and entropy stability for nonlinear equations that support a convex extension. The new closures are developed by enriching conventional approaches with additional boundary closure stencils and non-equidistant grid distributions at the domain boundaries. Greatly improved accuracy is achieved near the boundaries, as compared with traditional diagonal-norm operators of the same order. The superior accuracy of the new optimal diagonal-norm SBP operators is demonstrated for linear hyperbolic systems in one dimension and for the nonlinear compressible Euler equations in two dimensions.

Oracle Tractability of Skew Bisubmodular Functions

In this paper we consider skew bisubmodular functions as recently introduced by the authors and Powell. We construct a convex extension of a skew bisubmodular function which we call Lovász extension in correspondence to the submodular case. We use this extension to show that skew bisubmodular functions given by an oracle can be minimized in polynomial time.

Generalized skew bisubmodularity: A characterization and a min–max theorem

Huber, Krokhin, and Powell (2013) introduced a concept of skew bisubmodularity, as a generalization of bisubmodularity, in their complexity dichotomy theorem for valued constraint satisfaction problems over the three-value domain. In this paper we consider a natural generalization of the concept of skew bisubmodularity and show a connection between the generalized skew bisubmodularity and a convex extension over rectangles. We also analyze the dual polyhedra, called skew bisubmodular polyhedra, associated with generalized skew bisubmodular functions and derive a min–max theorem that characterizes the minimum value of a generalized skew bisubmodular function in terms of a minimum-norm point in the associated skew bisubmodular polyhedron.

Weighted inversion of general Dirichlet series

Inversion theorems of Wiener type are essential tools in analysis and number theory. We derive a weighted version of an inversion theorem of Wiener type for general Dirichlet series from that of Edwards from 1957, and we outline an alternative proof based on the duality theory of convex cones and extension techniques for characters of semigroups. Variants and arithmetical applications are described, including the case of multidimensional weighted generalized Dirichlet series.

Maximum subarray algorithms for use in astronomical imaging

The maximum subarray problem is used to identify the subarray of a two-dimensional array, where the sum of elements is maximized. In terms of image processing, the solution has been used to find the brightest region within an image. Two parallel algorithms of the maximum subarray problem solve this problem in O ( n ) and O(log n ) time. A field programmable gate array implementation has verified theoretical maximum performance; however, extensive customization is required, restricting general application. A more convenient platform for this work is a graphics processor unit since it offers a flexible trade-off between hardware customization and performance. Implementation of the maximum subarray algorithm on a graphics processor unit is discussed in this article for rectangular solutions and convex extensions are explored.