Convexity Arguments Research Articles

Asymptotics for argmin processes: Convexity arguments

The convexity arguments developed by Pollard [D. Pollard, Asymptotics for least absolute deviation regression estimators, Econometric Theory 7 (1991) 186–199], Hjort and Pollard [N.L. Hjort, D. Pollard, Asymptotics for minimizers of convex processes, 1993 (unpublished manuscript)], and Geyer [C.J. Geyer, On the asymptotics of convex stochastic optimization, 1996 (unpublished manuscript)] are now basic tools for investigating the asymptotic behavior of M -estimators with non-differentiable convex objective functions. This paper extends the scope of convexity arguments to the case where estimators are obtained as stochastic processes. Our convexity arguments provide a simple proof for the asymptotic distribution of regression quantile processes. In addition to quantile regression, we apply our technique to LAD (least absolute deviation) inference for threshold regression.

Transport of heat and mass in a fluid with vanishing mobility

We study a model describing the compressible displacement of a mixture in a porous medium. The transport of heat and mass is described by a nonlinear, fully coupled and degenerate parabolic system. Using a series of compensated compactness and convexity arguments, we prove the existence of relevant weak solutions.

Multidimensional Hardy-type inequalities with general kernels

Some new multidimensional Hardy-type inequalities involving arithmetic mean operators with general positive kernels are derived. Our approach is mainly to use a convexity argument and the results obtained improve some known results in the literature and, in particular, some recent results in [S. Kaijser, L. Nikolova, L.-E. Persson, A. Wedestig, Hardy-type inequalities via convexity, Math. Inequal. Appl. 8 (3) (2005) 403–417] are generalized and complemented.

Information Consistency of Nonparametric Gaussian Process Methods

Bayesian nonparametric models are widely and successfully used for statistical prediction. While posterior consistency properties are well studied in quite general settings, results have been proved using abstract concepts such as metric entropy, and they come with subtle conditions which are hard to validate and not intuitive when applied to concrete models. Furthermore, convergence rates are difficult to obtain. By focussing on the concept of information consistency for Bayesian Gaussian process (GP)models, consistency results and convergence rates are obtained via a regret bound on cumulative log loss. These results depend strongly on the covariance function of the prior process, thereby giving a novel interpretation to penalization with reproducing kernel Hilbert space norms and to commonly used covariance function classes and their parameters. The proof of the main result employs elementary convexity arguments only. A theorem of Widom is used in order to obtain precise convergence rates for several covariance functions widely used in practice.

Global synchronization on the circle

The convexity arguments used in the consensus literature to prove synchronization in vector spaces can be applied to the circle only when all agents are initially located on a semicircle. Existing strategies for (almost-)global synchronization on the circle are either restricted to specific interconnection topologies or use auxiliary variables. The present paper first illustrates this problem by showing that weighted, directed interconnection topologies can be designed to make any reasonably chosen configuration of the agents on the circle a stable equilibrium of a basic continuous-time consensus algorithm. Then it proposes a so-called “gossip algorithm”, which achieves global asymptotic synchronization on the circle with probability 1 for a large class of interconnections, without using auxiliary variables, thanks to the introduction of randomness in the system.

On One-to-One Codes for Memoryless Cost Channels

A new twist to the classical problem of compressing a discrete source over noiseless channel alphabets where different code symbols may have different transmission costs is considered. One-to-one or nonsingular codes map each letter from the source alphabet into a distinct codeword consisting of a string of channel alphabet letters; they need not be uniquely decodable. They have been studied for binary code alphabets with equal letter costs since the early 1970s and are extended here to memoryless cost channel alphabets. The study of optimal one-to-one codes is linked in this paper to three other well-known source coding problems. The algorithm to construct the optimal one-to-one code is closely related to the Varn code and the Tunstall code. These relationships and convexity arguments are used to establish some upper and lower bounds on the expected cost per source symbol of the optimal one-to-one code. Finally, the average cost per symbol of the optimal one-to-one code is demonstrated to offer a new lower bound to the average cost per symbol of the optimal uniquely decodable code which is sometimes tighter than the classic bound from Shannon's landmark 1948 paper.

L p Estimates for Maxwell's Equations with Source Term

The goal of this paper is to establish interior and global L p -type estimates for the solutions of Maxwell's equations with source term in a domain filled with two different materials separated by a 𝒞2 interface. The usual elliptic estimates cannot be applied directly, due to the singularity of the dielectric permittivity. A special curl-div decomposition is introduced for the electric field to reduce the problem to an elliptic equation in divergence form with jump coefficients. The potential analysis and the jump condition lead to the interior L p estimates which are superior to the straightforward Nash-Moser estimates. The reduction procedure is expected to be useful for future numerical simulation. Because of the natural physical requirements, the boundary condition is nonlocal and involves a first order pseudo-differential operator, the boundary estimate is established by delicate new maximum principles and Riesz convexity arguments. These estimates are then employed to solve a nonlinear optics problem that arises in the modeling of surface enhanced second-harmonic generation of nonlinear diffractive optics in periodic structures (gratings).

Variational approach to non-linear boundary value problems for elasto-plastic incompressible bending plate

Non-linear boundary value problems for inelastic isotropic homogeneous incompressible bending plate, within the range of J 2 -deformation theory, are considered. An existence of the weak solution of the non-linear problem with clamped boundary condition is obtained in H 2 ( Ω ) by using monotone operator theory and Browder–Minty theorem. For linearization of the non-linear problem a monotone iteration scheme is constructed. It is shown that the sequence of potentials obtained from the sequence of approximate solutions (i.e. iterations), is a monotone decreasing one. Convergence of the iteration process in H 2 -norm is proved by using the convexity argument. Numerical solutions, based on finite-difference scheme, are given for linear bending problems with rigid clamped as well as simply supported boundary conditions. Further numerical examples are presented to illustrate the convergence of approximate solutions and monotonicity of the potentials as applied to the non-linear problems.

Talagrand Inequality for the Semicircular Law and Energy of the Eigenvalues of Beta Ensembles

We give a short proof of an extension of the free Talagrand transportation cost inequality to the semicircular which was originally proved in \cite{BV}. The proof is based on a convexity argument and is in the spirit of the original Talagrand's approach for the classical counterpart from \cite{T}. We also discuss the convergence, fluctuations and large deviations of the energy of the eigenvalues of $\beta$ ensembles, which, as an application of Talagrand inequality gives in particular yet another proof of the convergence of the eigenvalue distribution to the semicircle law.

Free vibrations of a polar body at elastic range

The purpose of this paper is to study certain features of the equations governing the time-harmonic free vibrations of a polar body at elastic range. The governing equations of micropolar elasticity are expressed in differential form, and then, the uniqueness of their solutions is investigated. The conditions sufficient for uniqueness are enumerated using the logarithmic convexity argument without any positive-definiteness assumptions of material elasticity. Applying a general principle of physics and modifying it through an involutory transformation, a unified variational principle is obtained that leads to all the governing equations of the free vibrations as its Euler-Lagrange equations. The governing equations are alternatively expressed in terms of the operators related to the kinetic and potential energies of the body. The basic properties of vibrations are studied and a variational principle in Rayleigh’s quotient is given. As an application, the high-frequency vibrations of an elastic plate are treated.

Convexity Arguments for Efficient Minimization of the Bethe and Kikuchi Free Energies

Loopy and generalized belief propagation are popular algorithms for approximate inference in Markov random fields and Bayesian networks. Fixed points of these algorithms have been shown to correspond to extrema of the Bethe and Kikuchi free energy, both of which are approximations of the exact Helmholtz free energy. However, belief propagation does not always converge, which motivates approaches that explicitly minimize the Kikuchi/Bethe free energy, such as CCCP and UPS. Here we describe a class of algorithms that solves this typically non-convex constrained minimization problem through a sequence of convex constrained minimizations of upper bounds on the Kikuchi free energy. Intuitively one would expect tighter bounds to lead to faster algorithms, which is indeed convincingly demonstrated in our simulations. Several ideas are applied to obtain tight convex bounds that yield dramatic speed-ups over CCCP.

Free vibrations of a polar elastic body

This paper concerns certain features of the equations governing the time-harmonic free vibrations of a polar elastic body. The governing equations of micropolar elasticity are expressed in differential form; then, the uniqueness in their solutions is investigated. The conditions sufficient for the uniqueness are enumerated using the logarithmic convexity argument without any positive-definiteness assumptions of material elasticity. Applying a general principle of physics and modifying it through an involutory transformation, a unified variational principle is obtained for the free vibrations. The principle leads to all the governing equations of the free vibrations, as its Euler-Lagrange equations. The governing equations are alternatively expressed in terms of the operators related to the kinetic and potential energies of the body. The basic properties of vibrations are studied and a variational principle in Rayleigh’s quotient is given, and the high-frequency vibrations of a micropolar plate [cf., Altay and Dkmeci, Int. J. Solids Struct. (2006)] are likewise treated. [Work supported by TUBA.]

A stable semi-discrete central scheme for the two-dimensional incompressible Euler equations

We derive a second-order, semi-discrete central-upwind scheme for the incompressible 2D Euler equations in the vorticity formulation. The reconstructed velocity field preserves an exact discrete incompressibility relation. We state a local maximum principle for a fully discrete version of the scheme and prove it using a convexity argument. We then show how similar convexity arguments can be used to prove that the scheme maps certain Orlicz spaces into themselves. The consequences of this result on the convergence of the scheme are discussed. Numerical simulations support the expected properties of the scheme.

Weak matrix majorization

Given X , Y ∈ R n × m we introduce the following notion of matrix majorization, called weak matrix majorization, X ≻ w Y if there exists a row - stochastic matrix A ∈ R n × n such that AX = Y , and consider the relations between this concept, strong majorization (≻ s ) and directional majorization (≻). It is verified that ≻ s ⇒ ≻ ⇒ ≻ w , but none of the reciprocal implications is true. Nevertheless, we study the implications ≻ w ⇒ ≻ s and ≻ ⇒ ≻ s under additional hypotheses. We give characterizations of strong, directional and weak matrix majorization in terms of convexity. We also introduce definitions for majorization between Abelian families of selfadjoint matrices, called joint majorizations. They are induced by the previously mentioned matrix majorizations. We obtain descriptions of these relations using convexity arguments.

Existence results for a degenerated nonlinear elliptic partial differential equation

The aim of this paper is to establish the existence of weak solutions to a steady state two-dimensional irrotational compressible flow around a thin profile. This flow is described by the small disturbance equations. If the speed of sound exceeds the fluid one, the governing equations remain elliptic. But when the fluid speed is beyond the sound one, the flow becomes locally hyperbolic and shock waves arise. For a modified elliptic model, using convexity arguments, we prove the existence of a solution which is the solution to the first model when the flow remains subsonic.

Worst closed-loop controlled bulk distributions of stochastic arrival processes for queue performance

This paper presents basic queueing analysis contributing to teletraffc theory, with commonly accessible mathematical tools. This paper studies queueing systems with bulk arrivals. It is assumed that the number of arrivals and the expected number of arrivals in each bulk are bounded by some constraints B and λ, respectively. Subject to these constraints, convexity argument is used to show that the bulk-size probability distribution that results in the worst mean queue performance is an extremal distribution with support {1, B} and mean equal to λ. Furthermore, from the viewpoint of security against denial-of-service attacks, this distribution remains the worst even if an adversary were allowed to choose the bulk-size distribution at each arrival instant as a function of past queue lengths; that is, the adversary can produce as bad queueing performance with an open-loop strategy as with any closed-loop strategy. These results are proven for an arbitrary arrival process with bulk arrivals and a general service model.

Optimal Control of Structural Dynamic Systems in One Space Dimension Using a Maximum Principle

A maximum principle is developed for a class of problems involving the optimal control of a system of linear hyperbolic equations in one space dimension that are not necessarily separable. An index of performance is formulated, which consists of vector functions of the state variable, their first- and second-order space derivatives and first-order time derivative, and a penalty function involving the open-loop control force vector. The solution of the optimal control problem can easily be shown to be unique using convexity arguments. The given maximum principle involves a Hamiltonian, which contains an adjoint vector function as well as an admissible control vector function. The maximum principle can be used to compute the optimal control vector function and is particularly suitable for problems involving the active control of structural elements for vibration suppression. A numerical example is given which studies the active control of a beam under going flexural and torsional vibrations. A comparison of the energies of controlled and uncontrolled beams indicates that the proposed control method is quite effective in damping out the vibrations of structural systems.

APPROXIMATE MIN-MAX MPC FOR LINEAR HYBRID SYSTEMS

This paper presents a Receding Horizon Control (RHC) strategy to robustly control linear hybrid systems subject to bounded additive disturbances. First, the open-loop finite horizon min-max optimal control problem associated with the RHC strategy is presented, defining a mixed-integer optimization problem. The goal of the min-max formulation is to minimize the worst-case cost of the performance index, while guaranteeing performance/safety constraints for all possible disturbance realizations. Then it will be proved that feasibility at the initial time as well as convexity arguments and set-invariance assumptions are sufficient conditions to guarantee robust stability and performance of the closed-loop system. Simulation results for a perturbed hybrid system will demonstrate the potential of the proposed robust optimal control design.

A unified and elementary proof of serial and nonserial, univariate and multivariate, Chernoff–Savage results

We provide a simple proof that the Chernoff–Savage [H. Chernoff, I.R. Savage, Asymptotic normality and efficiency of certain nonparametric tests, Ann. Math. Statist. 29 (1958) 972–994] result, establishing the uniform dominance of normal-score rank procedures over their Gaussian competitors, also holds in a broad class of problems involving serial and/or multivariate observations. The non-admissibility of the corresponding everyday practice Gaussian procedures (multivariate least-squares estimators, multivariate t-tests and F-tests, correlogram-based methods, multivariate portmanteau and Durbin–Watson tests, etc.) follows. The proof, which generalizes to the multivariate—possibly serial—set-up the idea developed in J.L. Gastwirth, S.S. Wolff [An elementary method for obtaining lower bounds on the asymptotic power of rank tests, Ann. Math. Statist. 39 (1968) 2128–2130] in the context of univariate location problems, allows for avoiding technical convexity and variational arguments.

Convexity and Weighted Integral Inequalities for Energy Decay Rates of Nonlinear Dissipative Hyperbolic Systems

This work is concerned with the stabilization of hyperbolic systems by a nonlinear feedback which can be localized on a part of the boundary or locally distributed. We show that general weighted integral inequalities together with convexity arguments allow us to produce a general semi-explicit formula which leads to decay rates of the energy in terms of the behavior of the nonlinear feedback close to the origin. This formula allows us to unify for instance the cases where the feedback has a polynomial growth at the origin, with the cases where it goes exponentially fast to zero at the origin. We also give three other significant examples of nonpolynomial growth at the origin. Our work completes the work of [15] and improves the results of [21] and [22] (see also [23] and [10]). We also prove the optimality of our results for the one-dimensional wave equation with nonlinear boundary dissipation. The key property for obtaining our general energy decay formula is the understanding between convexity properties of an explicit function connected to the feedback and the dissipation of energy.