Vector Inequalities Research Articles

Power vector inequalities for operator pairs in Hilbert spaces and their applications

Abstract This study explores the power vector inequalities for a pair of operators ( B , C ) \left(B,C) in a Hilbert space. By utilizing a Mitrinović-Pečarić-Fink-type inequality for inner products and norms, we derive various power vector inequalities. Specifically, we consider the cases where ( B , C ) \left(B,C) is equal to ( A , A * ) \left(A,{A}^{* }) or ( Re ( A ) , Im ( A ) ) (\hspace{0.1em}\text{Re}\hspace{0.1em}\left(A),\hspace{0.1em}\text{Im}\hspace{0.1em}\left(A)) for an operator A A in B ( H ) B\left(H) , where H H is a Hilbert space. This leads to the derivation of vector, norm, and numerical radius inequalities for a single operator. Furthermore, we obtain power inequalities for the s s - r r -norm and s s - r r -numerical radius of the operator pair ( B , C ) ∈ B ( H ) \left(B,C)\in B\left(H) , which generalizes the Euclidean norm and Euclidean numerical radius. Finally, we apply these results to derive the corresponding inequalities for a single operator A ∈ B ( H ) A\in B\left(H) .

On the von Bahr–Esseen inequality for pairwise independent random vectors in Hilbert spaces with applications to mean convergence

Abstract In this correspondence, we prove the von Bahr–Esseen moment inequality for pairwise independent random vectors in Hilbert spaces. Our constant in the von Bahr–Esseen moment inequality is better than that obtained for the real-valued random variables by Chen et al. [The von Bahr–Esseen moment inequality for pairwise independent random variables and applications, J. Math. Anal. Appl. 419 (2014), 1290–1302], and Chen and Sung [Generalized Marcinkiewicz–Zygmund type inequalities for random variables and applications, J. Math. Inequal. 10(3) (2016), 837–848]. The result is then applied to obtain mean convergence theorems for triangular arrays of rowwise and pairwise independent random vectors in Hilbert spaces. Some results in the literature are extended.

A new super-predefined-time convergence and noise-tolerant RNN for solving time-variant linear matrix–vector inequality in noisy environment and its application to robot arm

A new super-predefined-time convergence and noise-tolerant RNN for solving time-variant linear matrix–vector inequality in noisy environment and its application to robot arm

Minimax Principle for Eigenvalues of Dual Quaternion Hermitian Matrices and Generalized Inverses of Dual Quaternion Matrices

Dual quaternions can represent rigid body motion in 3D spaces, and have found wide applications in robotics, 3D motion modelling and control, and computer graphics. In this paper, we introduce three different right linear independency concepts for a set of dual quaternion vectors, and study some related basic properties for dual quaternion vectors and dual quaternion matrices. We present a minimax principle for eigenvalues of dual quaternion Hermitian matrices. Based upon a newly established Cauchy-Schwarz inequality for dual quaternion vectors and singular value decomposition of dual quaternion matrices, we propose an inequality for singular values of dual quaternion matrices. Finally, we introduce the concept of generalized inverses of dual quaternion matrices, and present necessary and sufficient conditions for a dual quaternion matrix to be one of four types of generalized inverses of another dual quaternion matrix.

Inner Moreau Envelope of Nonsmooth Conic Chance-Constrained Optimization Problems

Optimization problems with uncertainty in the constraints occur in many applications. Particularly, probability functions present a natural form to deal with this situation. Nevertheless, in some cases, the resulting probability functions are nonsmooth, which motivates us to propose a regularization employing the Moreau envelope of a scalar representation of the vector inequality. More precisely, we consider a probability function that covers most of the general classes of probabilistic constraints: [Formula: see text]where [Formula: see text] is a convex cone of a Banach space. The conic inclusion [Formula: see text] represents an abstract system of inequalities, and ξ is a random vector. We propose a regularization by applying the Moreau envelope to the scalarization of the function [Formula: see text]. In this paper, we demonstrate, under mild assumptions, the smoothness of such a regularization and that it satisfies a type of variational convergence to the original probability function. Consequently, when considering an appropriately structured problem involving probabilistic constraints, we can, thus, entail the convergence of the minimizers of the regularized approximate problems to the minimizers of the original problem. Finally, we illustrate our results with examples and applications in the field of (nonsmooth) joint, semidefinite, and probust chance-constrained optimization problems. Funding: P. Pérez-Aros was supported by Centro de Modelamiento Matemático [Grants ACE210010 and FB210005] and BASAL funds for center of excellence and ANID-Chile grant Fondecyt Regular [Grants 1200283 and 1190110] and Fondecyt Exploración [Grant 13220097]. C. Soto was supported by the National Agency for Research and Development (ANID)/Scholarship Program/Doctorado Nacional Chile [Grant 2017-21170428]. E. Vilches was supported by Centro de Modelamiento Matemático [Grants ACE210010 and FB210005] and BASAL funds for center of excellence and Fondecyt Regular [Grant 1200283] and Fondecyt Exploración [Grant 13220097] from ANID-Chile.

Hölder's inequality and its reverse—A probabilistic point of view

AbstractIn this article, we take a probabilistic look at Hölder's inequality, considering the ratio of terms in the classical Hölder inequality for random vectors in . We prove a central limit theorem for this ratio, which then allows us to reverse the inequality up to a multiplicative constant with high probability. The models of randomness include the uniform distribution on balls and spheres. We also provide a Berry–Esseen–type result and prove a large and a moderate deviation principle for the suitably normalized Hölder ratio.

Almost output tracking control for switched positive systems: A hysteresis switching strategy

AbstractThis work studies the almost output tracking control problem via the co‐design of controllers and a switching strategy for switched positive systems. For devising a hysteresis switching strategy, the nonnegative state space is divided into some subdomains, where each subdomain corresponds to an activated subsystem. The main advantage of the proposed methodology is that the switching frequency can be reduced in contrast to the min‐switching based designs, and the almost output tracking control problem is allowed to be unsolvable for each subsystem. The almost output tracking criteria, based on multiple linear copositive Lyapunov functions are established in linear vector inequalities form, which ensures asymptotic convergence of the output tracking error to zero, the positivity as well as a prescribed ‐gain index in presence of disturbances. Finally, the theoretical findings are applied to the output tracking control of the boost converter circuit.

Dwell-time-dependent asynchronous dissipative control for discrete-time switched positive systems with linear supply rates

This article investigates the issues of dissipativity analysis and synthesis of discrete-time switched positive systems, in the presence of asynchronous switching. A switching retard, between activation of positive subsystems and activation of candidate controllers, is taken into account. The dwell-time dependent linear copositive storage functions and linear supply rates are adopted to capture the property of dissipativity, where the state of each subsystem is required to stay in positive orthant. Sufficient conditions are also attained such that the closed-loop system with time-varying asynchronous state-feedback controllers maintains positivity and meets strict (q,r)-α-dissipative performance. Furthermore, all derived criteria are presented in the form of linear vector inequalities. Finally, two examples including the compartmental model are offered to illustrate the validity of derived results.

Stabilization of Positive Systems With Time Delay via the Takagi-Sugeno Fuzzy Impulsive Control.

In this study, the Takagi-Sugeno (T-S) fuzzy impulsive control problem is investigated for a class of nonlinear positive systems with time delay. The time delay under consideration is both in the continuous-time dynamics and at the impulsive instants, which can model practical systems more accurately. An impulse-time-dependent copositive Lyapunov function (IDCLF) is constructed, and the Razumikhin technique is adopted to develop conditions that ensure the globally exponential stability of T-S fuzzy positive systems with delayed impulses. The size constraint between the impulse delay and the bound of impulsive intervals is removed. A T-S fuzzy impulsive controller is designed in terms of the solutions to certain vector inequalities that are readily solvable. Numerical examples and a practical example of lipoprotein metabolism and potassium ion transfer model are given to demonstrate the effectiveness, advantages, and practicality of the proposed results.

On Slater like inequality for vectors transformed by a doubly stochastic matrix with control function

In this work, we deal with a Slater type inequality designed for a symmetric convex function and for a collection of vectors transformed by a doubly stochastic matrix. In doing so, we use an additional convex control function. In the case when the composition of the control function and of the underlying convex function is Schur-concave, such an approach leads to a refinement of the standard Slater inequality. Special cases are also considered.

Vector-valued Maclaurin inequalities

We investigate a Maclaurin inequality for vectors and its connection to an Aleksandrov-type inequality for parallelepipeds.

Anti-concentration for subgraph counts in random graphs

Fix a graph H and some p∈(0,1), and let XH be the number of copies of H in a random graph G(n,p). Random variables of this form have been intensively studied since the foundational work of Erdős and Rényi. There has been a great deal of progress over the years on the large-scale behaviour of XH, but the more challenging problem of understanding the small-ball probabilities has remained poorly understood until now. More precisely, how likely can it be that XH falls in some small interval or is equal to some particular value? In this paper, we prove the almost-optimal result that if H is connected then for any x∈N we have Pr(XH=x)≤n1−v(H)+o(1). Our proof proceeds by iteratively breaking XH into different components which fluctuate at “different scales”, and relies on a new anti-concentration inequality for random vectors that behave “almost linearly.”

A supplement to the laws of large numbers and the large deviations

Let 0<p<2. Let be a sequence of independent and identically distributed -valued random variables and set . In this paper, an analogue of large deviation principle is established under assumption only. The main tools employed in proving this result are the symmetrization technique and three powerful inequalities established by Hoffmann-Jørgensen [Sums of independent Banach space valued random variables, Studia Math. 52 (1974), pp. 159–186], de Acosta [Inequalities for B-valued random vectors with applications to the law of large numbers, Ann. Probab. 9 (1981), pp. 157–161] and Ledoux and Talagrand [Probability in Banach Spaces: Isoperimetry and Processes, Springer-Verlag, Berlin, 1991], respectively. As a special case of this result, the main results of Hu and Nyrhinen [Large deviations view points for heavy-tailed random walks, J. Theoret. Probab. 17 (2004), pp. 761–768] are not only improved, but also extended.

Scalarization and Optimality Conditions for the Approximate Solutions to Vector Variational Inequalities in Banach Spaces

In this paper, we present the notion of approximate solutions for vector variational inequalities in Banach spaces, which extends the existing approximate solutions. It is shown that, under the cone subconvexlikeness, our new approximate solutions can be characterized by the approximate solutions of linear scalar problem by means of the convex separation theorem. For the nonconvex cases, based on the nonconvex separation theorem with a special scalar functional introduced by Hiriart–Urruty, we get the nonlinear scalarization results for the approximate solutions. In order to establish optimality conditions without convexity assumption, we calculate the subdifferential of the Hiriart–Urruty nonlinear scalar functional. And based on the scalar characterizations, optimality conditions are obtained by using Ekeland variational principle and Fermat rule for scalar optimization problems. Finally, we consider the special case of vector inequality problems in finite dimensional Euclidean space, and establish some relations between the approximate solutions of vector inequality problems and nonsmooth vector optimization problems.

Algebraic Solution to Constrained Bi-Criteria Decision Problem of Rating Alternatives through Pairwise Comparisons

We consider a decision-making problem to evaluate absolute ratings of alternatives from the results of their pairwise comparisons according to two criteria, subject to constraints on the ratings. We formulate the problem as a bi-objective optimization problem of constrained matrix approximation in the Chebyshev sense in logarithmic scale. The problem is to approximate the pairwise comparison matrices for each criterion simultaneously by a common consistent matrix of unit rank, which determines the vector of ratings. We represent and solve the optimization problem in the framework of tropical (idempotent) algebra, which deals with the theory and applications of idempotent semirings and semifields. The solution involves the introduction of two parameters that represent the minimum values of approximation error for each matrix and thereby describe the Pareto frontier for the bi-objective problem. The optimization problem then reduces to a parametrized vector inequality. The necessary and sufficient conditions for solutions of the inequality serve to derive the Pareto frontier for the problem. All solutions of the inequality, which correspond to the Pareto frontier, are taken as a complete Pareto-optimal solution to the problem. We apply these results to the decision problem of interest and present illustrative examples.

Input-to-state stability of impulsive reaction–diffusion neural networks with infinite distributed delays

In this paper, we focus on the global existence–uniqueness and input-to-state stability of the mild solution of impulsive reaction–diffusion neural networks with infinite distributed delays. First, the model of the impulsive reaction–diffusion neural networks with infinite distributed delays is reformulated in terms of an abstract impulsive functional differential equation in Hilbert space and the local existence–uniqueness of the mild solution on impulsive time interval is proven by the Picard sequence and semigroup theory. Then, the diffusion–dependent conditions for the global existence–uniqueness and input-to-state stability are established by the vector Lyapunov function and M-matrix where the infinite distributed delays are handled by a novel vector inequality. It shows that the ISS properties can be retained for the destabilizing impulses if there are no too short intervals between the impulses. Finally, three numerical examples verify the effectiveness of the theoretical results and that the reaction–diffusion benefits the input-to-state stability of the neural-network system.

On some Hölder type trace inequalities for operator weighted geometric mean

We obtain some Hölder type trace inequalities for operator weighted geometric mean. Some vector inequalities are also given.

Hanson–Wright inequality in Hilbert spaces with application to $K$-means clustering for non-Euclidean data

We derive a dimension-free Hanson–Wright inequality for quadratic forms of independent sub-gaussian random variables in a separable Hilbert space. Our inequality is an infinite-dimensional generalization of the classical Hanson–Wright inequality for finite-dimensional Euclidean random vectors. We illustrate an application to the generalized $K$-means clustering problem for non-Euclidean data. Specifically, we establish the exponential rate of convergence for a semidefinite relaxation of the generalized $K$-means, which together with a simple rounding algorithm imply the exact recovery of the true clustering structure.

English

We examine the problem of finding all solutions of two-sided vector inequalities given in the tropical algebra setting, where the unknown vector multiplied by known matrices appears on both sides of the inequality. We offer a solution that uses sparse matrices to simplify the problem and to construct a family of solution sets, each defined by a sparse matrix obtained from one of the given matrices by setting some of its entries to zero. All solutions are then combined to present the result in a parametric form in terms of a matrix whose columns form a complete system of generators for the solution. We describe the computational technique proposed to solve the problem, remark on its computational complexity and illustrate this technique with numerical examples.

Stability of the Shannon\u2013Stam inequality via the F\xf6llmer process

We prove stability estimates for the Shannon–Stam inequality (also known as the entropy-power inequality) for log-concave random vectors in terms of entropy and transportation distance. In particular, we give the first stability estimate for general log-concave random vectors in the following form: for log-concave random vectors X,Y in {mathbb {R}}^d, the deficit in the Shannon–Stam inequality is bounded from below by the expression CDX||G+DY||G,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} C \\left( \\mathrm {D}\\left( X||G\\right) + \\mathrm {D}\\left( Y||G\\right) \\right) , \\end{aligned}$$\\end{document}where mathrm {D}left( cdot ~ ||Gright) denotes the relative entropy with respect to the standard Gaussian and the constant C depends only on the covariance structures and the spectral gaps of X and Y. In the case of uniformly log-concave vectors our analysis gives dimension-free bounds. Our proofs are based on a new approach which uses an entropy-minimizing process from stochastic control theory.