Convex Functions Research Articles

Convex Regularized Recursive Minimum Error Entropy Algorithm

It is well known that the recursive least squares (RLS) algorithm is renowned for its rapid convergence and excellent tracking capability. However, its performance is significantly compromised when the system is sparse or when the input signals are contaminated by impulse noise. Therefore, in this paper, the minimum error entropy (MEE) criterion is introduced into the cost function of the RLS algorithm in this paper, with the aim of counteracting the interference from impulse noise. To address the sparse characteristics of the system, we employ a universally applicable convex function to regularize the cost function. The resulting new algorithm is named the convex regularization recursive minimum error entropy (CR-RMEE) algorithm. Simulation results indicate that the performance of the CR-RMEE algorithm surpasses that of other similar algorithms, and the new algorithm excels not only in scenarios with sparse systems but also demonstrates strong robustness against pulse noise.

Study on the dynamic response of aluminum‐based curved panels coated with polyurea under impact loads

AbstractMetal curved panels are commonly used in engineering structures, but they may be more susceptible to damage under impact loads compared to flat panels. Polyurea coating has been proposed as a solution to improve impact resistance. Therefore, this study investigates the impact resistance performance of curved composite panels by applying polyurea coating to aluminum‐based panels with different curvatures. First, a numerical model of polyurea‐coated aluminum composite panels was established using the finite element software LS‐DYNA, and the sensitivity of model mesh and the numerical model validity were carried out. Then, the impact resistance performance of polyurea‐coated aluminum‐based flat panels was studied from the aspects such as deformation modes and energy dissipation, and the influence of polyurea layer thickness and coating position on the impact resistance performance of polyurea‐coated aluminum composite panels was analyzed. Based on the above results, numerical models of polyurea‐coated aluminum‐based curved panels under different curvatures are further established, and the impact resistance performance of composite curved panels and the structural forms for effective protection against impact loads are comprehensively analyzed. The research results indicate that double‐sided polyurea coating is the optimal coating method for both aluminum flat panels and aluminum curved panels. After the coating thickness reaches 4 mm, there is no significant improvement in the structural impact resistance performance. The impact deformation of curved panels can be divided into four stages: dynamic buckling, plastic indentation, rebound deformation, and vibration. The concave surface has strong impact and deformation resistance, and the convex surface has strong impact resistance. The research results have important theoretical implications for the safe engineering design of protective structures using metal curved panels.

Remarks on the continuity of convex functions in geodesic spaces

We explore the behavior of convex functions defined on locally polyhedral type spaces. As a relevant example of such spaces we show that locally compact $\mathbb{R}$-trees with the geodesic extension property around any point are locally polyhedral spaces.

On Functions with Monotonic Differences

Abstract Motivated by the Szostok problem on functions with monotonic differences (2005, 2007), we consider a-Wright convex functions as a generalization of Wright convex functions. An application of these results to obtain new proofs of known results as well as new results is presented.

Distributed Stochastic Consensus Optimization Using Communication-Censoring Strategy

In this paper, a novel communication-efficient distributed stochastic algorithm (referred as CO-DSA) is proposed for solving large-scale consensus optimization problems. As compared to the existing relevant work where only a sublinear convergence rate is obtained for strongly convex and smooth objective functions, CO-DSA achieves a linear convergence rate even in the presence of an event-triggered based communication-censoring strategy. Moreover, by properly setting the threshold function of the event-triggered communication scheme, CO-DSA maintains the same convergence rate as the algorithm without event-triggered communication. This means CO-DSA theoretically yields communication efficiency for free. Numerical experiments verify the theoretical findings and also show the excellent communication saving effect of CO-DSA in large distributed networks.

Distributed Stochastic Proximal Algorithm With Random Reshuffling for Nonsmooth Finite-Sum Optimization.

The nonsmooth finite-sum minimization is a fundamental problem in machine learning. This article develops a distributed stochastic proximal-gradient algorithm with random reshuffling to solve the finite-sum minimization over time-varying multiagent networks. The objective function is a sum of differentiable convex functions and nonsmooth regularization. Each agent in the network updates local variables by local information exchange and cooperates to seek an optimal solution. We prove that local variable estimates generated by the proposed algorithm achieve consensus and are attracted to a neighborhood of the optimal solution with an O((1/T)+(1/√T)) convergence rate, where T is the total number of iterations. Finally, some comparative simulations are provided to verify the convergence performance of the proposed algorithm.

Conjugation-Based Approach to the ε\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\varepsilon $\\end{document}-Subdifferential of Convex Suprema

We provide new characterizations of the ε\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\varepsilon $\\end{document}-subdifferential of the supremum of an arbitrary family of convex functions. The resulting formulas only involve approximate subdifferentials of adequate convex combinations of the data functions. Families of convex functions with a concavity-like property are introduced and their relationship with affine models is studied. The role of the lower semi-continuity of the data functions is also analyzed.

Intraoperative ECoG in bottom-of-the-sulcus syndrome using a novel flexible strip electrode.

The recording of epileptiform discharges from bottom-of-sulcus focal cortical dysplasia (BOSD) is often difficult during intraoperative electrocorticography (ECoG) due to the deep localization. We describe the use in this scenario of a new-generation electrode strip with high flexibility, easily adapted to cortical gyri and sulci. A right-handed 20-year-old male with drug-resistant focal epilepsy due to BOSD of the inferior frontal gyrus and daily focal aware seizures was evaluated for epilepsy surgery. Based on electroclinical and neuroimaging results, a focal cortectomy guided by ECoG was proposed. ECoG recordings were performed with new-generation cortical strips (Wise Cortical Strip; WCS®) and standard cortical strips. ECoG, performed on the convexity of the frontal cortical surface, recorded only sporadic spikes with both types of strips. Then, after microsurgical trans-sulcal dissection, WCS was molded along the sulcal surface of the suspected BOSD based on 3D-imaging reconstruction, showing continuous/subcontinuous 3-4-Hz rhythmic spike activity from the deepest electrode. Registration after resection of the BOSD did not show any epileptiform activity. Pathology showed dysmorphic neurons and gliosis. No surgical complications occurred. The patient is seizure-free after 12 months. This single case experience shows that highly flexible electrode strips with adaptability to cortical gyrations can identify IEDs originating from deep location and could therefore be useful in cases of bottom of the sulcus dysplasia.

A multi-objectives framework for secure blockchain in fog–cloud network of vehicle-to-infrastructure applications

The Intelligent Transport System (ITS) is an emerging paradigm that offers numerous services at the infrastructure level for vehicle applications. Vehicle-to-infrastructure (V2I) is an advanced form of ITS where diverse vehicle services are deployed on the roadside unit. V2I consists of distributed computing nodes where transport applications are parallel processed. Many research challenges exist in the presented V2I paradigms regarding security, cyber-attacks, and application processing among heterogeneous nodes. These cyber-attacks, Sybil attacks, and their attempts cause a lack of security and degrade the V2I performance in the presented paradigms. This paper presents a new secure blockchain framework that handles cyber-attacks, as mentioned earlier. This paper formulates this complex problem as a combinatorial problem, encompassing concave and convex problems. The convex function minimizes the given constraints, such as time and security risk, and the concave function improves performance and accuracy. Therefore, numerous constraints, such as time, energy, malware detection accuracy, and application deadlines, require optimization for the considered problem. Combining the jointly non-dominated sorting genetic algorithm (NSGA-II) and long short-term memory (LSTM) schemes is the best way to meet the problem’s limitations. In this study, the paper designed a malware dataset with known and unknown malware. The different kinds of malware lists (e.g., cyber-attacks) are considered in the form of known and unknown malware lists with the characteristics, size of code, where malware comes from, attack on which data, and current status of the workload after being attacked by the malware. Our main idea is to present blockchain, NSGA-II, and LSTM schemes that handle phishing, routing, Sybil, and 51% of cyber-attacks without compromising application performance. Simulation results show that the study reduces delay and energy, improves accuracy, and minimizes security risks for vehicular applications.

A new class of convex functions and applications in entropy and analysis

In this article, we introduce the concepts of k-harmonic mean and k-harmonically convex functions. As an application of k-harmonic mean, we present a model in physics. Also, we prove Jensen type, Hermite–Hadamard type and Mercer type inequalities for these functions. Further, using this results, we give new bounds for Shannon entropy and geometrically mean.

Subordination properties and coefficient problems for a novel class of convex functions

Abstract In this study, a novel family of analytical functions connected to convex functions in the open unit is introduced and investigated. Additionally, relationships between this class and other subclasses of analytic functions are deduced. Further, different results for the mentioned class and several new interesting properties are obtained.

Generalized strongly n-polynomial convex functions and related inequalities

This paper focuses on introducing and examining the class of generalized strongly n-polynomial convex functions. Relationships between these functions and other types of convex functions are explored. The Hermite–Hadamard inequality is established for generalized strongly n-polynomial convex functions. Additionally, new integral inequalities of Hermite–Hadamard type are derived for this class of functions using the Hölder–İşcan integral inequality. The results obtained in this paper are compared with those known in the literature, demonstrating the superiority of the new results. Finally, some applications for special means are provided.

Divergences Induced by the Cumulant and Partition Functions of Exponential Families and Their Deformations Induced by Comparative Convexity.

Exponential families are statistical models which are the workhorses in statistics, information theory, and machine learning, among others. An exponential family can either be normalized subtractively by its cumulant or free energy function, or equivalently normalized divisively by its partition function. Both the cumulant and partition functions are strictly convex and smooth functions inducing corresponding pairs of Bregman and Jensen divergences. It is well known that skewed Bhattacharyya distances between the probability densities of an exponential family amount to skewed Jensen divergences induced by the cumulant function between their corresponding natural parameters, and that in limit cases the sided Kullback-Leibler divergences amount to reverse-sided Bregman divergences. In this work, we first show that the α-divergences between non-normalized densities of an exponential family amount to scaled α-skewed Jensen divergences induced by the partition function. We then show how comparative convexity with respect to a pair of quasi-arithmetical means allows both convex functions and their arguments to be deformed, thereby defining dually flat spaces with corresponding divergences when ordinary convexity is preserved.

Influence of wall curvature on evolution of converged flow generated by crossing shock wave/boundary layer interaction: An experimental study

The curved surfaces widely exist in three-dimensional inward-turning inlets and variable sectional isolators, which determine the downstream evolution of the converged flow generated by the crossing shock wave/boundary layer interaction, thereby affecting the start performance and combustion efficiency of the hypersonic vehicles. In this work, the converged flow subjected to the curved surfaces is experimentally studied with the nanoparticle-based planar laser scattering method and particle image velocimetry. Results show that the velocity profiles of the converged flow on the transverse plane can be normalized at different angles. The upstream separation size is found to be decreased by the downstream convex surface due to the reduction of backpressure, especially in large-radius cases, whereas the converged flow gradually becomes smooth ought to the dependence of the acceleration effect on the velocity. The concave surface shows an opposite effect on the converged flow, which further indicates that a downstream convex surface is preferred for hypersonic inlets to weakens the flow non-uniformity to improve the inlet performance.

Analysis of machined surface quality for multi-feature standard parts

Analyzing the machined surface quality of a multi-feature part helps to find out a comprehensive solution for machining quality promotion for complex multi-feature products. It is also significant to detect the machining accuracy and performance of the machine tools. In this paper, the surface of a standard part is adopted to investigate multi-feature part machining application, which has more than 5 types of features, such as plane, bevel, curved, cylindrical, etc. Three groups of finish machining experiments with different parameters are firstly conducted to conclude the effect of features on the machined surfaces. Then, surface roughness and morphology were measured and compared for five features under the same machining parameters to reveal machined quality differences between surfaces. The effects of inclination angle and curvature on the surface quality are also theoretically analyzed. It is found that the contact area between the tool and the part varies with the inclination angle of the bevel, leading to different roughness variations. Between the plane and curved surfaces, the concave surface has the highest roughness, while the convex surface has the lowest. Moreover, the wear of four tools with different machining time is examined. It is shown that, with the increase of machining time, the tool tip wears more severely than the rake and the flank. Furthermore, the correlation between tool wear and surface roughness is also studied by regression analysis, which confirms that the quality difference of some surfaces is related to the tool wear state.

An inexact proximal linearized DC algorithm with provably terminating inner loop

Standard approaches to difference-of-convex (DC) programs require exact solution to a convex subproblem at each iteration, which generally requires noiseless computation and infinite iterations of an inner iterative algorithm. To tackle these difficulties, inexact DC algorithms have been proposed, mostly by relaxing the convex subproblem to an approximate monotone inclusion problem. However, there is no guarantee that such relaxation can lead to a finitely terminating inner loop. In this paper, we point out the termination issue of existing inexact DC algorithms by presenting concrete counterexamples. Exploiting the notion of ϵ-subdifferential, we propose a novel inexact proximal linearized DC algorithm termed tPLDCA. Despite permission to a great extent of inexactness in computation, tPLDCA enjoys the same convergence guarantees as exact DC algorithms. Most noticeably, the inner loop of tPLDCA is guaranteed to terminate in finite iterations as long as the inner iterative algorithm converges to a solution of the proximal point subproblem, which makes an essential difference from prior arts. In addition, by assuming the first convex component of the DC function to be pointwise maximum of finitely many convex smooth functions, we propose a computational friendly surrogate of ϵ-subdifferential, whereby we develop a feasible implementation of tPLDCA. Numerical results demonstrate the effectiveness of the proposed implementation of tPLDCA.

Hybrid Modified Accelerated Gradient Method for Optimization Processes

This research reveals a hybrid variant of the modified accelerated gradient method. We prove that derived iteration is linearly convergent on the set of uniformly convex functions. Performance profiles of the introduced hybrid method were numerically compared with its non-hybrid version. The analyzed characteristics were CPU time, the number of iterations and the number of function evaluations. The results of the numerical experiments show a better performance in favor of the derived hybrid accelerated model compared with its forerunner.

Two Families of m-fold Symmetric Bi-univalent Functions Involving a Linear Combination of Bazilevic Starlike and Convex Functions

In the present paper, we define two new families $K M_{\Sigma_m}(\lambda, \gamma, \delta ; \alpha)$ and $K M_{\Sigma_m}^*(\lambda, \gamma, \delta ; \beta)$ of holomorphic and m-fold symmetric bi-univalent functions associated with the Bazilevic starlike and convex functions in the open unit disk U. We find upper bounds for the first two Taylor-Maclaurin $\left|a_{m+1}\right|$ and $\left|a_{2 m+1}\right|$ for functions in these families. Further, we point out several special cases for our results.

Game Models for Ordering and Channel Decisions of New and Differentiated Remanufactured Products in a Closed-Loop Supply Chain with Sales Efforts

Environmental responsibility and economic benefits have promoted the development of closed-loop supply chains (CLSCs), and shortages and channels are considered to be two important issues in a CLSC. This paper explores the ordering and channel decisions in a CLSC with new and differentiated remanufactured products; considers the price and sales-effort-dependent demands, as well as the proportion of emergency orders determined by emergency order costs and backorder losses; and establishes integrated and decentralized CLSC game models. We introduce a stochastic sales effort, which affects two types of products. The numerical results show that sales effort and the order quantity of new and remanufactured products exhibit concave and convex functions, respectively. The upper limit of sales effort has a greater impact on supply chain decisions. High sales efforts can serve as a means of coordinating dispersed supply chains. Moreover, in different cases, the decisions of an integrated channel are better than those of a decentralized channel. Finally, whether the supply chain adopts an emergency order strategy depends on the relative cost of emergency orders and out-of-stock costs. According to this research, some management insights are also provided.

Unified robust estimation

SummaryRobust estimation is primarily concerned with providing reliable parameter estimates in the presence of outliers. Numerous robust loss functions have been proposed in regression and classification, along with various computing algorithms. In modern penalised generalised linear models (GLMs), however, there is limited research on robust estimation that can provide weights to determine the outlier status of the observations. This article proposes a unified framework based on a large family of loss functions, a composite of concave and convex functions (CC‐family). Properties of the CC‐family are investigated, and CC‐estimation is innovatively conducted via the iteratively reweighted convex optimisation (IRCO), which is a generalisation of the iteratively reweighted least squares in robust linear regression. For robust GLM, the IRCO becomes the iteratively reweighted GLM. The unified framework contains penalised estimation and robust support vector machine (SVM) and is demonstrated with a variety of data applications.