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• Double bias fields are introduced into fidelity term to approximate image intensity inhomogeneity. • The proposed energy function is strictly convex, and allows flexible initialization. • A TV (total variation) regularization term is introduced to keep convex level-set function smooth. • The proposed method is robust against to noise and intensity inhomogeneity. The existing active contour models (ACMs) based on bias field (BF) correction mostly rely on a single BF assumption and lack in-depth discussion on the convexity of the energy functional, often leading to the problem of local minima. To address this issue, this paper introduces a dual BF and proposes a convex level-set (LS) method based on multiplicative-additive (MA) model to achieve global minima. Firstly, a MA model is adopted as the fidelity term, and a kernel function is introduced to adjust the size of the intensity inhomogeneous neighborhood, enhancing the adaptability to intensity inhomogeneity. Then, the convex LS function is embedded in the variational framework to ensure convexity of each variable in the energy functional. This transformation turns the segmentation problem into a convex optimization problem. By introducing the total variation regularization term to smooth the LS function, the model's resistance to noise is effectively enhanced. Finally, by minimizing the proposed energy functional, image segmentation and BF correction are successfully achieved. Experimental results validate the global minima property of our model, while also demonstrating good flexibility in the initial contour. The proposed model achieves superior segmentation results compared to other classical ACMs on various types of images. [ABSTRACT FROM AUTHOR]

In this paper, we study a class of nonconvex and nonsmooth fractional optimization problem, where the numerator of which is the sum of a nonsmooth and nonconvex function and a relative smooth nonconvex function, while the denominator is relative weakly convex nonsmooth function. We propose a Bregman proximal subgradient algorithm for solving this type of fractional optimization problems. Under moderate conditions, we prove that the subsequence generated by the proposed algorithm converges to a critical point, and the generated sequence globally converges to a critical point when the objective function satisfies the Kurdyka-Łojasiewicz property. We also obtain the convergence rate of the proposed algorithm. Finally, two numerical experiments illustrate the effectiveness and superiority of the algorithm. Our results give a positive answer to an open problem proposed by Bot et al. [14]. [ABSTRACT FROM AUTHOR]

In this paper, a novel non-convex fractional-order image denoising model is proposed to suppress the staircase effect produced by the TV model while maintaining a neat contour. The model combines ℓ q (0 < q < 1) quasi-norm and fractional-order regularization, and employs a diffusion coefficient with a faster convergence rate to preserve more image edges and details. Additionally, an adaptive regularization parameter is designed to adjust the denoising performance of the algorithm. To obtain the optimal approximate solution of the model, an enhanced primal-dual algorithm is adopted and the complexity and convergence of the algorithm are theoretically analyzed. Finally, the effectiveness of the proposed method is demonstrated through numerical experiments. • A new non-convex FOTV model is proposed. • A new diffusion coefficient is introduced to retain more edge details of the image. • The existing primal-dual algorithm is improved, and the convergence of this algorithm is analyzed. • An adaptive regularization parameter is designed. [ABSTRACT FROM AUTHOR]

The general first Zagreb index of a graph G , denoted by M 1 p (G) , is defined as the sum of powers d G p (u) over all vertices u of V (G) , where d G (u) denotes the degree of a vertex u in G. In this paper, we consider negative values of p and obtain sharp lower bounds on the general first Zagreb index of trees, unicyclic and bicyclic graphs in terms of their order and maximum vertex degrees. Also, the corresponding extremal graphs attaining the bounds are characterized. The results are then extended to other indices defined as sums over all vertices of contributions which are convex or concave functions of their degrees. [ABSTRACT FROM AUTHOR]

This paper is devoted to the stability analysis of a class of discrete-time switched systems under newly designed switching regularities. The utilized switching is extended from the primitive mode-dependent persistent dwell-time switching. By integrating the existing admissible edge-dependent average dwell-time switching into the mode-dependent persistent dwell-time regime, a novel admissible edge-dependent average persistent dwell-time strategy involving the notion of admissible transition edges is proposed. The integrated admissible edge-dependent average dwell-time switching has been confirmed to be superior to mode-dependent average dwell-time switching. This superiority makes the proposed switching more general than the relaxed mode-dependent persistent dwell-time switching in the literature. Even if the embedded admissible edge-dependent average dwell-time restrictions degrade to admissible edge-dependent dwell-time ones under special parameter settings, the resulting admissible edge-dependent persistent dwell-time switching retains the advantage of admissible transition edges to generalize the original mode-dependent persistent dwell-time switching. Meanwhile, to remove the switching information reliance in the existing multiple convex Lyapunov functions, an improved multiple convex Lyapunov function method is devised. The designed Lyapunov function can be used to implement broader feasible ranges and tighter lower bounds of admissible edge-dependent average persistent dwell-time (or admissible edge-dependent persistent dwell-time). Finally, the validity and efficiency of the presented approaches are illustrated by a three-phase inverter and a numerical example. • Admissible edge-dependent average persistent dwell-time switching is proposed. • Admissible edge-dependent persistent dwell-time switching is proposed. • A more widely applicable multiple convex Lyapunov function is raised. [ABSTRACT FROM AUTHOR]

The concept of jump system, introduced by Bouchet and Cunningham (1995), is a set of integer points satisfying a certain exchange property. We consider the minimization of a separable convex function on a jump system. It is known that the problem can be solved by a greedy algorithm. In this paper, we are interested in whether the greedy algorithm has the geodesic property, which means that the trajectory of the solutions generated by the algorithm is a geodesic from the initial solution to a nearest optimal solution. We show that a special implementation of the greedy algorithm enjoys the geodesic property, while the original algorithm does not. As a corollary to this, we present a new greedy algorithm for linear optimization on a delta-matroid and show that the algorithm has the geodesic property. [ABSTRACT FROM AUTHOR]

For a graph G = (V G , E G) , a subset S ⊆ V G is called a maximum dissociation set if the induced subgraph G [ S ] does not contain P 3 as its subgraph, and the subset has maximum cardinality. The dissociation number of G is the number of vertices in a maximum dissociation set of G. This paper mainly studies the problem of determining the maximum values of the vertex-degree function index H f (G) = ∑ v ∈ V G f (d (v)) and characterizing the corresponding extremal graphs among all trees and unicyclic graphs with fixed order and dissociation number when f (x) is a strictly convex function. Firstly, we describe all the trees having the maximum vertex-degree function index H f (T) among trees with given order and dissociation number when f (x) is a strictly convex function. Then we determine the graphs having the maximum vertex-degree function index H f (T) among unicyclic graphs with given order and dissociation number when f (x) is a strictly convex function and satisfies f (5) + 2 f (3) + f (2) > 3 f (4) + f (1). [ABSTRACT FROM AUTHOR]

• We propose a nonconvex prior for image smoothing. • The proposed algorithm of the nonconvex model has a convergence guarantee. • The performance of the prior outperforms the state-of-the-art. • The proposed method is flexible, convergent and has promising results. Image smoothing is a fundamental task in computer vision and graphics. This paper presents a new image smoothing method based on a truncated generalized Huber prior. The proposed model is neither convex nor concave and is hard to optimize. We first transform the prior into a concave one, then utilize the technique of half-quadratic minimization to get an equivalent convex surrogate function. Thus the numerical algorithm is obtained by solving a weighted least square problem and iteratively updating the weights. The convergence of the algorithm is theoretically guaranteed. The proposed method is flexible and powerful in preserving edge/structure and eliminating undesired information. The effectiveness of the proposed method is demonstrated by several applications, including scale space filtering, texture removal, and clip-art JPEG artifacts removal. [ABSTRACT FROM AUTHOR]

This paper proposes a deep learning approach for solving non-smooth convex optimization problems (NCOPs), which have broad applications in computer science, engineering, and physics. Our approach combines neurodynamic optimization with physics-informed neural networks (PINNs) to provide an efficient and accurate solution. We first use neurodynamic optimization to formulate an initial value problem (IVP) that involves a system of ordinary differential equations for the NCOP. We then introduce a modified PINN as an approximate state solution to the IVP. Finally, we develop a dedicated algorithm to train the model to solve the IVP and minimize the NCOP objective simultaneously. Unlike existing numerical integration methods, a key advantage of our approach is that it does not require the computation of a series of intermediate states to produce a prediction of the NCOP. Our experimental results show that this computational feature results in fewer iterations being required to produce more accurate prediction solutions. Furthermore, our approach is effective in finding feasible solutions that satisfy the NCOP constraint. [ABSTRACT FROM AUTHOR]

Conjugate gradient methods are much effective and widely used for large-scale unconstrained optimization problems by their simple computation, low memory requirement and strong global convergence property. Spectral gradient methods are also effective for large-scale problems. In this paper, a class of new descent spectral three-term conjugate gradient algorithms are proposed which automatically have the sufficient descent property and satisfy the Dai-Liao conjugate condition. Under the Wolfe line search technique and some standard conditions, the proposed methods are globally convergent for strongly convex functions and general nonlinear functions with the help of the modified secant equations. In numerical part, 732 problems with dimensions varying from 1500 to 150000 and three image restoration problems with three noise levels are considered. Numerical results indicate that the proposed algorithms are more efficient, reliable and robust than the other methods for the testing problems. • A class of spectral descent three-term conjugate gradient methods are proposed. • The proposed methods own the sufficient descent property and satisfy the Dai-Liao conjugate condition automatically. • The proposed methods are globally convergent for general functions without convex assumptions. • The proposed methods are applied to large-scale unconstrained optimization problems and some image restoration problems. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a three-operator splitting algorithm with deviations for solving the minimization problem composed of the sum of two convex functions minus a convex and smooth function in a real Hilbert space. The main feature of the proposed method is that two per-iteration deviation vectors provide additional degrees of freedom. We propose one-step and two step inertial three-operator splitting algorithms by selecting the deviations along a momentum direction. A numerical experiment for DC regularized sparse recovery problems shows that the proposed algorithms have better performance than the original three-operator splitting algorithm. [ABSTRACT FROM AUTHOR]

The present paper extends some previous works studying automorphisms in type-2 fuzzy sets. The framework for the analysis is the set of convex and normal functions from [ 0 , 1 ] to [ 0 , 1 ] (fuzzy truth values). The paper concentrates on those automorphisms that, in this framework, leave the constant function 1 fixed. This function is quite important since it defines the boundary between the functions that represent "TRUE" (increasing functions) and those that represent "FALSE" (decreasing functions), being at the same time the only normal function that is simultaneously increasing and decreasing. While C.L. Walker, E.A. Walker and J. Harding introduced in 2008 a family of functions leaving the constant function 1 fixed, the main goal of this paper is to prove that the functions of that family are in fact automorphisms, and moreover, that they are the only automorphisms (in the mentioned set of convex and normal functions from [ 0 , 1 ] to [ 0 , 1 ]) that preserve the function 1. [ABSTRACT FROM AUTHOR]

In many high-dimensional datasets, the phenomenon that features are relevant often occurs. Elastic-net regularization is widely used in support vector machines (SVMs) because it can automatically perform feature selection and encourage highly correlated features to be selected or removed together. Recently, some effective algorithms have been proposed to solve the elastic-net SVMs with different convex loss functions, such as hinge, squared hinge, huberized hinge, pinball and huberized pinball. In this paper, we develop a linearized alternating direction method of multipliers (LADMM) algorithm to solve above elastic-net SVMs. In addition, our algorithm can be applied to solve some new elastic-net SVMs such as elastic-net least squares SVM. Compared with some existing algorithms, our algorithm has comparable or better performances in terms of computational cost and accuracy. Under mild conditions, we prove the convergence and derive convergence rate of our algorithm. Furthermore, numerical experiments on synthetic and real datasets demonstrate the feasibility and validity of the proposed algorithm. • We develop a linearized ADMM algorithm for effectively solving a series of elastic-net SVMs. • The proposed algorithm is accompanied by convergence and complexity analysis. • A new elastic-net SVM called LESVM is proposed in this paper. [ABSTRACT FROM AUTHOR]

In the degree-constrained graph orientation problem , the input is an unweighted, undirected graph G = (V , E) and nonnegative integers a v and b v (with a v ≤ b v) for each v ∈ V , and the objective is to assign a direction to every edge in E in such a way that for each v ∈ V , the number of outgoing edges in the resulting directed graph lies in the interval [ a v , b v ]. When such an orientation does not exist, it is desirable to find an orientation that best fits the condition instead. In this paper, we consider the problem of finding an orientation that minimizes the total penalty ∑ v ∈ V c v , where c v is a penalty incurred whenever a vertex v violates its degree constraints. As penalty functions, convex, concave, and step functions are considered in this paper. We show that the problem with any convex penalty function can be solved in O (| E | 1.5 min ⁡ { log ⁡ (| E | ⋅ C) , | E | 0.5 log ⁡ Δ log ⁡ | E | }) time, where Δ and C are the maximum degree and the largest magnitude of a penalty, respectively. In contrast, we show APX-hardness of the problem with step or concave functions. For trees and graphs with treewidth τ , the problem with any penalty function can be solved exactly in O (| V | log ⁡ Δ) time and O (τ 2 Δ 2 τ + 2 | V |) time, respectively. Finally, we consider the generalization of the problem to edge-weighted graphs and establish strong bounds on its inapproximability that hold even in the special case of stars. On the positive side, we can extend our algorithms for unweighted version of the problem to obtain pseudo-polynomial-time algorithms for the edge-weighted problem variant when restricted to trees and graphs with bounded treewidth. Also, we design a PTAS and a linear-time algorithm for stars with further restrictions on the degree constraints and edge weights. [ABSTRACT FROM AUTHOR]

Dynamic networks are popularly used to describe networks that change with time. Although there have been a large number of research works on understanding dynamic networks using link prediction, node classification and community detection, there is rare work that is specially designed to address the challenge of big network size of dynamic networks. To this end, we study in this paper an emerging and challenging problem of network coarsening in dynamic networks. Network coarsening refers to a class of network "zoom-out" operations where node pairs and edges are grouped together for efficient analysis on big networks. However, existing network coarsening approaches can only handle static networks where network structure weights have been predefined before the coarsening calculation. Under the observation that big networks are highly dynamic and naturally change over time, we consider in this paper to embed information diffusion data which reflect the dynamics of networks for network coarsening. Specifically, we present a new Semi-NetCoarsen approach that jointly maximizes the likelihood of observing the information diffusion data and minimizes the network regularization with respect to the predefined network structural data. The learning function is convex and we use the accelerated proximal gradient algorithm to obtain the global optimal solution. We conduct experiments on two synthetic and five real-world data sets to validate the performance of the proposed method. [ABSTRACT FROM AUTHOR]

[Display omitted] • A model for multi-objective optimization operation of multiple water sources under uncertainties was proposed. • Uncertainties in inflow and water demand forecasts can be considered simultaneously, and the correlation between uncertainties can be considered quantitatively. • The impact of inflow-water demand forecast uncertainties and their correlations on water resource operations are explored for water resources managers. Uncertainties in inflow and water demand forecasts bring risks to water resources management. Therefore, the research on multi-objective optimization operation of multiple water sources under uncertainties holds critical research significance and application value. This paper proposes a model for the multi-objective optimization operation of multiple water sources that can consider the inflow and water demand forecast dual uncertainties. In handling uncertainties, this paper adopts a scenario generation method based on joint distribution-Monte Carlo (JDMC), which allows for a quantitative consideration of the correlation between uncertainties. Moreover, the variable weight method and LINGO 20.0 are employed for solving multi-objective optimization issues to obtain stable Pareto frontiers. The proposed model is applied to the water receiving area inside Jiangsu Province of the South-to-North Water Transfer East Route Project (SNWTERP) in China. The main findings are as follows: 1) The impact of the inflow and water demand forecast uncertainties on the operation results is synergistic rather than antagonistic. Considering only a single uncertainty may result in water receiving area facing more severe water shortages or purchasing spot water at high prices. 2) There is a strong spatial correlation between uncertainties. If we ignore or only qualitatively consider the correlation between uncertainties, the impact of uncertainties on operation results will be underestimated or overestimated. 3) The Pareto frontiers are convex functions, indicating that decision-makers should choose a solution in the middle part. [ABSTRACT FROM AUTHOR]

In this paper, we study the Differentially Private Empirical Risk Minimization (DP-ERM) problem, considering both convex and non-convex loss functions. For cases where DP-ERM involves smooth (strongly) convex loss functions with or without non-smooth regularization, we propose several novel methods. These methods achieve (near) optimal expected excess risks (i.e., utility bounds) while reducing the gradient complexity compared to existing approaches. When dealing with DP-ERM and smooth convex loss functions in high dimensions, we introduce an algorithm that achieves a superior upper bound with lower gradient complexity than previous solutions. In the second part of the paper, for DP-ERM with non-convex loss functions, we explore both low and high dimensional spaces. In the low dimensional case with a non-smooth regularizer, we extend an existing approach by measuring the utility using the ℓ 2 norm of the projected gradient. Furthermore, we introduce a novel error bound measurement, transitioning from empirical risk to population risk by employing the expected ℓ 2 norm of the gradient. For the high dimensional case, we demonstrate that by measuring utility with the Frank-Wolfe gap, we can bound the utility using the Gaussian Width of the constraint set, instead of the dimensionality (p) of the underlying space. We also show that the advantages of this approach can be achieved by measuring the ℓ 2 norm of the projected gradient. Finally, we reveal that the utility of certain special non-convex loss functions can be reduced to a level (depending only on log ⁡ p) similar to that of convex loss functions. [ABSTRACT FROM AUTHOR]

This paper investigates the bounded synchronization of the discrete-time master–slave neural networks (MSNNs) with uncertainty. To deal with the unknown parameter in the MSNNs, a parameter adaptive law combined with the impulsive mechanism is proposed to improve the estimation efficiency. Meanwhile, the impulsive method also is applied to the controller design for saving the energy. In addition, a novel time-varying Lyapunov functional candidate is employed to depict the impulsive dynamical characteristic of the MSNNs, wherein a convex function related to the impulsive interval is used to obtain a sufficient condition for bounded synchronization of the MSNNs. Based on the above condition, the controller gain is calculated utilizing an unitary matrix. An algorithm is proposed to reduce the boundary of the synchronization error by optimizing its parameters. Finally, a numerical example is provided to illustrate the correctness and the superiority of the developed results. [ABSTRACT FROM AUTHOR]

In this paper, a subgradient-based neurodynamic algorithm is presented to solve the nonsmooth nonconvex interval-valued optimization problem with both partial order and linear equality constraints, where the interval-valued objective function is nonconvex, and interval-valued partial order constraint functions are convex. The designed neurodynamic system is constructed by a differential inclusion with upper semicontinuous right-hand side, whose calculation load is reduced by relieving penalty parameters estimation and complex matrix inversion. Based on nonsmooth analysis and the extension theorem of the solution of differential inclusion, it is obtained that the global existence and boundedness of state solution of neurodynamic system, as well as the asymptotic convergence of state solution to the feasible region and the set of L U -critical points of interval-valued nonconvex optimization problem. Several numerical experiments and the applications to emergency supplies distribution and nondeterministic fractional continuous static games are solved to illustrate the applicability of the proposed neurodynamic algorithm. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the problem of recovering sparse initial conditions in the parabolic equations from final measurements. Since the problem is ill-posed, l 1 -weighted regularization is applied to solve it in a stable way. We prove that the misfit function is differentiable and strictly convex, and the regularized problem is wellposed in both continuous and discrete settings. As a result, the regularized problem is strictly convex, and its solution is sparse. Therefore, our approach is suitable to recover the sparse initial conditions. Furthermore, there are some available fast algorithms to solve the regularized problem efficiently. We will illustrate the advantages of our approach by some specific numerical examples. [ABSTRACT FROM AUTHOR]

In this paper, under the condition of granular differentiation, we consider the fuzzy optimization problems with the general fuzzy function as the objective function. Firstly, we introduce the concept of granular convexity, and propose the properties of the granular convex fuzzy functions. Secondly, we present the Karush-Kuhn-Tucker type optimality conditions of the fuzzy relative optimal solution of more general fuzzy programming problems and some test examples. Finally, the relationships between a class of variational inequalities and the fuzzy optimization problems are established. [ABSTRACT FROM AUTHOR]

In this paper, we investigate regularized online exponentially concave (abbr. exp-concave) optimization, in which each loss function consists of a time-varying exp-concave function and a fixed convex regularization. If the whole loss function is exp-concave, a classical method called online Newton step (ONS) enjoys an O (d log T) regret bound, where d is the dimensionality and T is the time horizon. However, in the regularized setting, the sum of an exp-concave function and a convex regularization is not necessarily an exp-concave function, which implies that ONS is not applicable. To address this problem, we propose the proximal online Newton step (ProxONS), and show that it can attain the same O (d log T) regret bound for any convex regularization. The main idea is to first perform an iteration of ONS with the exp-concave part in each loss function and then perform a proximal mapping with the regularization part. Furthermore, we demonstrate that by utilizing the standard online-to-batch conversion, our ProxONS can be extended to solve stochastic optimization with a regularized exp-concave objective, and enjoy an O (d log T / T) convergence rate with high probability. Experimental results on two real datasets verify the effectiveness of our ProxONS. [ABSTRACT FROM AUTHOR]

This paper presents two kinds of novel results about semiglobal synchronization in probability for stochastic systems. To this end, new criteria of semiglobal fixed-time synchronization in probability (SFXTSp) and semiglobal preassigned-time synchronization in probability (SPATSp) are proposed. In contrast to the existing results of semiglobal synchronization in probability, our criteria have wider applicability in SFXTSp and SPATSp due to their flexible forms. Meanwhile, based on the properties of convex functions and Jensen's inequality, our approaches overcome the issue arising from the non-equivalence between the Lyapunov function and its expectation, making our conclusions more rigorous. In addition, to achieve SFXTSp of stochastic neural networks with random delay (RDSNNs), a unified adaptive controller is developed. By modifying the parameters of the unified controller, RDSNNs reach SPATSp in a predetermined time. Ultimately, numerical simulations are conducted to confirm the practicality and accuracy of the derived outcomes. • Two novel semiglobal synchronization in probability results are proposed. • Our conclusions have wider applicability and address the existing challenges. • Various stochastic factors are considered, making our results more general. • Our unified controller is generalized to achieve two kinds of synchronization. [ABSTRACT FROM AUTHOR]

This paper introduces a distributed continuous-time algorithm that utilizes dual consensus to tackle the optimization problem involving time-varying (TV) local objective functions and TV coupled equality constraint. Here, the local objective functions can be any strongly convex functions. The optimum solution is represented by a trajectory rather than a fixed point, owing to the dynamic nature of the objective functions and the constraint. The initial step involves converting the studied problem into an equivalent saddle-point problem. Subsequently, we provide the optimal conditions for this transformed problem. Then a distributed continuous-time algorithm based on dual consensus is provided, guaranteeing that all agents possess the capability to discover and follow the optimal TV trajectories. It is noticeable that there are no limitations imposed on the information regarding local objective functions and the coupled equality constraint except for the strongly convexity of local objective functions. In addition, two simulation instances and the comparisons with state-of-the-art methods are performed in order to validate the proposed algorithm. • Dual consensus method is designed to address the time-varying optimization with coupled equality constraint. • The algorithm allows more flexibility for the local objective functions. • Bound restrictions on the information about local objective functions and the coupled equality constraint are removed. • The algorithm guarantees vanishing tracking errors. [ABSTRACT FROM AUTHOR]

The tensor completion(TC) problem has attracted wide attention due to its significant practical significance and broad application background. Various approaches have been used to solve this problem, including approximating the rank function through its convex envelope, the tensor nuclear norm. However, due to the gap between the rank function and its convex envelope, this approach often fails to achieve satisfactory recovery. Recent developments have highlighted that non-explicit non-convex functions, especially the one induced by the Smooth Hard(SH) shrinkage operator, can approximate the l 0 norm more effectively compared to other methods, and this function facilitates the reconstruction of noisy MRI data with minimal sampling rate requirements. In our paper, we implement the SH shrinkage operator within the TC framework, achieving accurate tensor data recovery at lower sampling rates. Moreover, we adopt a method that adaptively generates transformation matrices based on specific features of the data, which, compared to traditional TC models, can better exploit the inherent low-rank properties of tensor data. Additionally, our approach integrates a reweighting strategy to efficiently merge tensors recovered from various dimensions. Finally, a feasible minimization algorithm is proposed, which combines the non-explicit non-convex regularizer and self-adaptive transforms. It is proven that the constructed sequence of the proposed algorithm is a Cauchy sequence and can converge to a stable point. Extensive experimental results demonstrate that our proposed approach can achieve accurate tensor recovery at lower sampling rates compared to other methods. • We apply the SAT method to each dimension. • We combine the SH shrinkage operator with the SAT method. • we reweight the recovered tensors obtained from each dimension. [ABSTRACT FROM AUTHOR]

In recent years, a class of soft (deformable) robots has attracted the attention of researchers due to their continuum features being advantageous over rigid robots for dexterous and safe compliant motions. These distinct advantages are obtained from the elastomer hyperelastic material properties because continuum robots are manufactured and actuated through embedded pneumatic energy fields. In this way, the continuum deformation raises novel problems in modeling and control due to its highly nonlinear behavior. That is, the deformable shape of its components varies continuously, unlike rigid robots, suggesting that the controller should compensate for its varying continuum model. Motivated by the deformable soft robot subject to parametric uncertainties and unmodeled dynamics, this paper proposes an adaptive, instead of constant, tuning of feedback gains of a model-free controller. The tuning mechanism is based on a knowledge-based scheme using a discrete wavelet neural network (DWN) by an efficient input–output identification updating its parameters and weights online, with a gradient descent algorithm driven by the identification error. At the same time, the control feedback gains are self-tuned, minimizing a convex cost function of tracking errors. Dynamic simulations show the performance of the proposed approach considering the parameters of a real soft robot mimicking a gelatin-like behavior. Finally, some discussions on the proposed scheme's computational cost, feasibility, and viability are briefly addressed. [ABSTRACT FROM AUTHOR]

In this paper, we consider a secure full-duplex (FD) multi-user multiple-input multiple-output (MU-MIMO) network where an FD base station simultaneously communicates with users and transfers energy for untrusted energy harvesting (EH) devices. Considering that the EH users are untrusted and try to wiretap the signals, the base station adds artificial noise (AN) signals into its transmitted signals to enhance the system security. We focus on two single-objective optimization problems including (i) the sum secrecy rate maximization problem, (ii) the harvested energy maximization problem, and a multi-objective optimization problem of the secure information and EH trade-off. The goal is to jointly optimize the transmit covariance matrices of the precoding and AN jamming under the constraints on transmit power budgets, harvested energy requirements, and minimum secrecy rates. Three considered optimization problems are highly non-convex due to the non-convexity of the sum secrecy rate. To handle this challenge, we first recast the sum secrecy rate as the inner convex approximation one thanks to the difference of two convex functions (DC) approach. Then, the sequential convex programming (SCP) approach is utilized to iteratively find the optimal solution. The numerical simulations are provided to investigate the impacts of the system settings and residual self-interference on the achievable system performances in terms of the secrecy rate and harvested energy. Furthermore, the numerical results also reveal the interesting trade-offs between the secrecy rate and harvested energy. [ABSTRACT FROM AUTHOR]

This paper constructs a conjunction and a disjunction operator for type-2 fuzzy sets that cannot be expressed by the sup-t-norm extension principle applied to conjunction and disjunction of type-1 fuzzy sets by means of t-norms and t-conorms on the unit interval. This result answers affirmatively an open problem posed by Hernández et al. in 2015. The duality between the set-theoretic operators for type-2 fuzzy sets is obtained from the duality of set-theoretic operations on type-1 fuzzy sets. [ABSTRACT FROM AUTHOR]

Sparse optimization involving the L 0 -norm function as the regularization in objective function has a wide application in many fields. In this paper, we propose a projected neural network modeled by a differential equation to solve a class of these optimization problems, in which the objective function is the sum of a nonsmooth convex loss function and the regularization defined by the L 0 -norm function. This optimization problem is not only nonconvex, but also discontinuous. To simplify the structure of the proposed network and let it own better convergence properties, we use the smoothing method, where the new constructed smoothing function for the regularization term plays a key role. We prove that the solution to the proposed network is globally existent and unique, and any accumulation point of it is a critical point of the continuous relaxation model. Except for a special case, which can be easily justified, any critical point is a local minimizer of the considered sparse optimization problem. It is an interesting thing that all critical points own a promising lower bound property, which is satisfied by all global minimizers of the considered problem, but is not by all local minimizers. Finally, we use some numerical experiments to illustrate the efficiency and good performance of the proposed method for solving this class of sparse optimization problems, which include the most widely used models in feature selection of classification learning. [ABSTRACT FROM AUTHOR]

This paper addresses the problem of spacecraft six degree of freedom (6-DOF) pose tracking control with collision avoidance and field of view (FOV) pyramid-type constraints during the autonomous proximity maneuver. The constraints are modeled as pyramid envelopes, which can better represent some real cases with less conservativeness comparing with commonly used cone-shaped model. A novel modeling method is proposed to describe the pyramid-type constraints in the dual-quaternion frame. Based on the specific geometric property of the pyramid constraints, a new convex artificial potential function (APF) with only one global minimum is designed, which incorporates the pose constraints into the control design procedure. Then, an integrated APF based control law is presented to simultaneously control the rotational and translational motion of the spacecraft without violating the pyramid constraints. The stability of the closed-loop system is demonstrated through the Lyapunov theory, and numerical simulation results are carried out to show the effectiveness of the proposed control law. • The motion constraint is described as pyramid envelope with less system conservative. • A novel method is used to build the pyramid envelope model in dual quaternion frame. • A new convex potential function with clear meaning and simple form is designed. • An artificial potential function based feedback 6-DOF control law is designed. • Numerical simulations verify the effectiveness of the proposed control law. [ABSTRACT FROM AUTHOR]

In this paper, we present generalized Jensen-Mercer inequality for a generalized h -convex function on fractal sets. We proved Hermite-Hadamard-Mercer local fractional integral inequalities via integral operators pertaining Mittag-Leffler kernel. Also, we drive two new local fractional integral identities for differentiable functions. By employing these integral identities, we derive some new Hermite-Mercer type inequalities for generalized h -convex function in local fractional calculus settings. Finally, we give some examples to emphasize the applications of derived results. These results will be a significant addition to Jensen-type inequalities in the literature. [ABSTRACT FROM AUTHOR]

In this paper, we focus on the multiclass bi-criteria (time and toll) traffic assignment (MBTA) problem. The conventional MBTA model keeps multiple copies of class-specific variables to model user heterogeneity, which puts a great burden on memory storage and computational speed when dealing with real transportation networks. This paper proposes an alternative formulation for the MBTA problem without class-specific variables by exploiting the order information of paths and travelers, i.e., high value of time (VOT) travelers will prefer fast but expensive paths, while low VOT travelers will prefer slow but cheap paths. We prove the equivalence of the alternative formulation to the conventional MBTA model. To solve the alternative formulation with a nondifferentiable convex objective function, a path-based subgradient projection algorithm is developed utilizing the subgradient and available second-order information. We adopt a small network and several large networks to examine the detailed features and the computational performance of the proposed formulation and algorithm, respectively. The results show that the alternative formulation provides the same link flow pattern as that of the conventional MBTA model but uses much fewer variables, which can greatly relieve the burden on computer memory and computational speed in solving real transportation networks. • Propose an alternative formulation for the multiclass bi-criteria traffic assignment (MBTA) problem without class-specific variables. • Prove the equivalence between the alternative formulation and the conventional MBTA model. • Develop a subgradient projection (SGP) algorithm for solving the alternative MBTA formulation. • Demonstrate features of the alternative MBTA model and applicability of the SGP algorithm to large-scale networks. [ABSTRACT FROM AUTHOR]

As two important types of generalized convex functions, pseudoconvex and quasiconvex functions appear in many practical optimization problems. The lack of convexity poses some difficulties in solving pseudoconvex optimization with quasiconvex constraint functions. In this paper, we propose a one-layer recurrent neural network for solving such problems. We prove that the state of the proposed neural network is convergent from the feasible region to an optimal solution of the given optimization problem. We show that the proposed neural network has several advantages over the existing neural networks for pseudoconvex optimization. Specifically, the proposed neural network is applicable to optimization problems with quasiconvex inequality constraints as well as affine equality constraints. In addition, parameter matrix inversion is avoided and some assumptions on the objective function and inequality constraints in existing results are relaxed. We demonstrate the superior performance and characteristics of the proposed neural network with simulation results in three numerical examples. [ABSTRACT FROM AUTHOR]

In this paper, we establish a novel identity by leveraging fractional integral operators of Riemann-type. Subsequently, we employ this identity to investigate Bullen's fractional integral inequalities within the framework of classical convex functions. The validity of these innovative inequalities is verified through the visualization of 2D and 3D graphs. Furthermore, we obtain certain well-known results as special cases to our main findings and express them as remarks. Lastly, we derive differences in mean estimates, which serve as valuable tools for representing complex information, facilitating decision-making and aiding analysis in various fields. [ABSTRACT FROM AUTHOR]

In this paper, we present a multiplicative fractional integral identity. Based upon it, we establish the Hermite–Hadamard type inequalities for multiplicatively convex functions via multiplicative Riemann–Liouville fractional integrals. We also give some applications to special means of real numbers in the framework of multiplicative calculus. To make the results obtained here more intuitive for the readers, we provide some examples for suitable choices of multiplicative convex functions and their graphical descriptions. • A multiplicative fractional *integral identity is developed. • Fractional HH type inequalities for *differentiable functions are established. • Applications of the obtained results are considered as well. [ABSTRACT FROM AUTHOR]

This paper addresses the global optimal consensus problem with time-varying objective functions and state-coupled inequality constraints. By integrating the predefined-time consensus and prediction-correction schemes, a distributed optimization protocol is proposed for the cooperative interception scenarios involving the unmanned aerial vehicle (UAV) swarm. Specifically, the δ -exact penalty method is utilized to eliminate the coupled constraints. A prediction-correction term with the penalty function is then established to track the trajectory of the time-varying optimal solution. Additionally, an edge-based predefined-time consensus term is introduced to facilitate the state consensus among UAVs. Under the undirected and connected topology, the proposed protocol ensures the predefined-time state consensus and asymptotically minimizes the sum of individual convex objective functions. Furthermore, the protocol is extended to other scenarios involving switching topology, directed detail-balanced topology, and external disturbances. Numerical simulations illustrate that the UAV swarm can form the predesigned interception formation with a predefined-time frame and effectively track the target swarm. [ABSTRACT FROM AUTHOR]

In this paper, we show the connections of inequalities involving fractional integrals with Ohlin lemma and Levin-Stechkin theorem. First, we give a new proof of the fractional version of the inequality of Hermite Hadamard type and then we extend it in two directions. Thus we compare the fractional expression occurring in this inequality with the usual integral and we obtain stronger inequalities (one of them is related to Bullen inequality). • The use of stochastic orderings tools in the theory of inequalities for fractional integrals. • A new proof of the fractional version of the Hermite-Hadamard inequality. • Inequalities between the fractional integral and the regular one. • Refinements of Hermite-Hadamard inequality (in particular a fractional version of Bullen inequality). [ABSTRACT FROM AUTHOR]

• Establishment of a power-gas network system model for efficient energy absorption and smart grid load shifting. • Introduction of a novel ADMM-based cooperative regulation scheme, enhancing convergence through dynamic step size adjustment. • Development of an improved cooperative scheme, addressing inequality constraints with privacy guarantees and flexible initial conditions. This paper considers the optimal capacity allocation of integrated energy systems (IESs) with the power-gas systems for clean energy consumption. First, power-gas network models are established with equality and inequality constraints. Second, a novel full distributed cooperative optimal regulation scheme is designed to deal with the optimal capacity allocation problem. Moreover, to deal with the inequality constraints in IESs, a distributed projection operator is developed such that the system states can be guaranteed within the constraint conditions. Then the convergences of those distributed optimization algorithms were proved for strictly convex objective functions. And the initial conditions of systems for the algorithm are free. The simulation is provided to demonstrate the effectiveness of the distributed optimazation approach. [ABSTRACT FROM AUTHOR]

Combined heat and power economic dispatch (CHPED) can enhance energy efficiency compared with conventional economic dispatch (ED). From the optimization standpoint, the CHPED problem usually involves the nonlinear products of heat and power generation variables, nonconvex objective functions, and nonconvex feasible operating range. Thus, its solution method should be able to cope with the problematic nonconvex problem since finding a poor solution for the CHPED implies reducing the maximum achievable efficiency. This paper presents an effective method utilizing several mathematical transformations to cope with the nonlinear, nonconvex terms. The method transforms the nonconvex regions and nonlinear functions into convex polyhedrons and segments. Then, the method formulates the polyhedrons and segments with integer variables, logical constraints, and combinatorial restrictions. Thus, we derive a mixed integer model, which optimization software can better solve. Simulation results illustrate the effectiveness of the method presented and its advantages compared with existing CHPED solution techniques in the literature. [Display omitted] • Transforming nonconvex feasible operating ranges in CHP units into some polyhedrons. • Converting nonconvex cost functions into linear segments. • Presenting a robust solution method for the optimal operation of a CHP. [ABSTRACT FROM AUTHOR]

In this paper, we derive the time-fractional Cahn–Hilliard equation from continuum mixture theory with a modification of Fick's law of diffusion. This model describes the process of phase separation with nonlocal memory effects. We analyze the existence, uniqueness, and regularity of weak solutions of the time-fractional Cahn–Hilliard equation. In this regard, we consider degenerating mobility functions and free energies of Landau, Flory–Huggins and double-obstacle type. We apply the Faedo–Galerkin method to the system, derive energy estimates, and use compactness theorems to pass to the limit in the discrete form. In order to compensate for the missing chain rule of fractional derivatives, we prove a fractional chain inequality for semiconvex functions. The work concludes with numerical simulations and a sensitivity analysis showing the influence of the fractional power. Here, we consider a convolution quadrature scheme for the time-fractional component, and use a mixed finite element method for the space discretization. [ABSTRACT FROM AUTHOR]

In this paper the problem of maximizing vertex-degree function index H f (G) for trees and unicyclic graphs G of order n and independence number s is solved for strictly convex functions f (x). In the case of unicyclic graphs f (x) must also satisfy strict inequality f (4) + 3 f (2) > 3 f (3) + f (1). These conditions are fulfilled by general first Zagreb index 0 R α (G) for α > 2 , second multiplicative Zagreb index ∏ 2 (G) and sum lordeg index S L (G). The extremal graphs are unique and they are stars or have diameter equal to three or to four. The same results are valid for the corresponding minimization problem when f (x) is strictly concave and the inequality is reversed. [ABSTRACT FROM AUTHOR]

Convex optimization is explored in this paper to maximize the security against power analysis attacks while minimizing the power and performance overheads for a cryptographic circuit employs a random dynamic voltage and frequency scaling (RDVFS) technique. To realize this optimization goal, firstly, non-convex functions are used for modeling all the critical parameters related to the RDVFS technique to create a non-convex objective function. Subsequently, the non-convex objective function is appropriately converted into a convex objective function in order to grab the global optimum value. By transforming the acquired convex objective function into the Lagrange dual function and using Karush–Kuhn–Tucker (KKT) conditions to solve this optimization problem, the best RDVFS strategy can be uncovered. As shown in the result, after employing convex optimization for a cryptographic circuit with a RDVFS technique, the corresponding figure-of-merit (FOM) is enhanced by 43.4%. Moreover, in comparison with the conventional machine learning technique, the proposed convex optimization technique improves the overall FOM related with RDVFS technique by 17.6% while reducing 85.3% computational time. • Random dynamic voltage and frequency scaling (RDVFS) is beneficial for improving the robustness of cryptographic circuits. • RDVFS technique undermines the performance and power saving of cryptographic circuits. • Convex optimization can be utilized for improving the security of RDVFS technique without increasing the overall overhead significantly. • Convex optimization methodology has a better efficiency for optimizing RDVFS technique as compared to the training methodology related with MLP neural networks. [ABSTRACT FROM AUTHOR]

The spread of a matrix is defined as the maximum of the distances between any two eigenvalues of that matrix. In this paper we investigate spread maximization as a function on a compact convex subset of the set of real symmetric matrices. We provide some general results and we further study the spread maximizing problem on S n [ a , b ] (the set of symmetric matrices with entries restricted to the interval [ a , b ]). In particular we develop some results by X. Zhan (see [13]), S. M. Fallat and J. J. Xing (see [3]). [ABSTRACT FROM AUTHOR]

In this paper we present efficient algorithms to compute the radius, diameter, incenter, circumcenter, width and k dimensional enclosing cylinder for convex polyhedral and convex polyhedral offset distance functions in plane and in ℜ d for several types of inputs. We get optimal algorithms when the time complexities are linear. Let the size of input (convex polyhedron P /set S of points/convex polyhedra) be n / N , respectively, size of distance function convex polygon/polyhedron Q be m and size of constraint convex polyhedral region R be r in plane or ℜ d. The radius, incenter and circumcenter of the convex polygon P in ℜ 2 can be optimally computed in linear time, i.e. O (n + m) and in time O (n + m + r) , if the incenter or circumcenter is additionally constrained in a convex polygon/polyhedron R. In ℜ d , the radius, incenter and circumcenter of a convex polyhedron as well as of a set of convex polyhedra, unconstrained or constrained, can be computed in O (N m) and O (N m + r) time respectively, for convex constraint polyhedral region R. We also compute the minimum stabbing sphere for a set of convex polyhedra, unconstrained or constrained, for the above mentioned distance functions in O (N m) -time or O (N m + r) -time respectively for convex constraint polyhedral region R in ℜ d. The diameter of a convex polygon in plane can be computed in O (n + m) time and the diameter of a convex polyhedron in ℜ d can be computed in O (n m) time. The width of a convex polyhedron can be computed in O (n + m) time and O (n 2 m) time, for ℜ 2 and ℜ d , respectively. The diameter and width of a set of convex polyhedra can be computed in O (N m) -time and (N 2 m) , respectively, in any dimensions. We also show how the k dimensional minimum enclosing/stabbing cylinder for the convex polyhedron P can be computed in O (n d − k + 2 m 2) in ℜ d and in O (n d m (n + m)) in ℜ d for k = 1. The k dimensional minimum enclosing/stabbing cylinder for the set of convex polyhedra S in ℜ d can be computed in O (N d − k + 2 m 2) -time. The diameter and width problems can also be solved for a set of points in ℜ d , either in time O (n m) or in time O (m T (n)) , where T (n) is the time complexity of the best convex hull computation algorithm for the given set of points. [ABSTRACT FROM AUTHOR]

Due to the non-smoothness of the Hinge loss in SVM, it is difficult to obtain a faster convergence rate with modern optimization algorithms. In this paper, we introduce two smooth Hinge losses ψ G (α ; σ) and ψ M (α ; σ) which are infinitely differentiable and converge to the Hinge loss uniformly in α as σ tends to 0. By replacing the Hinge loss with these two smooth Hinge losses, we obtain two smooth support vector machines (SSVMs), respectively. Solving the SSVMs with the Trust Region Newton method (TRON) leads to two quadratically convergent algorithms. Experiments in text classification tasks show that the proposed SSVMs are effective in real-world applications. We also introduce a general smooth convex loss function to unify several commonly-used convex loss functions in machine learning. The general framework provides smooth approximation functions to non-smooth convex loss functions, which can be used to obtain smooth models that can be solved with faster convergent optimization algorithms. [ABSTRACT FROM AUTHOR]