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The classical perspective of a function is a construction which transforms a convex function into one that is jointly convex with respect to an auxiliary scaling variable. Motivated by applications in several areas of applied analysis, we investigate an extension of this construct in which the scaling variable is replaced by a nonlinear term. Our construction is placed in the general context of locally convex spaces and it generates a lower semicontinuous convex function under broad assumptions on the underlying functions. Various convex-analytical properties are established and closed-form expressions are derived. Several applications are presented. [ABSTRACT FROM AUTHOR]

Based on the results discussed by Meinhardt (The Pre-Kernel as a Tractable Solution for Cooperative Games: An Exercise in Algorithmic Game Theory, volume 45 of Theory and Decision Library: Series C, Springer, Heidelberg, 2013). which presents a dual characterization of the pre-kernel by a finite union of solution sets of a family of quadratic and convex objective functions, we could derive some results related to the single-valuedness of the pre-kernel. Rather than extending the knowledge of game classes for which the pre-kernel consists of a single point, we apply a different approach. We select a game from an arbitrary game class with a single pre-kernel element satisfying the non-empty interior condition of a payoff equivalence class and then establish that the set of related and linear independent games which are derived from this pre-kernel point of the default game replicates this point also as its sole pre-kernel element. Hence, a bargaining outcome related to this pre-kernel element is stable. Furthermore, we establish that on the restricted subset on the game space that is constituted by the convex hull of the default and the set of related games, the pre-kernel correspondence is single-valued; and consequently continuous. In addition, we provide sufficient conditions that preserve the pre-nucleolus property for related games even when the default game possesses not a single pre-kernel point. Finally, we apply the same techniques to related solutions of the pre-kernel, namely the modiclus, and anti-pre-kernel, to work out replication results for them. [ABSTRACT FROM AUTHOR]

This paper studies well-posedness and parameter sensitivity of the square root LASSO (SR-LASSO), an optimization model for recovering sparse solutions to linear inverse problems in finite dimension. An advantage of the SR-LASSO (e.g., over the standard LASSO) is that the optimal tuning of the regularization parameter is robust with respect to measurement noise. This paper provides three point-based regularity conditions at a solution of the SR-LASSO: the weak, intermediate, and strong assumptions. It is shown that the weak assumption implies uniqueness of the solution in question. The intermediate assumption yields a directionally differentiable and locally Lipschitz solution map (with explicit Lipschitz bounds), whereas the strong assumption gives continuous differentiability of said map around the point in question. Our analysis leads to new theoretical insights on the comparison between SR-LASSO and LASSO from the viewpoint of tuning parameter sensitivity: noise-robust optimal parameter choice for SR-LASSO comes at the "price" of elevated tuning parameter sensitivity. Numerical results support and showcase the theoretical findings. [ABSTRACT FROM AUTHOR]

This paper considers the problem of finding maximum volume (axis-aligned) inscribed boxes in a compact convex set, defined by a finite number of convex inequalities, and presents optimization and geometric approaches for solving them. Several optimization models are developed that can be easily generalized to find other inscribed geometric shapes such as triangles, rhombi, and squares. To find the largest volume axis-aligned inscribed rectangles in the higher dimensions, an interior-point method algorithm is presented and analyzed. For two-dimensional space, a parametrized optimization approach is developed to find the largest area (axis-aligned) inscribed rectangles in convex sets. The optimization approach provides a uniform framework for solving a wide variety of relevant problems. Finally, two computational geometric (1−ε) –approximation algorithms with sublinear running times are presented that improve the previous results. History: Accepted by Antonio Frangioni, Area Editor for Design & Analysis of Algorithms—Continuous. Funding: The author is grateful for the support of Northeastern University for this research. Supplemental Material: The software that supports the findings of this study is available within the paper and its Supplemental Information (https://pubsonline.informs.org/doi/suppl/10.1287/ijoc.2022.0239) as well as from the IJOC GitHub software repository (https://github.com/INFORMSJoC/2022.0239). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/. [ABSTRACT FROM AUTHOR]

In this paper, we provide an explicit expression for the proximity operator of a perspective of any proper lower semicontinuous convex function defined on a Hilbert space. Our computation enhances and generalizes known formulae for the case when the Fenchel conjugate of the convex function has an open domain or when it is radial. We show numerically that our approach is more efficient than previous approaches in the literature and we provide several examples of nonradial functions for which the domain of its conjugate is not open and we compute the proximity operators of their perspectives. [ABSTRACT FROM AUTHOR]

Dynamic spectral risk measures define a claim's valuation bounds as supremum and infimum of expectations of the claim's payoff over a dominated set of measures. The measures at which such extrema are attained are called extreme measures. We determine explicit expressions for their Radon-Nykodim derivatives with respect to the common dominating measure. Based on the formulas found, we estimate the extreme measures in two cases. First, the dominating measure is calibrated to mid prices of options and valuation bounds are given by options bid and ask prices. Second, the dominating measure is estimated from historical mid equity prices and valuation bounds are given by historical 5-day high and low prices. In both experiments, we find that the market determines upper bounds by testing scenarios in which losses are significantly lower than expected under the dominating measure, while lower bounds by ones in which gains are only slightly lower than in the base case. [ABSTRACT FROM AUTHOR]

In this paper via the so-called Fenchel–Lagrange duality, we provide necessary local optimality conditions for a reverse convex programming problem (P) . As is well known, this duality has been first defined for convex programming problems. So, since in general problem (P) is not convex even if the data is, we first proceed to a decomposition of an equivalent problem of (P) into a family of convex minimization subproblems. Then, by means of the decomposition and the Fenchel–Lagrange duality applied to the subproblems we provide necessary local optimality conditions for the initial problem (P) . [ABSTRACT FROM AUTHOR]

We propose the truncated nonsmooth Newton multigrid method (TNNMG) as a solver for the spatial problems of the small-strain brittle-fracture phase-field equations. TNNMG is a nonsmooth multigrid method that can solve biconvex, block-separably nonsmooth minimization problems with linear time complexity. It exploits the variational structure inherent in the problem, and handles the pointwise irreversibility constraint on the damage variable directly, without regularization or the introduction of a local history field. In the paper we introduce the method and show how it can be applied to several established models of phase-field brittle fracture. We then prove convergence of the solver to a solution of the nonsmooth Euler–Lagrange equations of the spatial problem for any load and initial iterate. On the way, we show several crucial convexity and regularity properties of the models considered here. Numerical comparisons to an operator-splitting algorithm show a considerable speed increase, without loss of robustness. [ABSTRACT FROM AUTHOR]

Barrier coverage is a fundamental application in wireless sensor networks, which are widely used for smart cities. In applications, the sensors form a barrier for the intruders and protect an area through intrusion detection. In this paper, we study a new branch of barrier coverage, namely warning barrier coverage (WBC). Different from the classic barrier coverage, WBC has the inverse protect direction, which moves the sensors surrounding a dangerous region and protects any unexpected visitors by warning them away from the dangers. WBC holds a promising prospect in many danger keep out applications for smart cities. For example, a WBC can enclose the debris area in the sea and alarm any approaching ships in order to avoid their damaging propellers. One special feature of WBC is that the target region is usually dangerous and its boundary is previously unknown. Hence, the scattered mobile nodes need to detect the boundary and form the barrier coverage themselves. It is challenging to form these distributed sensor nodes into a barrier because a node can sense only the local information and there is no global information of the unknown region or other nodes. To this end, in response to the newly proposed issue of the formation of barrier cover, we propose a novel solution AutoBar for mobile sensor nodes to automatically form a WBC for smart cities. Notably, this is the first work to trigger the coverage problem of the alarm barrier, wherein the regional information is not pre-known. To pursue the high coverage quality, we theoretically derive the optimal distribution pattern of sensor nodes using convex theory. Based on the analysis, we design a fully distributed algorithm that enables nodes to collaboratively move toward the optimal distribution pattern. In addition, AutoBar is able to reorganize the barrier even if any node is broken. To validate the feasibility of AutoBar, we develop the prototype of the specialized mobile node, which consists of two kinds of sensors: one for boundary detection and another for visitor detection. Based on the prototype, we conduct extensive real trace-driven simulations in various smart city scenarios. Performance results demonstrate that AutoBar outperforms the existing barrier coverage strategies in terms of coverage quality, formation duration, and communication overhead. [ABSTRACT FROM AUTHOR]

The main result concerns the second-order differentiability of the marginal function of a convex twice differentiable function. The inf-convolution and the proximal algorithm are revised according to the result as examples. [ABSTRACT FROM AUTHOR]

The paper proves the minimax theorem for a specific class of functions that are defined on branching polylines in a linear space, not on convex subsets of a linear space. The existence of a saddle point for such functions does not follow directly from the classical minimax theorem and needs individual consideration based both on convex analysis and graph theory. The paper presents a self-sufficient analysis of the problem. It contains everything that enables a plain understanding of the main result and its proof and avoids using concepts outside the scope of obligatory mathematical education of engineers. The paper is addressed to researchers in applied mechanics, engineering, and other applied sciences as well as to mathematicians who lecture convex analysis and optimization methods to non-mathematicians. [ABSTRACT FROM AUTHOR]

Robust convex constraints are difficult to handle, since finding the worst-case scenario is equivalent to maximizing a convex function. In this paper, we propose a new approach to deal with such constraints that unifies most approaches known in the literature and extends them in a significant way. The extension is either obtaining better solutions than the ones proposed in the literature, or obtaining solutions for classes of problems unaddressed by previous approaches. Our solution is based on an extension of the Reformulation-Linearization-Technique, and can be applied to general convex inequalities and general convex uncertainty sets. It generates a sequence of conservative approximations which can be used to obtain both upper- and lower- bounds for the optimal objective value. We illustrate the numerical benefit of our approach on a robust control and robust geometric optimization example. [ABSTRACT FROM AUTHOR]

Switching between finitely many continuous-time autonomous steepest descent dynamics for convex functions is considered. Convergence of complete solutions to common minimizers of the convex functions, if such minimizers exist, is shown. The convex functions need not be smooth and the dynamics may be subject to constraints. Since the common minimizers may represent consensus in a multi-agent system modeled by an undirected communication graph, several known results about asymptotic consensus are deduced as special cases. Extension to a time-varying case is included.

Switching between finitely many continuous-time autonomous steepest descent dynamics for convex functions is considered. Convergence of complete solutions to common minimizers of the convex functions, if such minimizers exist, is shown. The convex functions need not be smooth and the dynamics may be subject to constraints. Since the common minimizers may represent consensus in a multi-agent system modeled by an undirected communication graph, several known results about asymptotic consensus are deduced as special cases. Extension to a time-varying case is included.

The research on the learning performance of machine learning algorithms is one of the important contents of machine learning theory, and the selection of loss function is one of the important factors affecting the learning performance. In this paper, we introduce a parameterized loss function into the online learning algorithm and investigate the performance. By applying convex analysis techniques, the convergence of the learning sequence is proved and the convergence rate is provided in the expectation sense. The analysis results show that the convergence rate can be greatly improved by adjusting the parameter in the loss function.

In this paper, we establish necessary and sufficient conditions for the existence of line segments (or flats) in the sphere of the nuclear norm via the notion of simultaneous polarization and a refined expression for the subdifferential of the nuclear norm. This is then leveraged to provide (point-based) necessary and sufficient conditions for uniqueness of solutions for minimizing the nuclear norm over an affine subspace. We further establish an alternative set of sufficient conditions for uniqueness, based on the interplay of the subdifferential of the nuclear norm and the range of the problem-defining linear operator. Finally, we show how to transfer the uniqueness results for the original problem to a whole class of nuclear norm-regularized minimization problems with a strictly convex fidelity term. [ABSTRACT FROM AUTHOR]

This paper is devoted to studying the representation of measures of non-generalized compactness, in particular, measures of noncompactness, of non-weak compactness and of non-super weak compactness, etc., defined on Banach spaces and its applications. With the aid of a three-time order-preserving embedding theorem, we show that for every Banach space X, there exist a Banach function space C(K) for some compact Hausdorff space K and an order-preserving affine mapping from the super space of all the nonempty bounded subsets of X endowed with the Hausdorff metric to the positive cone C(K)+ of C(K) such that for every convex measure, in particular, the regular measure, the homogeneous measure and the sublinear measure of non-generalized compactness μ on X, there is a convex function F on the cone which is Lipschitzian on each bounded set of V such that As its applications, we show a class of basic integral inequalities related to an initial value problem in Banach spaces, and prove a solvability result of the initial value problem, which is an extension of some classical results due to Banaś and Goebel (1980), Goebel and Rzymowski (1970) and Rzymowski (1971). [ABSTRACT FROM AUTHOR]

We construct an original framework based on convex analysis to prove the existence and uniqueness of a solution to a class of implicit numerical schemes. We propose an application of this general framework in the case of a new non linear implicit scheme for the 1D Lagrangian gas dynamics equations. We provide numerical illustrations that corroborate our proof of unconditional stability for this non linear implicit scheme. [ABSTRACT FROM AUTHOR]

In this work we consider regularized Wasserstein barycenters (average in Wasserstein distance) in Fourier basis. We prove that random Fourier parameters of the barycenter converge to some Gaussian random vector in distribution. The convergence rate has been derived in finite-sample case with explicit dependence on measures count () and the dimension of parameters (). [ABSTRACT FROM AUTHOR]

Convex analysis and the theory of differential inclusions with maximal monotone right-hand sides suggests casting consensus algorithms as systems involving switching between such differential inclusions. Convergence of solutions to such switching systems is shown and applications to consensus are presented. Robustness of pointwise asymptotic stability for a single differential inclusion which has some monotonicity-related properties, but needs not be monotone, is shown.

Convex analysis and the theory of differential inclusions with maximal monotone right-hand sides suggests casting consensus algorithms as systems involving switching between such differential inclusions. Convergence of solutions to such switching systems is shown and applications to consensus are presented. Robustness of pointwise asymptotic stability for a single differential inclusion which has some monotonicity-related properties, but needs not be monotone, is shown.

The aim of this paper is to analyze the finite-time stability (FTS) of nonlinear time-varying systems with saturated impulse inputs and further analyze the finite-time contractive stability (FTCS). By employing theories of convex analysis, impulsive control, matrix inequality and Lyapunov function, several results about FTS and FTCS of nonlinear time-varying systems with saturated impulse inputs are obtained. The simulation examples show the validation of the proposed results. [ABSTRACT FROM AUTHOR]

The Brouwer fixed point theorem is a key ingredient in the proof that a periodic differential equation has a periodic solution in a set that satisfies a suitable tangency condition on its boundary. The main goal of this note is to show that both results are in fact equivalent. [ABSTRACT FROM AUTHOR]

We propose to derive deviation measures through the Minkowski gauge of a given set of acceptable positions. We show that, given a suitable acceptance set, any positive homogeneous deviation measure can be accommodated in our framework. In doing so, we provide a new interpretation for such measures, namely, that they quantify how much one must shrink or deleverage a financial position for it to become acceptable. In particular, the Minkowski Deviation of a set which is convex, translation insensitive, and radially bounded at non-constants, is a generalized deviation measure in the sense of [R. T. Rockafellar, S. Uryasev and M. Zabarankin, Generalized deviations in risk analysis, Finance Stoch. 10 2006, 1, 51–74]. Furthermore, we explore the converse relations from properties of a Minkowski Deviation to its sub-level sets, introducing the notion of acceptance sets for deviations. Hence, we fill a gap existing in the literature, namely the lack of a well-defined concept of acceptance sets for deviation measures. Dual characterizations in terms of polar sets and support functionals are provided. [ABSTRACT FROM AUTHOR]

In a self contained presentation we study the boundedness of the Laplace transform on the rearrangement invariant Lebesgue Lp-space, we use real analysis techniques provided by convex analysis and the K-method. [ABSTRACT FROM AUTHOR]

This paper addresses the study and applications of polyhedral duality in locally convex topological vector (LCTV) spaces. We first revisit the classical Rockafellar's proper separation theorem for two convex sets one of which is polyhedral and then present its LCTV extension replacing the relative interior by its quasi-relative interior counterpart. Then we apply this result to derive enhanced calculus rules for normals to convex sets, coderivatives of convex set-valued mappings, and subgradients of extended-real-valued functions under certain polyhedrality requirements in LCTV spaces by developing a geometric approach. We also establish in this way new results on conjugate calculus and duality in convex optimization with relaxed qualification conditions in polyhedral settings. Our developments contain significant improvements to a number of existing results obtained by Ng and Song (Nonlinear Anal. 55, 845–858, 12). [ABSTRACT FROM AUTHOR]

Barrier coverage is a fundamental application in wireless sensor networks, which are widely used for smart cities. In applications, the sensors form a barrier for the intruders and protect an area through intrusion detection. In this paper, we study a new branch of barrier coverage, namely warning barrier coverage (WBC). Different from the classic barrier coverage, WBC has the inverse protect direction, which moves the sensors surrounding a dangerous region and protects any unexpected visitors by warning them away from the dangers. WBC holds a promising prospect in many danger keep out applications for smart cities. For example, a WBC can enclose the debris area in the sea and alarm any approaching ships in order to avoid their damaging propellers. One special feature of WBC is that the target region is usually dangerous and its boundary is previously unknown. Hence, the scattered mobile nodes need to detect the boundary and form the barrier coverage themselves. It is challenging to form these distributed sensor nodes into a barrier because a node can sense only the local information and there is no global information of the unknown region or other nodes. To this end, in response to the newly proposed issue of the formation of barrier cover, we propose a novel solution AutoBar for mobile sensor nodes to automatically form a WBC for smart cities. Notably, this is the first work to trigger the coverage problem of the alarm barrier, wherein the regional information is not pre-known. To pursue the high coverage quality, we theoretically derive the optimal distribution pattern of sensor nodes using convex theory. Based on the analysis, we design a fully distributed algorithm that enables nodes to collaboratively move toward the optimal distribution pattern. In addition, AutoBar is able to reorganize the barrier even if any node is broken. To validate the feasibility of AutoBar, we develop the prototype of the specialized mobile node, which consists of two kinds of sensors: one for boundary detection and another for visitor detection. Based on the prototype, we conduct extensive real trace-driven simulations in various smart city scenarios. Performance results demonstrate that AutoBar outperforms the existing barrier coverage strategies in terms of coverage quality, formation duration, and communication overhead.

Optimal transport is a mathematical tool that has been a widely used to measure the distance between two probability distributions. To mitigate the cubic computational complexity of the vanilla formulation of the optimal transport problem, regularized optimal transport has received attention in recent years, which is a convex program to minimize the linear transport cost with an added convex regularizer. Sinkhorn optimal transport is the most prominent one regularized with negative Shannon entropy, leading to densely supported solutions, which are often undesirable in light of the interpretability of transport plans. In this paper, we report that a deformed entropy designed by q-algebra, a popular generalization of the standard algebra studied in Tsallis statistical mechanics, makes optimal transport solutions supported sparsely. This entropy with a deformation parameter q interpolates the negative Shannon entropy ( q = 1 ) and the squared 2-norm ( q = 0 ), and the solution becomes more sparse as q tends to zero. Our theoretical analysis reveals that a larger q leads to a faster convergence when optimized with the Broyden–Fletcher–Goldfarb–Shanno (BFGS) algorithm. In summary, the deformation induces a trade-off between the sparsity and convergence speed. [ABSTRACT FROM AUTHOR]

This article is concerned with the approximation of unbounded convex sets by polyhedra. While there is an abundance of literature investigating this task for compact sets, results on the unbounded case are scarce. We first point out the connections between existing results before introducing a new notion of polyhedral approximation called ε , δ -approximation that integrates the unbounded case in a meaningful way. Some basic results about ε , δ -approximations are proved for general convex sets. In the last section, an algorithm for the computation of ε , δ -approximations of spectrahedra is presented. Correctness and finiteness of the algorithm are proved. [ABSTRACT FROM AUTHOR]

We present a set of necessary and sufficient conditions for a class of optimal control problems with pure state constraints for which the objective function is linear in the state variable but the objective function is only required to be upper semicontinuous in the control variable. We apply those conditions to economic environments in contract theory where discontinuities in objectives prevail. Examples of applications include nonlinear pricing of digital goods and nonlinear pricing under competitive threat. [ABSTRACT FROM AUTHOR]

In this paper, we consider a class of evolutionary quasi-variational inequalities arising in the study of contact problems for viscoelastic materials. Based on convex analysis methods and fixed point arguments, we prove the well-posedness and regularity of solutions in a general framework. Then, we establish the correspondence between evolutionary variational inequalities and implicit sweeping processes, which enables us to prove an existence result for history-dependent implicit sweeping processes. Finally, our theoretical results are illustrated with an application to contact mechanics problems for viscoelastic materials with a short memory. [ABSTRACT FROM AUTHOR]

We introduce notions of concavity for functions on balanced polyhedral spaces, and we show that concave functions on such spaces satisfy several strong continuity properties. [ABSTRACT FROM AUTHOR]

Motivated by classical work on the numerical integration of ordinary differential equations we present a ResNet-styled neural network architecture that encodes non-expansive (1-Lipschitz) operators, as long as the spectral norms of the weights are appropriately constrained. This is to be contrasted with the ordinary ResNet architecture which, even if the spectral norms of the weights are constrained, has a Lipschitz constant that, in the worst case, grows exponentially with the depth of the network. Further analysis of the proposed architecture shows that the spectral norms of the weights can be further constrained to ensure that the network is an averaged operator, making it a natural candidate for a learned denoiser in Plug-and-Play algorithms. Using a novel adaptive way of enforcing the spectral norm constraints, we show that, even with these constraints, it is possible to train performant networks. The proposed architecture is applied to the problem of adversarially robust image classification, to image denoising, and finally to the inverse problem of deblurring. • New neural network architectures are proposed with provable stability guarantees. • The proposed architectures are based on gradient flows in convex potentials. • The methods are tested on robust classification and image reconstruction tasks. • The methods attain high performance, while having robustness guarantees. [ABSTRACT FROM AUTHOR]

This article studies the formation control problem of second-order multiagent networks, in which cooperative and antagonistic interactions of the agents spontaneously coexist in the communication process. Based on the convex analysis theory, several convex polytopes that do not require some kinds of system constraints are constructed in the presence of these interactions. Then, the matrix perturbation theory and some mathematical techniques are utilized to analyze these convex polytopes. The obtained results show that the agents with cooperative interactions monotonously converge to their own specified formation shape while maintaining the desired relative position of the other agents with antagonistic interactions. Subsequently, two numerical examples are presented to illustrate the obtained results. [ABSTRACT FROM AUTHOR]

The problem of optimal quantizer design that maximizes the mutual information between the input and the quantized output of a communication channel is considered. It is shown that an optimal quantizer exists which employs convex polytopes as its decision regions in the Euclidean space of likelihood ratios. This complements the previous results in the literature by presenting a new and intuitive proof, establishing a unified treatment of the most general case, extending the solution to continuous-input channels, and providing a characterization for the form of optimal decision rule based on likelihood ratios, whereas the previous results are expressed in terms of the posterior probabilities and depend on the channel input distribution. The result is corroborated with an analytical example employing the transmission of a spherically symmetric random vector source over an additive white Gaussian noise channel. [ABSTRACT FROM AUTHOR]

The scientific activity of Professor Gábor Kassay (1956-2021), one of the most prolific mathematician in Cluj-Napoca (Romania) is presented, through the memories of some co-authors, with whom he collaborated throughout the four decades of his scientific activity. [ABSTRACT FROM AUTHOR]

Non-convex sparsity-inducing penalties outperform their convex counterparts, but generally sacrifice the cost function convexity. As a middle ground, we propose the sharpening sparse regularizers (SSR) framework to design non-separable non-convex penalties that induce sparsity more effectively than convex penalties such as $\ell _1$ and nuclear norms, but without sacrificing the cost function convexity. The overall problem convexity is preserved by exploiting the data fidelity relative strong convexity. The framework constructs penalties as the difference of convex functions, namely the difference between convex sparsity-inducing penalties and their smoothed versions. We propose a generalized infimal convolution smoothing technique to obtain the smoothed versions. Furthermore, SSR recovers and generalizes several non-convex penalties in the literature as special cases. The SSR framework is applicable to any sparsity regularized least squares ill-posed linear inverse problem. Beyond regularized least squares, the SSR framework can be extended to accommodate Bregman divergence, and other sparsity structures such as low-rankness. The SSR optimization problem can be formulated as a saddle point problem, and solved by a scalable forward-backward splitting algorithm. The effectiveness of the SSR framework is demonstrated by numerical experiments in different applications.

As a crucial technique for identifying irregular samples or outlier patterns, anomaly detection has broad applications in many fields. Convex analysis (CA) is one of the fundamental methods used in anomaly detection, which contributes to the robust approximation of algebra and geometry, efficient computation to a unique global solution, and mathematical optimization for modeling. Despite the essential role and evergrowing research in CA-based anomaly detection algorithms, little work has realized a comprehensive survey of it. To fill this gap, we summarize the CA techniques used in anomaly detection and classify them into four categories of density estimation methods, matrix factorization methods, machine learning methods, and the others. The theoretical background, sub-categories of methods, typical applications as well as strengths and limitations for each category are introduced. This paper sheds light on a succinct and structured framework and provides researchers with new insights into both anomaly detection and CA. With the remarkable progress made in the techniques of big data and machine learning, CA-based anomaly detection holds great promise for more expeditious, accurate and intelligent detection capacities.

By employing the notion of exceptional family of elements, we establish existence results for the mixed tensor variational inequalities. We show that the nonexistence of an exceptional family of elements is a sufficient condition for the solvability of mixed tensor variational inequality. For positive semidefinite mixed tensor variational inequalities, the nonexistence of an exceptional family of elements is proved to be an equivalent characterization of the nonemptiness of the solution sets. We derive several sufficient conditions of the nonemptiness and compactness of the solution sets for the mixed tensor variational inequalities with some special structured tensors. Finally, we show that the mixed tensor variational inequalities can be defined as a class of convex optimization problems. [ABSTRACT FROM AUTHOR]