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Using a matrix decomposition for sectorial matrices and an AM-GM inequality for singular values, we present a new Rotfel'd type inequality for sectorial matrices, which generalizes and partially improves some results by Zhang. We also give alternative simpler proofs of a main result of a recent paper by Yang, Lu and Chen as well as that by Fu and Liu. [ABSTRACT FROM AUTHOR]

In this paper we deal with an eigenvalue problem in an interface elliptic equation. We characterize the set of principal eigenvalues as a level set of a concave and regular function. As application, we study a problem arising in population dynamics. In these problems each species lives in a subdomain, and they interact in a common border, which acts as a geographical barrier; but unlike previous results, we consider the case of different growth rates in each subdomain. [ABSTRACT FROM AUTHOR]

In this paper, we prove the stability of the Brunn–Minkowski inequality for multiple convex bodies in terms of the concept of relative asymmetry. Using these stability results and the relationship of the compact support of functions, we establish the stability of the Borell–Brascamp–Lieb inequality for multiple power concave functions via relative asymmetry. [ABSTRACT FROM AUTHOR]

We treat the exterior Dirichlet problem for a class of fully nonlinear elliptic equations of the form f (λ (D 2 u)) = g (x) with prescribed asymptotic behavior at infinity. The equations of this type had been studied extensively by Caffarelli et al. [The Dirichlet problem for nonlinear second-order elliptic equations. III. Functions of the eigenvalues of the Hessian, Acta Math. 155 (1985) 261–301], Trudinger [On the Dirichlet problem for Hessian equations, Acta Math. 175 (1995) 151–164] and many others, and there had been significant discussions on the solvability of the classical Dirichlet problem via the continuity method, under the assumption that f is a concave function. In this paper, based on Perron's method, we establish an exterior existence and uniqueness result for viscosity solutions of the equations, by assuming f to satisfy certain structure conditions as in [L. Caffarelli, L. Nirenberg and J. Spruck, The Dirichlet problem for nonlinear second-order elliptic equations. III. Functions of the eigenvalues of the Hessian, Acta Math. 155 (1985) 261–301; N. S. Trudinger, On the Dirichlet problem for Hessian equations, Acta Math. 175 (1995) 151–164] but without requiring the concavity of f. The equations in our setting may embrace the well-known Monge–Ampère equations, Hessian equations and Hessian quotient equations as special cases. [ABSTRACT FROM AUTHOR]

There exist two major subclasses in the class of superquadratic functions, one comprises concave and decreasing functions, while the other consists of convex and monotone increasing functions. Leveraging this distinction, we introduce eigenvalue inequalities for each case. The characteristics of these functions allow us to advance our findings in two ways: firstly, by refining existing results related to eigenvalues for convex functions, and secondly, by deriving complementary inequalities for other function types. To bolster our claims, we will provide illustrative examples. [ABSTRACT FROM AUTHOR]

In this paper, we prove that for certain class of initial data, the corresponding solutions to the 3-D viscous primitive equations blow up in finite time. Specifically, we find a special solution to simplify the 3-D systems, assuming that the pressure function p (x , y , t) is a concave function. We also consider the equations on the line x = 0 , y = 0 . [ABSTRACT FROM AUTHOR]

Fractional calculus has been a concept used to obtain new variants of some well-known integral inequalities. In this study, our main goal is to establish the new fractional Hermite–Hadamard, and Simpson's type estimates by employing a differentiable function. Furthermore, a novel class of fractional integral related to prominent fractional operator (Caputo–Fabrizio) for differentiable convex functions of first order is proven. Then, taking this equality into account as an auxiliary result, some new estimation of the Hermite–Hadamard and Simpson's type inequalities as generalization is presented. Moreover, few inequalities for concave function are obtained as well. It is observed that newly established outcomes are the extension of comparable inequalities existing in the literature. Additionally, we discuss the applications to special means, matrix inequalities, and the q-digamma function. [ABSTRACT FROM AUTHOR]

This study investigates a novel intervention approach to network games, in which players are delinquents whose payoffs depend on the actions of their network neighbors. The social planner aims to manipulate the delinquency costs of players, seeking to minimize the total delinquency level. We consider two intervention scenarios. First, we consider binary interventions, where the planner can either increase the cost of an offender by a fixed amount; or leave its cost unchanged. The optimal intervention problem involves maximizing a submodular function. We establish a connection between cost and structural intervention in networks. Next, we consider continuous levels of intervention, where the planner can choose how much to increase the cost of an offender. We show that the optimal intervention problem is a tractable convex optimization if the intervention function is concave. We provide a characterization of the optimal intervention which is highly related to players' centralities in the network. We further discuss the interior solution and apply our result to nested split graphs. [ABSTRACT FROM AUTHOR]

Summary: Robust 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. [ABSTRACT FROM AUTHOR]

Given a strictly increasing continuous function ϕ : R ≥ 0 ⟶ R ∪ { - ∞ } with lim t → ∞ ϕ (t) = ∞ , a function f : [ a , b ] ⟶ R ≥ 0 is said to be ϕ -concave if ϕ ∘ f is concave. When ϕ (t) = t p , p > 0 , this notion is that of p-concavity whereas for ϕ (t) = log (t) it leads to the so-called log-concavity. In this work, we show that if the cross-sections volume function of a compact set K ⊂ R n (of positive volume) w.r.t. some hyperplane H passing through its centroid is ϕ -concave, then one can find a sharp lower bound for the ratio vol (K -) / vol (K) , where K - denotes the intersection of K with a halfspace bounded by H. When K is convex, this inequality recovers a classical result by Grünbaum. Moreover, other related results for ϕ -concave functions (and involving the centroid) are shown. [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]

In this article, we first define and then propose to systematically study some new subclasses of the class of analytic and bi-concave functions in the open unit disk. For this purpose, we make use of a combination of the binomial series and the confluent hypergeometric function. Among some other properties and results, we derive the estimates on the initial Taylor-Maclaurin coefficients | a 2 | and | a 3 | for functions in these analytic and bi-concave function classes, which are introduced in this paper. We also derive a number of corollaries and consequences of our main results in this paper. [ABSTRACT FROM AUTHOR]

We prove the (generalized) principal pivot transform is matrix monotone, in the sense of the Löwner ordering, under minimal hypotheses. This improves on the recent results of Pascoe and Tully-Doyle (2022) [69] in two ways. First, we use the "generalized" principal pivot transform, where matrix inverses in the classical definition of the principal pivot transform are replaced with Moore-Penrose pseudoinverses. Second, the hypotheses they used to prove the monotonicity is relaxed and, in particular, we find the weakest hypotheses possible for which it can be true. We also prove the principal pivot transform is a matrix concave function on positive semi-definite matrices that have the same kernel (and, in particular, on positive definite matrices). Our proof is a corollary of a minimization variational principle for the principal pivot transform. [ABSTRACT FROM AUTHOR]

Convex functions and their analogues have been powerful tools in almost all mathematical fields, including optimization, fractional calculus, mathematical analysis, functional analysis, operator theory, and mathematical physics. It is well established in the literature that a convex function f : [0,∞) → [0,∞) with f(0) = 0 is necessarily superadditive, while a concave function f : [0,∞) → [0,∞) is subadditive. The converses of these two assertions are not valid in general. The main target of this article is to study the subadditivity and superadditivity of convex and superquadratic functions. In particular, we obtain several results extending, refining, and reversing some known inequalities in this direction. Further discussion of superquadratic functions in this line will be given. [ABSTRACT FROM AUTHOR]

This paper proposes a path-based approach for the minimization of a continuously differentiable function over sparse symmetric sets, which is a hard problem that exhibits a restrictiveness-hierarchy of necessary optimality conditions. To achieve the more restrictive conditions in the hierarchy, state-of-the-art algorithms require a support optimization oracle that must exactly solve the problem in smaller dimensions. The path-based approach developed in this study produces a path-based optimality condition, which is placed well in the restrictiveness-hierarchy, and a method to achieve it that does not require a support optimization oracle and, moreover, is projection-free. In the development process, new results are derived for the regularized linear minimization problem over sparse symmetric sets, which give additional means to identify optimal solutions for convex and concave objective functions. We complement our results with numerical examples. [ABSTRACT FROM AUTHOR]

We develop structure preserving schemes for a class of nonlinear mobility continuity equation. When the mobility is a concave function, this equation admits a form of gradient flow with respect to a Wasserstein-like transport metric. Our numerical schemes build upon such formulation and utilize modern large-scale optimization algorithms. There are two distinctive features of our approach compared to previous ones. On the one hand, the essential properties of the solution, including positivity, global bounds, mass conservation, and energy dissipation, are all guaranteed by construction. On the other hand, our approach enjoys sufficient flexibility when applied to a large variety of problems including different free energy functionals, general wetting boundary conditions, and degenerate mobilities. The performance of our methods is demonstrated through a suite of examples. [ABSTRACT FROM AUTHOR]

The cost of ornamentation is often measured experimentally to study the relative importance of sexual and viability selection for ornamentation, but these experiments can lead to a misleading conclusion when compensatory trait is ignored. For example, a classic experiment on the outermost tail feathers in the barn swallow Hirundo rustica explains that the concave (or U‐shaped) aerodynamic performance cost of the outermost tail feathers would be the evolutionary outcome through viability selection for optimal tail length, but this conclusion depends on the assumption that compensatory traits do not cause reduced performance. Using a simple "toy model" experiment, I demonstrated that ornamentation evolved purely though sexual selection can produce a concave cost function under the presence of compensatory traits, which was further reinforced by a simple mathematical model. Therefore, concave cost function (and the low performance of individuals with reduced ornaments) cannot be used to infer the evolutionary force favoring ornamentation, due to a previously overlooked concept, "overcompensation," which can worsen the whole body performance. [ABSTRACT FROM AUTHOR]

Aim: Distribution maps for Amazonian birds are often limited to species‐level taxonomy even though many subspecies represent biological species. We provide digital maps depicting subspecies‐level distributions of Neotropical birds in the Amazonas River basin and adjoining areas. Location: Amazon region, which includes the entire Amazon basin, the east slope of the tropical Andes, French Guiana, Suriname, Guyana, south Venezuela (Bolivar and Amazonas departments), parts of the Brazilian Cerrado and the Araguaia‐Tocantins basin. Taxon: Birds (Class Aves). Methods: To build the distribution maps, we compiled a point‐locality database of 620,000 records, 90% of which represent specimens vouchered in research collections. After manually cleaning and optimising the quality of the point localities, we generated extent‐of‐occurrence polygons at the subspecies level using a concave hull function. We corrected each polygon based on the literature and expert knowledge. We used this data set to define zoogeographical regions based on multidimensional scaling and clustering analyses, and then compared subspecies‐defined regions to zoogeographic regions inferred from species‐level data. Results: We inferred range polygons for 3990 subspecies, representing 2043 species from 65 families. The average distribution size was 955,739 km2, with half of the taxa having range sizes smaller than 230,484 km2 (median value) and nearly one‐quarter of all subspecies having ranges smaller than 50,000 km2. We identified 10 zoogeographical regions from the subspecies data set in comparison with four areas based on a species‐level analysis of the same data. The 10 zoogeographical regions are largely congruent with previously identified areas of endemism. Main Conclusions: The new maps of Amazonian bird subspecies distributions allow biodiversity studies (e.g. macroecology, evolution, and conservation) to be conducted at the subspecies level, and facilitate biogeographic, ecological, evolutionary and conservation research on birds in the most biologically diverse region in the world. [ABSTRACT FROM AUTHOR]

In this article, we develop automated inference methods for "local" parameters in a collection of convexity constrained models based on the natural constrained tuning-free estimators. A canonical example is given by the univariate convex regression model, in which automated inference is drawn for the function value, the function derivative at a fixed interior point, and the anti-mode of the convex regression function, based on the widely used tuning-free, piecewise linear convex least squares estimator (LSE). The key to our inference proposal in this model is a pivotal joint limit distribution theory for the LS estimates of the local parameters, normalized appropriately by the length of certain data-driven linear piece of the convex LSE. Such a pivotal limiting distribution instantly gives rise to confidence intervals for these local parameters, whose construction requires almost no more effort than computing the convex LSE itself. This inference method in the convex regression model is a special case of a general inference machinery that covers a number of convexity constrained models in which a limit distribution theory is available for model-specific estimators. Concrete models include: (i) log-concave density estimation, (ii) s-concave density estimation, (iii) convex nonincreasing density estimation, (iv) concave bathtub-shaped hazard function estimation, and (v) concave distribution function estimation from corrupted data. The proposed confidence intervals for all these models are proved to have asymptotically exact coverage and oracle length, and require no further information than the estimator itself. We provide extensive simulation evidence that validates our theoretical results. Real data applications and comparisons with competing methods are given to illustrate the usefulness of our inference proposals. for this article are available online. [ABSTRACT FROM AUTHOR]

John's fundamental theorem characterizing the largest volume ellipsoid contained in a convex body |$K$| in |$\mathbb{R}^{d}$| has seen several generalizations and extensions. One direction, initiated by V. Milman is to replace ellipsoids by positions (affine images) of another body |$L$|⁠. Another, more recent direction is to consider logarithmically concave functions on |$\mathbb{R}^{d}$| instead of convex bodies: we designate some special, radially symmetric log-concave function |$g$| as the analogue of the Euclidean ball, and want to find its largest integral position under the constraint that it is pointwise below some given log-concave function |$f$|⁠. We follow both directions simultaneously: we consider the functional question, and allow essentially any meaningful function to play the role of |$g$| above. Our general theorems jointly extend known results in both directions. The dual problem in the setting of convex bodies asks for the smallest volume ellipsoid, called Löwner's ellipsoid , containing |$K$|⁠. We consider the analogous problem for functions: we characterize the solutions of the optimization problem of finding a smallest integral position of some log-concave function |$g$| under the constraint that it is pointwise above |$f$|⁠. It turns out that in the functional setting, the relationship between the John and the Löwner problems is more intricate than it is in the setting of convex bodies. [ABSTRACT FROM AUTHOR]

We study the problem of fairly allocating indivisible goods to agents with weights corresponding to their entitlements. Previous work has shown that, when agents have binary additive valuations, the maximum weighted Nash welfare rule is resource-, population-, and weight-monotone, satisfies group-strategyproofness, and can be implemented in polynomial time. We generalize these results to the class of weighted additive welfarist rules with concave functions and agents with matroid-rank (also known as binary submodular) valuations. • We consider fairly allocating indivisible goods to agents with varying entitlements. • Prior work has shown desirable properties for the maximum weighted Nash welfare rule. • We generalize this to weighted additive welfarist rules with concave functions. • Our results permit matroid-rank valuations, which include binary additive valuations. [ABSTRACT FROM AUTHOR]

In this paper, we study various Rogers–Shephard-type inequalities for the lattice point enumerator G n (⋅) on ℝ n . In particular, for any non-empty convex bounded sets K , L ⊂ ℝ n , we show that G n (K + L) G n (K ∩ (− L)) ≤ 2 n n G n (K + (− 1 , 1) n) G n (L + (− 1 , 1) n) and G n − k (P H ⊥ K) G k (K ∩ H) ≤ n k G n (K + (− 1 , 1) n) , for H = lin { e 1 , ... , e k } , k ∈ { 1 , ... , n − 1 }. Additionally, a discrete counterpart to a classical result by Berwald for concave functions, from which other discrete Rogers–Shephard-type inequalities may be derived, is shown. Furthermore, we prove that these new discrete analogues for G n (⋅) imply the corresponding results involving the Lebesgue measure. [ABSTRACT FROM AUTHOR]

We consider the following problem stated in 1993 by Buttazzo and Kawohl (Math Intell 15:7–12, 1993): minimize the functional ∫ ∫ Ω (1 + | ∇ u (x , y) | 2 ) - 1 d x d y in the class of concave functions u : Ω → [0,M], where Ω ⊂ R 2 is a convex domain and M > 0. It generalizes the classical minimization problem, which was initially stated by I. Newton in 1687 in the more restricted class of radial functions. The problem is not solved until now; there is even nothing known about the structure of singular points of a solution. In this paper we, first, solve a family of auxiliary 2D least resistance problems and, second, apply the obtained results to study singular points of a solution to our original problem. More precisely, we derive a necessary condition for a point being a ridge singular point of a solution and prove, in particular, that all ridge singular points with horizontal edge lie on the top level and zero level sets. [ABSTRACT FROM AUTHOR]

The aggressive scaling down of the feature size in modern ICs has introduced major bottlenecks in printing layouts using a conventional 193 nm immersion lithography system. As different mitigation techniques have reached their limitations, Next Generation Lithography (NGL) techniques have been gaining popularity. Extreme Ultra-violet Lithography (EUVL) system which uses light having 13. 5 nm wavelength is one of the popular NGL for printing features below 20 nm. However, EUVL faces the problem of flare caused by irregular reflection from clear-field mask surfaces used. It leads to distortion of critical dimension (CD) during layout printing which in turn degrades the performance of the circuit. In this paper, we have formulated a delay-aware global routing problem in order to minimize flare without sacrificing the delay. Two different solvers, namely ILP and SAT-SMT, are employed and the results are compared. Experimental results show a significant reduction in flare without much compromise on delay. • Flare is a phenomenon in Extreme Ultra-violet lithography, which causes layout distortions. • In order to reduce flare, it is required to distribute the layout features conforming to the global flare distribution which can be modelled as a 2D concave function. In addition to that, it is important to take care of the delay during redistribution of the layout features. • For global routing, a set of delay optimized Steiner trees are to be constructed for each of the nets. • The Steiner tree that might lead to the best possible wire distribution for minimizing flare has to be selected for each net. • We have proposed a method based on mathematical programming that can select concurrently a delay-aware Steiner tree for each net such that flare over the layout area can be minimized. We have validated our proposed approach on IBM benchmark circuits to show the effectiveness of delay-aware flare reduction at the global routing stage. [ABSTRACT FROM AUTHOR]

We propose a new nonconvex regularization scheme to improve the performance of the stochastic dual dynamic programming (SDDP) algorithm for solving large-scale multistage stochastic programs. Specifically, we use a class of nonconvex regularization functions, namely folded concave penalty functions, to improve solution quality and the convergence rate of the SDDP procedure. We develop a strategy based on mixed-integer programming to guarantee global optimality of the nonconvex regularization problem. Moreover, we establish provable convergence guarantees for our customized SDDP algorithm. The benefits of our regularization scheme are demonstrated by solving large-scale instances of two multistage stochastic optimization problems. History: Pascal Van Hentenryck, Area Editor for Computational Modeling: Methods & Analysis. 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.2021.0255) as well as from the IJOC GitHub software repository (https://github.com/INFORMSJoC/2021.0255). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/. [ABSTRACT FROM AUTHOR]

Let S(p) be the class of meromorphic univalent functions f in the unit disk D with a simple pole at p ∊ (0, 1), CO(p,α) be the subclass of S(p) such that C \ f(D) is a convex domain of order α. In this paper, some characterizations of functions in CO(p, α) are given and the Livingston conjecture of f ∊ CO(p, α) is considered. [ABSTRACT FROM AUTHOR]

We study the polyhedral convex hull structure of a mixed-integer set which arises in a class of cardinality-constrained concave submodular minimization problems. This class of problems has an objective function in the form of f (a ⊤ x) , where f is a univariate concave function, a is a non-negative vector, and x is a binary vector of appropriate dimension. Such minimization problems frequently appear in applications that involve risk-aversion or economies of scale. We propose three classes of strong valid linear inequalities for this convex hull and specify their facet conditions when a has two distinct values. We show how to use these inequalities to obtain valid inequalities for general a that contains multiple values. We further provide a complete linear convex hull description for this mixed-integer set when a contains two distinct values and the cardinality constraint upper bound is two. Our computational experiments on the mean-risk optimization problem demonstrate the effectiveness of the proposed inequalities in a branch-and-cut framework. [ABSTRACT FROM AUTHOR]

This study presents an extended vector polar histogram (EVPH) method for efficient robot navigation using omni-directional LiDAR data. Although the conventional vector polar histogram (VPH) method is a powerful technique suitable for LiDAR sensors, it is limited in its sensing range by the single LiDAR sensor to a semicircle. To address this limitation, the EVPH method incorporates multiple LiDAR sensor's data for omni-directional sensing. First off, in the EVPH method, the LiDAR sensor coordinate systems are directly transformed into the robot coordinate system to obtain an omni-directional polar histogram. Several techniques are also employed in this process, such as minimum value selection and linear interpolation, to generate a uniform omni-directional polar histogram. The resulting histogram is modified to represent the robot as a single point. Subsequently, consecutive points in the histogram are grouped to construct a symbol function for excluding concave blocks and a threshold function for safety. These functions are combined to determine the maximum cost value that generates the robot's next heading angle. Robot backward motion is made feasible based on the determined heading angle, enabling the calculation of the velocity vector for time-efficient and collision-free navigation. To assess the efficacy of the proposed EVPH method, experiments were carried out in two environments where humans and obstacles coexist. The results showed that, compared to the conventional method, the robot traveled safely and efficiently in terms of the accumulated amount of rotations, total traveling distance, and time using the EVPH method. In the future, our plan includes enhancing the robustness of the proposed method in congested environments by integrating parameter adaptation and dynamic object estimation methods. [ABSTRACT FROM AUTHOR]

In this work, we investigate a novel intelligent surface-assisted multiuser multiple-input single-output multiple-eavesdropper (MU-MISOME) secure communication network where an intelligent reflecting surface (IRS) is deployed to enhance the secrecy performance and an intelligent transmission surface (ITS)-based transmitter is utilized to perform energy-efficient beamforming. A weighted sum secrecy rate (WSSR) maximization problem is developed by jointly optimizing transmit power allocation, ITS beamforming, and IRS phase shift. To solve this problem, we transform the objective function into an approximated concave form by using the successive convex approximation (SCA) technique. Then, we propose an efficient alternating optimization (AO) algorithm to solve the reformulated problem in an iterative way, where Karush–Kuhn–Tucker (KKT) conditions, the alternating direction method of the multiplier (ADMM), and majorization–minimization (MM) methods are adopted to derive the closed-form solution for each subproblem. Finally, simulation results are given to verify the convergence and secrecy performance of the proposed schemes. [ABSTRACT FROM AUTHOR]

Consider a quasiconcave, upper semicontinuous and homogeneous of degree γ function f. This paper shows that the reciprocal of the degree of homogeneity, 1 / γ , can be interpreted as a measure of the degree of concavity of f. As a direct implication of this result, it is also shown that f is harmonically concave if γ ≤ - 1 or γ ≥ 0 , concave if 0 ≤ γ ≤ 1 and logconcave if γ ≥ 0 . Some relevant applications to economic theory are given. For example, it is shown that a quasiconcave and homogeneous production function is concave if it displays nonincreasing returns to scale and logconcave if it displays increasing returns to scale. [ABSTRACT FROM AUTHOR]

Nonnegative matrix factorization is the following problem: given a nonnegative input matrix V and a factorization rank K, compute two nonnegative matrices, W with K columns and H with K rows, such that WH approximates V as well as possible. In this paper, we propose two new approaches for computing high-quality NMF solutions using conic optimization. These approaches rely on the same two steps. First, we reformulate NMF as minimizing a concave function over a product of convex cones – one approach is based on the exponential cone and the other on the second-order cone. Then, we solve these reformulations iteratively: at each step, we minimize exactly, over the feasible set, a majorization of the objective functions obtained via linearization at the current iterate. Hence these subproblems are convex conic programs and can be solved efficiently using dedicated algorithms. We prove that our approaches reach a stationary point with an accuracy decreasing as O (1 i) , where i denotes the iteration number. To the best of our knowledge, our analysis is the first to provide a convergence rate to stationary points for NMF. Furthermore, in the particular cases of rank-1 factorizations (i.e. K = 1), we show that one of our formulations can be expressed as a convex optimization problem, implying that optimal rank-1 approximations can be computed efficiently. Finally, we show on several numerical examples that our approaches are able to frequently compute exact NMFs (i.e. with V = WH) and compete favourably with the state of the art. [ABSTRACT FROM AUTHOR]

We study all-pay auctions under incomplete information with n contestants who have non-linear cost functions. The designer may award two kinds of subsidy (taxation): one that decreases (increases) each contestant's marginal cost of effort and another that increases (decreases) each contestant's value of winning. The designer's expected payoff is the contestants' expected total effort minus the cost of subsidy or, alternatively, plus the tax payment. We show that when the resource of subsidy (the marginal taxation rate) is relatively small and the cost function is concave (convex), the designer's expected payoff in all-pay auctions with both kinds of subsidy (taxation) is higher than in the same contest without any subsidy (taxation). [ABSTRACT FROM AUTHOR]

We propose the concept of Riemann diamond-alpha integrals for time scales interval-valued functions. We first give the definition and some properties of the interval Riemann diamond-alpha integral that are naturally investigated as an extension of interval Riemann nabla and delta integrals. With the help of the interval Riemann diamond-alpha integral, we present interval variants of Jensen inequalities for convex and concave interval-valued functions on an arbitrary time scale. Moreover, diamond-alpha Hölder's and Minkowski's interval inequalities are proved. Also, several numerical examples are provided in order to illustrate our main results. [ABSTRACT FROM AUTHOR]

In this paper, we first propose online and offline channel incentive models (CIM) with consideration of the consumers’ rational choices to solve and simulate the channel incentive problem (CIP) for supplier. We investigate whether the increase in demand along with channel incentive activities is enough to compensate for the decrease in the supplier's marginal revenue and retailers could benefit from the increase in market demand when retail channel information reference factor satisfies a certain threshold value. Our results show that decision preference of channel members is influenced by the reference factor and marginal revenue. Furthermore, the numerical achievements indicate that there is a unique optimal channel incentive coefficient related to the rational choices in benchmark channel incentive model (BCIM) for omnichannel. Both suppliers and retailers would benefit from the increased orders. Channel efficiency is improved from 72 % to 79 %. And the supplier profit function is a concave function of supplier's input in the offline channel incentive in the offline CIM. [ABSTRACT FROM AUTHOR]

We study the asymptotic behavior of solutions to the constrained MFG system as the time horizon T goes to infinity. For this purpose, we analyze first Hamilton–Jacobi equations with state constraints from the viewpoint of weak KAM theory, constructing a Mather measure for the associated variational problem. Using these results, we show that a solution to the constrained ergodic mean field games system exists and the ergodic constant is unique. Finally, we prove that any solution of the first-order constrained MFG problem on [0, T] converges to the solution of the ergodic system as T goes to infinity. [ABSTRACT FROM AUTHOR]

In this paper, we propose several calculus rules for the generalized concave Kurdyka–Łojasiewicz (KL) property, which generalize Li and Pong's results for KL exponents. The optimal concave desingularizing function has various forms and may be nondifferentiable. Our calculus rules do not assume desingularizing functions to have any specific form nor differentiable, while the known results do. Several examples are also given to show that our calculus rules are applicable to a broader class of functions than the known ones. [ABSTRACT FROM AUTHOR]

Let T = T 11 T 12 T 21 T 22 be accretive-dissipative, where T 11 , T 12 , T 21 , and T 22 are n × n complex matrices. Let f be a non-negative function on [ 0 , ∞) such that f (0) = 0 , and let α , β ∈ (0 , 1) such that α + β = 1 . For every unitarily invariant norm · , it is shown that ∑ j = 1 2 f T jj + (2 α - 1) T jj ∗ 2 2 + f α β 2 T jj ∗ ≤ 2 max (α , β) f (T) whenever t → f t is convex and ∑ j = 1 2 α f T jj + (2 α - 1) T jj ∗ 2 α + β f 2 α T jj ∗ ≤ 4 f max (α , β) T whenever f is concave. [ABSTRACT FROM AUTHOR]

In this paper, we construct the metric tensor and volume for the manifold of purifications associated with an arbitrary reduced density operator ρ S . We also define a quantum coarse-graining (CG) to study the volume where macrostates are the manifolds of purifications, which we call surfaces of ignorance (SOI), and microstates are the purifications of ρ S . In this context, the volume functions as a multiplicity of the macrostates that quantifies the amount of information missing from ρ S . Using examples where the SOI are generated using representations of S U (2) , S O (3) , and S O (N) , we show two features of the CG: (1) A system beginning in an atypical macrostate of smaller volume evolves to macrostates of greater volume until it reaches the equilibrium macrostate in a process in which the system and environment become strictly more entangled, and (2) the equilibrium macrostate takes up the vast majority of the coarse-grained space especially as the dimension of the total system becomes large. Here, the equilibrium macrostate corresponds to a maximum entanglement between the system and the environment. To demonstrate feature (1) for the examples considered, we show that the volume behaves like the von Neumann entropy in that it is zero for pure states, maximal for maximally mixed states, and is a concave function with respect to the purity of ρ S . These two features are essential to typicality arguments regarding thermalization and Boltzmann's original CG. [ABSTRACT FROM AUTHOR]

By considering the stock market's fuzzy uncertainty and investors' psychological factors, this paper studies the portfolio performance evaluation problems with different risk attitudes (optimistic, pessimistic, and neutral) by the Data Envelopment Analysis (DEA) approach. In this work, the return rates of assets are characterized as trapezoidal fuzzy numbers, whose membership functions with risk attitude parameters are described by exponential expression. Firstly, these characteristics with risk attitude are strictly derived including the possibilistic mean, variance, semi-variance, and semi-absolute deviation based on possibility theory. Secondly, three portfolio models (mean-variance, mean-semi-variance, and mean-semi-absolute-deviation) with different risk attitudes are proposed. Thirdly, we prove the real frontiers determined by our models are concave functions through mathematical theoretical derivation. In addition, two novel indicators are defined by difference and ratio formulas to characterize the correlation between DEA efficiency and portfolio efficiency. Finally, numerical examples are given to verify the feasibility and effectiveness of our model. No matter what risk attitude an investor holds, the DEA can generate approximate real frontiers. Correlation analysis indicates that our proposed approach outperforms in evaluating portfolios with risk attitudes. At the same time, our model is an improvement of Tsaur's work (2013) which did not study the different risk measures, and an extension of Chen et al.'s work (2018) which only considered risk-neutral attitude. [ABSTRACT FROM AUTHOR]

For a positive integer s, the projection body of an s-concave function f : ℝ n → [ 0 , + ∞) , a convex body in the (n + s)-dimensional Euclidean space ℝ n + s , is introduced. Associated inequalities for s-concave functions, such as, the functional isoperimetric inequality, the functional Petty projection inequality and the functional Loomis-Whitney inequality are obtained. [ABSTRACT FROM AUTHOR]

Let M be a finite von Neumann algebra with a normal faithful finite trace τ , L 0 (M) be the set of all measurable operators with respect to (M , τ) and μ t (x) be the generalized singular number of x ∈ L 0 (M). Set L 0 (M) + = { x : x ∈ L 0 (M) , x ≥ 0 } and M + + = { x : x ∈ M , x ≥ 0 and x is invertible }. We prove that if f : [ 0 , ∞) → [ 0 , ∞) is an operator concave function, 0 < p ≤ 1 , 0 < s ≤ 1 p and Φ j is a continuous positive linear map from L 0 (M j) to L 0 (M) with Φ j (M j) ⊂ M , where M j is finite von Neumann algebra, j = 1 , 2 , ⋯ , n , then for 0 ≤ t < τ (1) ∫ t τ (1) μ v ((∑ j = 1 n Φ j (f (x j p))) s) d v and ∫ t τ (1) μ v ((∑ j = 1 n Φ j (f (x j) p)) s) d v are jointly concave in (x 1 , x 2 , ⋯ , x n) ∈ L 0 (M 1) + × L 0 (M 2) + × ⋯ × L 0 (M n) +. We also prove that if f : (0 , ∞) → (0 , ∞) is an operator concave function, Φ j is a strictly positive linear map from finite von Neumann algebra M j to M , j = 1 , 2 , ⋯ , n , 0 < p ≤ 1 and 0 < s ≤ 1 p , then for 0 ≤ t < τ (1) , ∫ t τ (1) μ v ((∑ j = 1 n Φ j (f (x j − p))) − s) d v and ∫ t τ (1) μ v ((∑ j = 1 n Φ j (f (x j) − p)) − s) d v are jointly concave in (x 1 , x 2 , ⋯ , x n) ∈ M 1 + + × M 2 + + × ⋯ × M n + +. [ABSTRACT FROM AUTHOR]

Yau [Math. Ann. 207 (1974), pp. 269–270] proved that: There is no non-trivial continuous concave function on a complete manifold with finite volume. We prove analogue theorems for several metric spaces, including Alexandrov spaces with curvature bounded below/above, C^{\alpha }-Hölder Riemannian manifolds. [ABSTRACT FROM AUTHOR]

We investigate how basic probability inequalities can be extended to an imprecise framework, where (precise) probabilities and expectations are replaced by imprecise probabilities and lower/upper previsions. We focus on inequalities giving information on a single bounded random variable X , considering either convex/concave functions of X (Jensen's inequalities) or one-sided bounds such as (X ≥ c) or (X ≤ c) (Markov's and Cantelli's inequalities). As for the consistency of the relevant imprecise uncertainty measures, our analysis considers coherence as well as weaker requirements, notably 2-coherence, which proves to be often sufficient. Jensen-like inequalities are introduced, as well as a generalisation of a recent improvement to Jensen's inequality. Some of their applications are proposed: extensions of Lyapunov's inequality and inferential problems. After discussing upper and lower Markov's inequalities, Cantelli-like inequalities are proven with different degrees of consistency for the related lower/upper previsions. In the case of coherent imprecise previsions, the corresponding Cantelli's inequalities make use of Walley's lower and upper variances, generally ensuring better bounds. [ABSTRACT FROM AUTHOR]

Networks are widely used to describe relational data among objects in a complex system. As network data often exhibit clustering structures, research interest often focuses on discovering clusters of nodes. We develop a novel concordance-based method for node clustering in networks, where a linear model is imposed on the latent position of each node with respect to a node-specific center and its covariates via linear transformation. By maximizing a triangular concordance function with a concave pairwise penalty, the latent positions are estimated so that each node would be more likely to be close to its neighbors in contrast to non-neighbors and nodes are clustered by their node-specific centers. We develop an alternating direction method of multipliers algorithm for parameter estimation and an intimacy score between unlinked nodes for link prediction. Our method takes into account common characteristics of network data (i.e., assortativity, link pattern similarity, node heterogeneity and link transitivity), while it does not require the number of clusters to be known. The clustering effectiveness and link prediction accuracy of our method are demonstrated in simulated and real networks. for this article are available online. [ABSTRACT FROM AUTHOR]

The paper is devoted to the study of the uniqueness and certain qualitative properties of the solution of a class of integral equations with sum-difference kernel on the positive half-line and with a convex nonlinearity. This class of equations arises in a particular case in the dynamical theory of -adic closed-open strings for the scalar field of tachyons. Such equations also play a very important role in the study of the existence and uniqueness of solutions of nonlinear integral equations in the mathematical theory of the geographical distribution of an epidemic within the framework of the Diekmann–Kaper model. We prove the uniqueness theorem for the solution of the equation under consideration for a class of nonnegative (nonzero) and bounded functions on , thereby obtaining a definitive solution of Vladimirov's open problem on the uniqueness of rolling solutions of nonlinear -adic equations. Under an additional constraint on the kernel of the equation, we also prove that the solution is a concave function on whose derivative belongs to the space . At the end of the paper, we give specific model equations from the above-mentioned applications, to which our results are applied. [ABSTRACT FROM AUTHOR]

This paper studies several versions of the sparse optimization problem in statistical estimation defined by a pairwise separation objective. The sparsity (i.e., ℓ 0 ) function is approximated by a folded concave function; the pairwise separation gives rise to an objective of the Z-type. After presenting several realistic estimation problems to illustrate the Z-structure, we introduce a linear-step inner-outer loop algorithm for computing a directional stationary solution of the nonconvex nondifferentiable folded concave sparsity problem. When specialized to a quadratic loss function with a Z-matrix and a piecewise quadratic folded concave sparsity function, the overall complexity of the algorithm is a low-order polynomial in the number of variables of the problem; thus the algorithm is strongly polynomial in this quadratic case. We also consider the parametric version of the problem that has a weighted ℓ 1 -regularizer and a quadratic loss function with a (hidden) Z-matrix. We present a linear-step algorithm in two cases depending on whether the variables have prescribed signs or with unknown signs. In both cases, a parametric algorithm is presented and its strong polynomiality is established under suitable conditions on the weights. Such a parametric algorithm can be combined with an interval search scheme for choosing the parameter to optimize a secondary objective function in a bilevel setting. The analysis makes use of a least-element property of a Z-function, and, for the case of a quadratic loss function, the strongly polynomial solvability of a linear complementarity problem with a hidden Z-matrix. The origin of the latter class of matrices can be traced to an inspirational paper of Olvi Mangasarian to whom we dedicate our present work. [ABSTRACT FROM AUTHOR]

The Jensen inequality is one of the most favorable inequalities during the last few decades due to its expressive characteristics and properties. This inequality has gained a very dominant position in the several fields of science as a result of its extensive applications. In the present note, we use an interesting approach for the determination of estimates for the Jensen difference. We acquire several estimates for the Jensen difference while utilizing the convex function definition, Jensen's inequality for concave functions, power mean, and Hölder inequalities. For the real visualizations of the acquired estimates, we provide some particular examples. We get different improvements for the Hölder inequality while taking specified functions in the received estimates. Moreover, we deduce some more results from the main work in the form of improvements for the Hermite‐Hadamard inequality. Furthermore, various relations for the famous quasi‐arithmetic and power means are acquired with the help of established estimates. At the end, we present various applications of the main work in information theory. The intended applications provide some new bounds for the Csiszár and Rényi divergences, Shannon entropy, and Bhattacharyya co‐efficient. [ABSTRACT FROM AUTHOR]

We prove Lieb type convexity and concavity results for trace functionals associated with positive operator monotone (decreasing) functions and certain monotone concave functions, together with strictly positive linear maps of matrices. This gives a partial generalization of Hiai's recent work on trace functionals associated with power functions, by allowing positive operator monotone decreasing functions instead of power maps.Our proof is based on variational formula for trace functionals based on the Legendre transform, and a strengthened convexity of positive operator monotone decreasing functions in a previous work of Kirihata and the second named author. We also provide the generalization to the framework of unital tracial C -algebras based on Petz's work. [ABSTRACT FROM AUTHOR]

In 2019, Livshyts studied the Minkowski problem of measures in ℝ n with positive homogeneous and positive concave density functions. After that, Wu studied the L p -Minkowski problem with p ≥ 1 under the corresponding even measure. In this paper, we further study the existence of solution for L p -Minkowski problem of measures with positive homogeneous and positive concave density functions for 0 < p < 1. [ABSTRACT FROM AUTHOR]