Convex Sets Research Articles

Disk-Like Surfaces of Section and Symplectic Capacities

AbstractWe prove that the cylindrical capacity of a dynamically convex domain in ${\mathbb{R}}^{4}$ agrees with the least symplectic area of a disk-like global surface of section of the Reeb flow on the boundary of the domain. Moreover, we prove the strong Viterbo conjecture for all convex domains in ${\mathbb{R}}^{4}$ which are sufficiently C3 close to the round ball. This generalizes a result of Abbondandolo-Bramham-Hryniewicz-Salomão establishing a systolic inequality for such domains.

High dynamic range land wavefield reconstruction from randomized acquisition

Compressive sensing (CS) is an alternative to regular Shannon sampling that captures similar information from reduced measurements. It relies on randomized sampling patterns and a sparse data representation to reconstruct the regularly sampled object. CS is an important ingredient in afford- able seismic acquisition which can lead to improvements in the near surface mapping and in noise suppression for land data. However, the near surface traps the majority of the source-generated energy, resulting in data that are rich in high-wavenumber content and have amplitudes spanning several orders of magnitude. When dealing with such high dynamic range non-stationary data, the Fourier domain is not optimal for providing a sparse representation - a necessary condition for successful application of CS. In contrast, a discrete complex wavelet transform can localize high energy features, has good directional selectivity, and is near-shift invariant. Combined, these properties allow complex wavelets to represent detail-rich wavefields in a compact form. To leverage these features and achieve good CS reconstructions, we develop a scale- and orientation- dependent iterative soft thresholding scheme (IST) for reconstructing high dynamic range wavefields. Our approach requires little parametrization, is easy to implement, and robust to reconstructed wave- field sampling grid and dynamic range. We test IST on different wavefields with randomly missing traces, and compare the data reconstructions to the spectral projected gradient solver and projection onto convex sets. We quantify the reconstructions by a direct comparison of Fourier coefficients between fully sampled and reconstructed wavefields. Taking log10 of Fourier coefficients prior to computing the quality metric de-emphasizes the importance of magnitude match while highlighting Fourier coefficient support accuracy which usually translates into good structural fidelity of reconstructed data. We find that IST performs consistently among all examples, yielding a good phase match while performing gentle denoising.

Generalizations of Rolle’s Theorem

The classical Rolle’s theorem establishes the existence of (at least) one zero of the derivative of a continuous one-variable function on a compact interval in the real line, which attains the same value at the extremes, and it is differentiable in the interior of the interval. In this paper, we generalize the statement in four ways. First, we provide a version for functions whose domain is in a locally convex topological Hausdorff vector space, which can possibly be infinite-dimensional. Then, we deal with the functions defined in a real interval: we consider the case of unbounded intervals, the case of functions endowed with a weak derivative, and, finally, we consider the case of distributions over an open interval in the real line.

A dynamic multiple proximal bundle algorithm for locally Lipschitz multiobjective optimization problems

We propose a method for nonsmooth nonconvex multiobjective optimization problems that is based on a bundle-type approach. The proposed method employs an extension of the redistributed proximal bundle algorithm and uses an augmented improvement function to handle different objectives. In which, at each iteration, a common piecewise linear model is used to approximate the augmented improvement function. Contrary to many existing multiobjective optimization methods, this algorithm works directly with objective functions, without using any kind of a priori chosen parameters or employing any scalarization. Under appropriate assumptions, we discuss convergence to points which satisfy a necessary condition for Pareto optimality. We provide numerical results for a set of nonsmooth convex and nonconvex multiobjective optimization problems in the form of tables, figures and performance profiles. The numerical results confirm the superiority of the proposed algorithm in the considered test problems compared to other multiobjective solvers, in the computational effort necessary to compute weakly Pareto stationary points.

Noncoercive convex sweeping processes with velocity constraints

In this paper, we investigate noncoercive monotone convex sweeping processes with velocity constraints, a topic not previously investigated. Using some fundamental results on Bochner integration, the Tikhonov regularization method, a solution existence result for coercive convex sweeping processes with velocity constraints and a useful fact on measurable single-valued mappings, we prove the solution existence and properties of the solution set of noncoercive convex sweeping processes with velocity constraints under suitable conditions. These results represent a significant contribution, addressing a specific aspect of an open question posed in our recent work [Adly et al., Convex and nonconvex sweeping processes with velocity constraints: well-posedness and insights. Appl Math Optim. 2023;88(Paper No. 45)]. Additionally, we resolve two other open questions from the same paper concerning the behaviour of the regularized trajectories, relying on the Dominated Convergence Theorem for Bochner integration.

Tropical convexity in location problems

AbstractWe investigate location problems where the optimal solution is found within the tropical convex hull of the given input points. Our initial focus is on geodesically star-convex sets, using the asymmetric tropical distance. We introduce the concept of tropically quasiconvex functions, which have sub-level sets with this shape, and are closely related to monotonic functions. Our findings demonstrate that location problems using tropically quasiconvex functions as distance measures will result in an optimal solution within the tropical convex hull of the input points. We also extend this result to cases where the input points are replaced with tropically convex sets. Finally, we explore the applications of our research in phylogenetics, highlighting the properties of consensus methods that arise from our class of location problems.

Efficient targets and reference sets in selectively convex technologies

Conventional models of data envelopment analysis (DEA) typically assume that the underlying production technology is a convex set. It is known that such assumption may be clearly unsubstantiated in certain cases. Examples include studies in which some inputs or outputs are stated as proportions or percentages, or are represented by categorical measures. Excluding such “problematic” inputs and outputs from the assumption of convexity while assuming the latter for the remaining measures leads to the notion of selective convexity. Further examples of selectively convex technologies include technologies parameterized by an environmental factor and technologies in which only the input or output sets are convex. In this paper, we consider the identification of efficient targets and reference sets of decision making units in a selectively convex technology, which has not yet been explored in the literature. We show that, for such technologies, the conventional method based on the solution of the additive DEA model may not correctly identify the reference sets and needs an adjustment.

Geometric estimates and comparability of Eisenman volume elements with the Bergman kernel on ([formula omitted]-)convex domains

We establish geometric upper and lower estimates for the Carathéodory and Kobayashi-Eisenman volume elements on the class of non-degenerate convex domains, as well as on the more general class of non-degenerate C-convex domains. As a consequence, we obtain explicit universal lower bounds for the quotient invariant both on non-degenerate convex and C-convex domains. Here the bounds we derive, for the above mentioned classes in Cn, only depend on the dimension n for a fixed n≥2. Finally, it is shown that the Bergman kernel is comparable with these volume elements up to small/large constants depending only on n.

Characterizing the geometry of the Kirkwood–Dirac-positive states

The Kirkwood–Dirac (KD) quasiprobability distribution can describe any quantum state with respect to the eigenbases of two observables A and B. KD distributions behave similarly to classical joint probability distributions but can assume negative and nonreal values. In recent years, KD distributions have proven instrumental in mapping out nonclassical phenomena and quantum advantages. These quantum features have been connected to nonpositive entries of KD distributions. Consequently, it is important to understand the geometry of the KD-positive and -nonpositive states. Until now, there has been no thorough analysis of the KD positivity of mixed states. Here, we investigate the dependence of the full convex set of states with positive KD distributions on the eigenbases of A and B and on the dimension d of the Hilbert space. In particular, we identify three regimes where convex combinations of the eigenprojectors of A and B constitute the only KD-positive states: (i) any system in dimension 2; (ii) an open and dense probability one set of bases in dimension d = 3; and (iii) the discrete-Fourier-transform bases in prime dimension. Finally, we show that, if for example d = 2m, there exist, for suitable choices of A and B, mixed KD-positive states that cannot be written as convex combinations of pure KD-positive states. We further explicitly construct such states for a spin-1 system.

Exponential stabilization of linear systems with feedback optimization and its application to microgrids control

This paper proposes a feedback-optimization-based control method for linear time-invariant systems, which is aimed to exponentially stabilize the system and, meanwhile, drive the system output to an approximate solution of an optimization problem with convex set constraints and affine inequality constraints. To ensure the exponential stability of the closed-loop system, the original optimization problem is first reformulated into a counterpart that has only convex set constraints. It is shown that the optimal solution of the new optimization problem is an approximate optimal solution of the original problem. Then, based on this new optimization problem, the projected primal–dual gradient dynamics algorithm is used to design the controller. By using the singular perturbation method, sufficient conditions are provided to ensure the exponential stability of the closed-loop system. The proposed method is also applied to microgrid control.

Isometries to Analyze the Stability of Norm-Based Functional Equations in p-Uniformly Convex Spaces

Over the past two decades, significant advancements have been made in understanding the stability according to Hyers–Ulam involving different functional equations (FEs). This study investigates the generalized stability of norm-based (norm-additive) FEs within the framework of arbitrary (noncommutative) groups and p-uniformly convex spaces. Specifically, we analyze two key functional equations, ∥η(gh)∥=∥η(g)+η(h)∥ and ∥η(gh−1)∥=∥η(g)−η(h)∥foreveryg,h∈G, where (G,·) denotes an arbitrary group and B is considered to be a p-uniformly convex space. The surjectivity of the function η:G→B is a critical assumption in our analysis. Drawing upon the foundational works of L. Cheng and M. Sarfraz, this paper applies the large perturbation method tailored for p-uniformly convex spaces, where p≥1. This study extends previous research by offering a deeper exploration of the conditions under which these functional equations demonstrate Hyers–Ulam stability. In this study, the additive functional equation demonstrates a fundamental form of symmetry, where the order of operands does not affect the results. This symmetry under permutation of arguments is crucial for the analysis of stability. In the context of norm-additive FEs, this stability criterion investigates how small changes in the inputs of a functional equation affect the outputs, especially when the function is expected to follow an additive form.

Optimized backstepping‐based finite‐time containment control for nonlinear multi‐agent systems with prescribed performance

AbstractIn this article, a finite‐time optimal containment control method is proposed for nonlinear multi‐agent systems with prescribed performance. First, a neural network‐based reinforcement learning algorithm is developed under the optimized backstepping framework. The algorithm employs an identifier‐critic‐actor architecture, where the identifiers, critics and actors are used to estimate the unknown dynamics, evaluate the system performance, and optimize the system, respectively. Subsequently, in order to guarantee the transient performance of the tracking error, the original system is converted into an equivalent unconstrained system. Then, the tracking errors are allowed to converge to a prescribed set of residuals in finite time by combining prescribed performance control and finite‐time optimal control techniques. Furthermore, by using the Lyapunov stability theorem, it is verified that all signals are semi‐globally practical finite‐time stable, and all followers can converge to a convex region formed by multiple leaders. Finally, the effectiveness of the proposed scheme is demonstrated by a practical example.

Carathéodory Number of P 3 -Convexity of Claw-Free Graphs

Given a connected simple undirected graph G = ( V , E ) , a subset S of V is P 3 -convex if each vertex of G not in S has at most one neighbor in S . The P 3 -convex hull ⟨ S ⟩ of S is the smallest P 3 -convex set containing S . A Carathéodory set of G is a set S ⊆ V such that ⟨ S ⟩ ∖ ⋃ w ∈ S ⟨ S ∖ { w } ⟩ is non-empty. The Carathéodory number of G , denoted by C ( G ) , is the largest cardinality of a Carathéodory set of G . In this paper, we settle the conjecture posed by Barbosa et al. appeared in [SIAM J. Discrete Math. 26 (2012) 929–939] in the affirmative, which states that for a claw-free graph G of order n ( G ) , the Carathéodory number C ( G ) of the P 3 -convexity satisfies C ( G ) ≤ 2 n ( G ) + 6 5 . Furthermore, we determine all graphs attaining the bound.

An Advanced Lung Carcinoma Prediction and Risk Screening Model Using Transfer Learning.

Lung cancer, also known as lung carcinoma, has a high death rate, but an early diagnosis can substantially reduce this risk. In the current era, prediction models face challenges such as low accuracy, excessive noise, and low contrast. To resolve these problems, an advanced lung carcinoma prediction and risk screening model using transfer learning is proposed. Our proposed model initially preprocesses lung computed tomography images for noise removal, contrast stretching, convex hull lung region extraction, and edge enhancement. The next phase segments the preprocessed images using the modified Bates distribution coati optimization (B-RGS) algorithm to extract key features. The PResNet classifier then categorizes the cancer as normal or abnormal. For abnormal cases, further risk screening determines whether the risk is low or high. Experimental results depict that our proposed model performs at levels similar to other state-of-the-art models, achieving enhanced accuracy, precision, and recall rates of 98.21%, 98.71%, and 97.46%, respectively. These results validate the efficiency and effectiveness of our suggested methodology in early lung carcinoma prediction and risk assessment.

Verification of the person’s dynamic signature on a limited number of samples

Objectives. The goal of the research is to develop a new person-dependent method for verification of a signature of one person made on a tablet with a stylus in the presence of a limited number of signature samples of this person.Methods. The paper shows how to construct an individual pattern of the dynamic signatures of any person, which is described by points in a multidimensional feature space and is intended for subsequent verification of the authenticity of the signatures of a given person. It is constructed using 5<N<20 samples of genuine human signatures. The pattern forms a convex object in a multidimensional feature space. It describes the peculiar properties of a signature performed by a specific person.Results . The dynamics of signature execution is represented by three discrete parametric functions: coordinates of the stylus X, Y and its pressure on the tablet P, recorded at fixed time intervals. In the process of research, a number of secondary functions-features were selected and calculated from them. Since these data sets have different lengths, the dynamic time warping algorithm is used to compare them. The results of this transformation are distances between the dynamic features of two signatures, which serve as coordinates of a point in the feature space that describes the similarity of these signatures. The set of such points describes similarity of all pairs of genuine human signatures presented for verification in a multidimensional feature space. The convex hull of the cloud of these points is used as a pattern of a particular person's signature. The genuine signatures of any person are always different from each other; significant differences between them can distort the verification result.Conclusion. Experimental studies performed on genuine and fake signatures of 498 people from the largest available database of dynamic signatures, DeepSignDB, showed a verification accuracy of about 98 % when analyzing 24,900 signatures. Half of them are genuine, half are fake

Fixed Point of α-Modular Nonexpanive Mappings in Modular Vector Spaces 𝓁p(·)

Let C denote a convex subset within the vector space 𝓁p(·), and let T represent a mapping from C onto itself. Assume α=(α1,⋯,αn) is a multi-index in [0,1]n such that ∑i=1nαi=1, where α1>0 and αn>0. We define Tα:C→C as Tα=∑i=1nαiTi, known as the mean average of the mapping T. While every fixed point of T remains fixed for Tα, the reverse is not always true. This paper examines necessary and sufficient conditions for the existence of fixed points for T, relating them to the existence of fixed points for Tα and the behavior of T-orbits of points in T’s domain. The primary approach involves a detailed analysis of recurrent sequences in R. Our focus then shifts to variable exponent modular vector spaces 𝓁p(·), where we explore the essential conditions that guarantee the existence of fixed points for these mappings. This investigation marks the first instance of such results in this framework.

Polynomials with exponents in compact convex sets and associated weighted extremal functions: The Siciak-Zakharyuta theorem

Polynomials with exponents in compact convex sets and associated weighted extremal functions: The Siciak-Zakharyuta theorem

On proper separation of convex sets

On proper separation of convex sets

Projection free methods on product domains

AbstractProjection-free block-coordinate methods avoid high computational cost per iteration, and at the same time exploit the particular problem structure of product domains. Frank–Wolfe-like approaches rank among the most popular ones of this type. However, as observed in the literature, there was a gap between the classical Frank–Wolfe theory and the block-coordinate case, with no guarantees of linear convergence rates even for strongly convex objectives in the latter. Moreover, most of previous research concentrated on convex objectives. This study now deals also with the non-convex case and reduces above-mentioned theory gap, in combining a new, fully developed convergence theory with novel active set identification results which ensure that inherent sparsity of solutions can be exploited in an efficient way. Preliminary numerical experiments seem to justify our approach and also show promising results for obtaining global solutions in the non-convex case.

Rigidity for the perimeter inequality under Schwarz symmetrization

In this paper, we give necessary and sufficient conditions for the rigidity of the perimeter inequality under Schwarz symmetrization. The term rigidity refers to the situation in which the equality cases are only obtained by translations of the symmetric set. In particular, we prove that the sufficient conditions for rigidity provided in M. Barchiesi, F. Cagnetti and N. Fusco [Stability of the Steiner symmetrization of convex sets. J. Eur. Math. Soc. 15 (2013), 1245-1278.] are also necessary.