Convex Sets Research Articles

Minimally fine exhausters

In this article, we study upper exhausters of positively homogenous functions defined on locally convex space. We prove the existence of minimally fine exhauster finer than the given one. Such exhauster has to be unique for two-element exhauster. We show that a positively homogenous function is upper semicontinuous if and only if it has an upper exhauster. We prove that Clarke subdifferential of a positively homogenous function is the closed convex hull of the union of its minimally fine exhauster. We also present a number of appropriate examples.

Unconditional MBP preservation and energy stability of the stabilized exponential time differencing schemes for the vector-valued Allen–Cahn equations

The vector-valued Allen–Cahn equations have been extensively applied to simulate the multiphase flow models. In this paper, we consider the maximum bound principle (MBP) and corresponding numerical schemes for the vector-valued Allen–Cahn equations. We firstly formulate the stabilized equations via utilizing the linear stabilization technique, and then focus on the bounding constant of the nonlinear function based on the fact that the extremes of a constrained problem will occur in the bounded and convex domain. Later the first- and second-order stabilized exponential time differencing schemes are adopted for temporal integration, which are linear and unconditionally preserve the discrete MBP in the time discrete sense. Moreover, the proposed schemes can be proven to dissipate the original energy instead of the modified energy. Their convergence analysis is also presented. Various numerical examples in two and three dimensions are performed to verify these theoretical results and demonstrate the efficiency of the proposed schemes.

Effect of concave–convex degree of substrate surface on thermal shock performance of Cr coating

ABSTRACT The thermal shock resistance of electroplating Cr coating on three typical concave–convex gun substrate at 900℃ is investigated. The microstructure and crystal phase of Cr coating are analysed by SEM, EDS and XRD. Compared with the concave inside corner of the surface of the three substrates, electroplating Cr coating at the convex outside corner shows better thermal shock performance. During the test, it is found that the concave–convex degree on the surface of substrate has a significant effect on the thermal shock performance of the surface coating. With the increase of the transitional angle of concave–convex region on the surface of substrate, the thermal shock resistance of the surface coating is better. The oxygen contact area and thermal stress concentration are the main reasons for the difference of thermal shock performance of the three substrate coatings.

Continuous Equality Knapsack with Probit-Style Objectives

We study continuous, equality knapsack problems with uniform separable, non-convex objective functions that are continuous, antisymmetric about a point, and have concave and convex regions. For example, this model captures a simple allocation problem with the goal of optimizing an expected value where the objective is a sum of cumulative distribution functions of identically distributed normal distributions (i.e., a sum of inverse probit functions). We prove structural results of this model under general assumptions and provide two algorithms for efficient optimization: (1) running in linear time and (2) running in a constant number of operations given preprocessing of the objective function.

Unsupervised 3D seismic data reconstruction using a weighted-attentive deep learning framework

The physical and cost limitations of seismic data acquisition often result in spatial undersampling of data, which has a detrimental impact on subsequent data processing. Seismic data reconstruction plays a crucial role in recovering missing traces arising from the acquisition process. In recent years, deep learning (DL) has emerged as an intelligent solution for this purpose. We introduce an innovative approach called the weighted-attentive DL framework for unsupervised 3D seismic data reconstruction, which combines an attentive transformer network (ATNet) with the conventional projection onto convex sets (POCS) algorithm. The proposed framework follows the plug-and-play concept, requiring only the original sub-sampled data to achieve robust performance. It accomplishes this by iteratively updating ATNet parameters, where ATNet serves as the reconstruction operator during the DL process. This approach allows us to simultaneously recover missing traces and filter out random noise, with a key feature being its enhanced attention to the observed signal compared to convolutional neural networks. Additionally, we incorporate the POCS algorithm into our framework to introduce a linear decay weight for the aliasing space containing both noise and signal during the reconstruction process. We rigorously evaluate the proposed method against four existing approaches: adaptive POCS (APOCS), fast dictionary learning sequential generalized K-means (SGK), optimally damped rank-reduction (ODRR), and DenseNet methods. Our comparative experiments, conducted on synthetic and field data examples, clearly demonstrate the substantial improvements achieved by the proposed method in terms of reconstruction and denoising when compared to the four benchmarked methods.

On locally Lipschitz convex functions defined on subsets with empty interior of locally convex spaces

The generic Fréchet and/or Gateaux differentiability of a continuous convex function on an open convex subset of a Banach space is well studied and has important theoretical implications. An example of Rainwater [Yet more on the differentiability of convex functions. Proc Amer Math Soc. 1988;103(3):773–778] shows that similar results are not true when the openness of the domain is not ensured, but such properties can be obtained if the respective convex function is assumed to be locally Lipschitz, as shown by Verona [More on the differentiability of convex functions. Proc Amer Math Soc. 1988;103(1):137–140]; Rainwater [Yet more on the differentiability of convex functions. Proc Amer Math Soc. 1988;103(3):773–778]; Noll [Generic Gâteaux-differentiability of convex functions on small sets. J Math Anal Appl. 1990;147(2):531–544] and others. It is our aim to extend such results for convex functions defined on (not necessarily open) convex subsets of locally convex spaces. In the same framework we extend Gale's duality theorem (1967), as well as a result of H. Ergin & T. Sarver on the unique dual representation of a convex function (2010). Moreover, a detailed study of the Rainwater example is done.

Almost optimal polynomial approximation on convex sets in $$\mathbb {C}$$

Almost optimal polynomial approximation on convex sets in $$\mathbb {C}$$

Polygonal Approximation Learning for Convex Object Segmentation in Biomedical Images With Bounding Box Supervision.

As a common and critical medical image analysis task, deep learning based biomedical image segmentation is hindered by the dependence on costly fine-grained annotations. To alleviate this data dependence, in this article, a novel approach, called Polygonal Approximation Learning (PAL), is proposed for convex object instance segmentation with only bounding-box supervision. The key idea behind PAL is that the detection model for convex objects already contains the necessary information for segmenting them since their convex hulls, which can be generated approximately by the intersection of bounding boxes, are equivalent to the masks representing the objects. To extract the essential information from the detection model, a repeated detection approach is employed on biomedical images where various rotation angles are applied and a dice loss with the projection of the rotated detection results is utilized as a supervised signal in training our segmentation model. In biomedical imaging tasks involving convex objects, such as nuclei instance segmentation, PAL outperforms the known models (e.g., BoxInst) that rely solely on box supervision. Furthermore, PAL achieves comparable performance with mask-supervised models including Mask R-CNN and Cascade Mask R-CNN. Interestingly, PAL also demonstrates remarkable performance on non-convex object instance segmentation tasks, for example, surgical instrument and organ instance segmentation.

Well-posedness of nonautonomous abstract Cauchy problems under dissipativity conditions expressed by metric-like functionals

Under a dissipativity condition expressed by a metric-like functional, the well-posedness of a nonautonomous abstract Cauchy problem is established. The main result not only extends Martin's result on the generation of an evolution operator in a class of uniformly convex Banach spaces but also can be applied to mixed problems for nonautonomous evolution equations of Kirchhoff type with which Kato's quasilinear theory or the theory of quasi-contractive semigroups cannot directly deal. The main result also gives an affirmative answer to Kōmura's conjecture that an evolution operator of linear operators with moving domains can be generated even under a weak stability condition rather than Kato's stability condition, and the result provides an effective means for solving degenerate wave equations.

A Hessian-Based Technique for Specular Reflection Detection and Inpainting in Colonoscopy Images.

In the field of Computer-Aided Detection (CADx), the use of AI-based algorithms for disease detection in endoscopy images, especially colonoscopy images, is on the rise. However, these algorithms often encounter performance issues due to obstructions like specular reflection, resulting in false positives. This paper presents a novel algorithm specifically designed to tackle the challenges posed by high specular reflection regions in colonoscopy images. The proposed algorithm identifies these regions and applies precise inpainting for restoration. The process entails converting the input image from RGB to HSV color space and focusing on the Saturation (S) component in convex regions detected using a Hessian-based method. This step creates a binary mask that pinpoints areas of specular reflection. The inpainting function then uses this mask to guide the restoration of these identified regions and their borders. To ensure a seamless blend of the restored regions with the background and adjacent pixels, a feathering process is applied to the repaired regions. This enhances both the accuracy and aesthetic coherence of the inpainted images. The performance of our algorithm was rigorously tested on five unique colonoscopy datasets and various endoscopy images from the Kvasir dataset, using an extensive set of evaluation metrics and a comparative analysis with existing methods consistently highlighted the superior performance of our algorithm.

Zone Model Predictive Control with Ellipsoid Softing Target in Chemical Processes

A zone control algorithm is proposed that considers both economic performance indicators and control performance indicators. Unlike classic set point control, zone control expands the control target into a convex set. In this study, an ellipsoid is used as the control target, and the advantages of the ellipsoid target are explained in terms of overall stability and computational load. After defining the distance measurement function and appropriate terminal constraints, an objective function that considers both control performance and optimization performance is constructed. A theoretical analysis shows that the proposed control algorithm satisfies the Lyapunov stability criterion. The superiority of the ellipsoid control target in handling complex multivariable control tasks is also demonstrated. This method has significant potential value in practical industrial applications, helping to unleash the potential control performance and economic benefits of zone control systems. Finally, the feasibility and stability of the algorithm are verified through a typical chemical process simulation.

Modified Inertial Krasnosel'skii-Mann Type Method for Solving Fixed PointProblems in Real Uniformly Convex Banach Spaces

We present an altered version of the inertial Krasnosel'skii-Mann algorithm and demonstrate convergence outcomes for mappings that are asymptotically nonexpansive within real, uniformly convex Banach spaces. To achieve our results, we skillfully construct the inequality in equation \eqref{6} and apply it accordingly. Our findings support and broadly generalize a number of significant findings from the literature. We demonstrate, as an application, the generation of maximal monotone operators' zeros via fixed point methods in Hilbert spaces. Additionally, we solve convex minimization issues using our fixed-point techniques.

The natural ordering cone of a polyhedral convex set-valued objective mapping

ABSTRACT For a given polyhedral convex set-valued mapping we define a polyhedral convex cone which we call the natural ordering cone. We show that the solution behaviour of a polyhedral convex set optimization problem can be characterized by this cone. Under appropriate assumptions, the natural ordering cone is proven to be the smallest ordering cone which makes a polyhedral convex set optimization problem solvable.

Matrix extreme points and free extreme points of free spectrahedra

Free spectrahedra are dimension free solution sets to linear matrix inequalities of the form L A ( X ) = I d ⊗ I n + A 1 ⊗ X 1 + A 2 ⊗ X 2 + ⋯ + A g ⊗ X g ⪰ 0 , where the A i and X i are symmetric matrices and the X i have any size n × n . Free spectrahedra are ubiquitous in systems engineering, operator algebras, and the theory of matrix convex sets. Matrix and free extreme points of free spectrahedra are particularly important. We present theoretical, algorithmic, and experimental results illuminating basic properties of extreme points. For example, though many authors have studied matrix and free extreme points, it has until now been unknown if these two types of extreme points are actually different. This paper settles that issue. We also present and analyze several algorithms. Namely, we perfect an algorithm for computing an expansion of an element of a free spectrahedron in terms of free extreme points. We also give algorithms for testing if a point is matrix extreme and for computing matrix extreme points that are not free extreme.

A new concept of semistrict quasiconvexity for vector functions

We establish a new concept of semistrict quasiconvexity for vector functions defined on a nonempty convex set in a real linear space X that take values in some real topological linear space Y, partially ordered by a proper solid convex cone C. The so-called semistrict C-quasiconvexity notion recovers the classical concept of semistrict quasiconvexity of scalar functions when Y = R and C = R + . Additionally, analogous to the scalar scenario, if the cone C is closed, a vector function is both semistrictly C-quasiconvex and C-quasiconvex (in the sense of Luc, 1989) if and only if it is explicitly C-quasiconvex (in the sense of Popovici, 2007). Finally, we convey a characterization of semistrictly C-quasiconvex functions by means of scalar semistrictly quasiconvex functions that are compositions of the nonlinear scalarization functions introduced by Gerstewitz (Tammer) in 1983 with the initial vector function. In light of this characterization, the new concept of semistrict C-quasiconvexity seems to be a natural vector counterpart for the scalar concept of semistrict quasiconvexity.

The Convex Sets in Banach Spaces and Polynomial Approximation

The Convex Sets in Banach Spaces and Polynomial Approximation

Shi arrangements and low elements in Coxeter groups

AbstractGiven an arbitrary Coxeter system and a non‐negative integer , the ‐Shi arrangement of is a subarrangement of the Coxeter hyperplane arrangement of . The classical Shi arrangement () was introduced in the case of affine Weyl groups by Shi to study Kazhdan–Lusztig cells for . As two key results, Shi showed that each region of the Shi arrangement contains exactly one element of minimal length in and that the union of their inverses form a convex subset of the Coxeter complex. The set of ‐low elements in were introduced to study the word problem of the corresponding Artin–Tits (braid) group and they turn out to produce automata to study the combinatorics of reduced words in . In this article, we generalize and extend Shi's results to any Coxeter system for any : (1) the set of minimal length elements of the regions in a ‐Shi arrangement is precisely the set of ‐low elements, settling a conjecture of the first and third authors in this case; (2) the union of the inverses of the (0‐)low elements form a convex subset in the Coxeter complex, settling a conjecture by the third author, Nadeau and Williams.

Characterizations of minimal elements of upper support with applications in minimizing DC functions

Abstract In this study, we discuss on the problem of minimizing the differences of two non-positive valued increasing, co-radiant and quasi-concave (ICRQC) functions defined on X X (where X X is a real locally convex topological vector space). For this purpose, we first gave different characterizations of the upper support set’s minimal elements of non-positive co-radiant functions. Then, we presented sufficient and necessary conditions for the global minimizers of the differences of two non-positive ICRQC functions.

A New Class of Univalent Functions Defined by Differential Operator

In the present work, we submit and study a new class $\mathcal{S}_{\lambda, \alpha, b}^{z, m, t}(\beta, \gamma, \omega, \mu)$ containing analytic univalent functions defined by new differential operator $D_{\lambda, \alpha}^{z, m, t}$ in the open unit disk $E=\{s \in \mathbb{C}:|s|<1\}$. We get some geometric properties, such as, coefficient estimate, growth and distortion theorems, convex set, radii of convexity and starlikeness, weighted mean, arithmetic mean and partial sums for functions belonging to the class $\mathcal{S}_{\lambda, \alpha, b}^{z, m, t}(\beta, \gamma, \omega, \mu)$.

Spatial syntax analysis on the renewal of the external space of mining heritage buildings —— A case study of Zollverein Coal Mine Industrial Complex

ABSTRACT The renovation of external spaces of mining heritage buildings should respect their historical and cultural value and be adapted to the specific characteristics of the mining area for spatial restructuring and functional replacement. Current research often relies on subjective experience and focuses on qualitative analysis, lacking in-depth quantitative analysis. Building upon the summary of research on mining heritage exterior spaces and spatial syntax studies, this paper proposes a research methodology that quantitatively analyzes the accessibility and integration of mining heritage exterior spaces using spatial syntax convex space and axis map models. Taking the exterior connecting spaces and characteristic spaces before and after the renovation of the German Zollverein Coal Mine Industrial Complex as the research objects, the impact of spatial location on different circulation organizations and functions of spatial places is visually expressed, providing a more intuitive explanation of the complex relationships in mining heritage exterior spaces. The research findings reveal that the spatial syntax model is adaptable in evaluating the renovation of mining heritage exterior spaces, with spaces exhibiting high integration being essentially consistent with key and highly utilized exterior spaces within the renovated mining area. In summary, it can be concluded that the integration of mining heritage exterior spaces and consequently the utilization rate and value can be improved through spatial “Connection”, “Growth”, and “Darning”, thereby offering direct reference for subsequent research and practical organization of mining heritage architectural exterior space renewal.