Convex Cone Research Articles

Set-Theoretic Inequalities Based on Convex Multi-Argument Approximate Functions via Set Inclusion

Hypersoft set is a novel area of study which is established as an extension of soft set to handle its limitations. It employs a new approximate mapping called multi-argument approximate function which considers the Cartesian product of attribute-valued disjoint sets as its domain and the power set of universe as its co-domain. The domain of this function is broader as compared to the domain of soft approximate function. It is capable of handling the scenario where sub-attribute-valued sets are considered more significant than taking merely single set of attributes. In this study, notions of set inclusion, convex (concave) sets, strongly convex (concave) sets, strictly convex (concave) sets, convex hull, and convex cone are conceptualized for the multi-argument approximate function. Based on these characterized notions, some set-theoretic inequalities are established with generalized properties and results. The set-theoretic version of classical Jensen’s type inequalities is also discussed with the help of proposed notions.

The structure of production technologies with ratio inputs and outputs

Applications of efficiency and productivity analysis in which some inputs and outputs are given in the form of percentages, averages and other types of ratio measures are sufficiently common in the literature. In two recent papers, the authors developed the variable and constant returns-to-scale technologies with both volume and ratio types of inputs and outputs, referred to as the R-VRS and R-CRS technologies. These technologies are generally nonconvex and have a complex structure. In this paper we explore this in detail. We show that the R-VRS technology can be stated as the union of a finite number of specially constructed standard VRS technologies. Similarly, the R-CRS technology in which all ratio inputs and outputs are of the fixed type, which are typically used to represent environmental and quality factors, can be stated as the union of a finite number of partial polyhedral cones. We show that these results have important theoretical, including conceptual, implications.

Invariant integrals on topological groups

We generalize the fixed-point property for discrete groups acting on convex cones given by Monod in [23] to topological groups. At first, we focus on describing this fixed-point property from a functional point of view, and then we look at the class of groups that have it. Finally, we go through some applications of this fixed-point property. To accomplish these tasks, we introduce a new class of normed Riesz spaces that depend on group representation.

Terracini convexity

We present a generalization of the notion of neighborliness to non-polyhedral convex cones. Although a definition of neighborliness is available in the non-polyhedral case in the literature, it is fairly restrictive as it requires all the low-dimensional faces to be polyhedral. Our approach is more flexible and includes, for example, the cone of positive-semidefinite matrices as a special case (this cone is not neighborly in general). We term our generalization Terracini convexity due to its conceptual similarity with the conclusion of Terracini’s lemma from algebraic geometry. Polyhedral cones are Terracini convex if and only if they are neighborly. More broadly, we derive many families of non-polyhedral Terracini convex cones based on neighborly cones, linear images of cones of positive-semidefinite matrices, and derivative relaxations of Terracini convex hyperbolicity cones. As a demonstration of the utility of our framework in the non-polyhedral case, we give a characterization based on Terracini convexity of the tightness of semidefinite relaxations for certain inverse problems.

A new projection neural network for linear and convex quadratic second-order cone programming

A new projection neural network approach is presented for the linear and convex quadratic second-order cone programming. In the method, the optimal conditions of the linear and convex second-order cone programming are equivalent to the cone projection equations. A Lyapunov function is given based on the G-norm distance function. Based on the cone projection function, the descent direction of Lyapunov function is used to design the new projection neural network. For the proposed neural network, we give the Lyapunov stability analysis and prove the global convergence. Finally, some numerical examples and two kinds of grasping force optimization problems are used to test the efficiency of the proposed neural network. The simulation results show that the proposed neural network is efficient for solving some linear and convex quadratic second-order cone programming problems. Especially, the proposed neural network can overcome the oscillating trajectory of the exist projection neural network for some linear second-order cone programming examples and the min-max grasping force optimization problem.

New representations of epigraphs of conjugate mappings and Lagrange, Fenchel–Lagrange duality for vector optimization problems

In this paper, we concern the vector problem: where X, Y, Z are locally convex Hausdorff topological vector spaces, and are proper mappings, C is a nonempty convex subset of X, and S is a non-empty closed, convex cone in Z. Several new presentations of epigraphs of composite conjugate mappings associated to (VP) are established under variant qualifying conditions. The significance of these representations is twofold: Firstly, they play a key role in establishing new kinds of vector Farkas lemmas which serve as tools in the study of vector optimization problems; secondly, they pay the way to define Lagrange and two new kinds of Fenchel–Lagrange dual problems for (VP). Strong and stable strong duality results corresponding to these mentioned dual problems of (VP) are established using the new Farkas-type results just obtained. It is shown that in the special case where , the Lagrange and Fenchel–Lagrange dual problems for (VP), go back to Lagrange, and Fenchel–Lagrange dual problems for scalar problems, and the resulting duality results cover, and in some cases, extend the corresponding ones for scalar problems in the literature.

The singular set of minimal surfaces near polyhedral cones

We adapt the method of Simon [26] to prove a $C^{1,\alpha}$-regularity theorem for minimal varifolds which resemble a cone $\mathbf{C}^2_0$ over an equiangular geodesic net. For varifold classes admitting a “no-hole” condition on the singular set, we additionally establish $C^{1,\alpha}$-regularity near the cone $\mathbf{C}^2_0 \times \mathbb{R}^m$. Combined with work of Allard [2], Simon [26], Taylor [29], and Naber–Valtorta [21], our result implies a $C^{1,\alpha}$-structure for the top three strata of minimizing clusters and size-minimizing currents, and a Lipschitz structure on the $(n-3)$-stratum.

Preorders on Subharmonic Functions and Measures with Applications to the Distribution of Zeros of Holomorphic Functions

Let $$X$$ be a class of extended numerical functions on a domain $$D$$ of $$d$$ -dimensional Euclidean space $$\mathbb{R}^{d}$$ , $$H\subset X$$ . Given $$u\in X$$ and $$M\in X$$ , we write $$u\prec_{H}M$$ if there is a function $$h\in H$$ such that $$u+h\leq M$$ on $$D$$ . We consider this special preorder $$\prec_{H}$$ for a pair of subharmonic functions $$u$$ and $$M$$ on $$D$$ in cases when $$H$$ is the space of all harmonic functions on $$D$$ or $$H$$ is the convex cone of all subharmonic functions $$h\not\equiv-\infty$$ on $$D$$ . Main results are dual equivalent forms for this preorder $$\prec_{H}$$ in terms of balayage processes for Riesz measures of subharmonic functions $$u$$ and $$M$$ , for Jensen and Arens–Singer (representing) measures, for potentials of these measures, and for special test functions generated by subharmonic functions on complements $$D\backslash S$$ of non-empty precompact subsets $$S\Subset D$$ . Applications to holomorphic functions $$f$$ on a domain $$D\subset\mathbb{C}^{n}$$ relate to the distribution of zero sets of functions $$f$$ under upper restrictions $$|f|\leq\exp M$$ on $$D$$ . If a domain $$D\subset\mathbb{C}$$ is a finitely connected domain with non-empty exterior or a simply connected domain with two different points on the boundary of $$D$$ , then our conditions for the distribution of zeros of $$f\neq 0$$ with $$|f|\leq\exp M$$ on $$D$$ are both necessary and sufficient.

Optimal operation of regional meshed power–gas systems: A tightened conic relaxation based approach

Under the reliability consideration of the multi-energy system, more and more regional networks are in meshed topology. This paper presents a novel method for the regional meshed power–gas system based on the reformulated and tightened cone relaxations. The proposed model defines two energy flows, which are relaxed as convex constraints in the optimization. In the power network, considering the phase angle of each node in meshed networks, a series of conic relaxation-based constraints are employed to represent the alternating current power flow. In the gas network, the conic relaxations are imposed with convex hulls and cones to the gas flow, taking the transmission directions and line-pack constraints into account. To enhance the quality of the relaxation, we propose an approach to tighten the feasible region through two steps. The first step is to tighten the lower and upper bounds of the variables through an optimality-based linear program. The second step is to solve the relaxed tightened optimization with new bounds through a mixed-integer second-order cone program. The advantage and applicability of the proposed approach are tested over a real case study, which shows the improvement of solution quality and optimization speed.

Geometric inequalities for anti-blocking bodies

We study the class of (locally) anti-blocking bodies as well as some associated classes of convex bodies. For these bodies, we prove geometric inequalities regarding volumes and mixed volumes, including Godbersen’s conjecture, near-optimal bounds on Mahler volumes, Saint-Raymond-type inequalities on mixed volumes, and reverse Kleitman inequalities for mixed volumes. We apply our results to the combinatorics of posets and prove Sidorenko-type inequalities for linear extensions of pairs of [Formula: see text]-dimensional posets. The results rely on some elegant decompositions of differences of anti-blocking bodies, which turn out to hold for anti-blocking bodies with respect to general polyhedral cones.

On the Conic Convex Approximation to Locate and Size Fixed-Step Capacitor Banks in Distribution Networks

The problem of the optimal siting and sizing of fixed-step capacitor banks is studied in this research from the standpoint of convex optimization. This problem is formulated through a mixed-integer nonlinear programming (MINLP) model, in which its binary/integer variables are related to the nodes where the capacitors will be installed. Simultaneously, the continuous variables are mainly associated with the power flow solution. The main contribution of this research is the reformulation of the exact MINLP model through a mixed-integer second-order cone programming model (MI-SOCP). This mixed-integer conic model maintains the nonlinearities of the original MINLP model; however, it can be solved efficiently with the branch & bound method combined with the interior point method adapted for conic programming models. The main advantage of the proposed MI-SOCP model is the possibility of finding the global optimum based on the convex nature of the power flow problem for each binary/integer variable combination in the branch & bound search tree. The numerical results in the IEEE 33- and IEEE 69-bus systems demonstrate the effectiveness and robustness of the proposed MI-SOCP model compared to different metaheuristic approaches. The MI-SOCP model finds the final power losses of the IEEE 33- and IEEE 69-bus systems of 138.416kW and 145.397kW, which improves the best literature results reached with the flower pollination algorithm, i.e., 139.075 kW, and 145.860kW, respectively. The simulations are carried out in MATLAB software using its convex optimizer tool known as CVX with the Gurobi solver.

Slater Condition for Tangent Derivatives

Noting that the existing Slater condition, as a fundamental constraint qualification in optimization, is only applicable in the convex setting, we introduce and study the Slater condition for the Bouligand and Clarke tangent derivatives of a general vector-valued function F with respect to a closed convex cone K. Without any assumption, it is proved that the Slater condition for the Clarke (respectively, Bouligand) tangent derivative with respect to K is always stable when the objective function F undergoes small Lipschitz (calm) perturbations. Based on this, we prove that if the Clarke (Bouligand) tangent derivative of F satisfies the Slater condition (with respect to K) then the conic inequality determined by F has a stable metric subregularity when F undergoes small Lipschitz (calm regular) perturbations. In the composite-convexity case, the converse implication is also proved to be true. Moreover, under the Slater condition for the tangent derivative of F, it is proved that the normal cone to the sublevel set of F can be formulated by the subdifferential of F, which improves the corresponding results in either the smooth or convex case. As applications, without any qualification assumption, we improve and generalize formulas for the normal cone to a convex sublevel set by Cabot and Thibault [(2014), Sequential formulae for the normal cone to sublevel sets. Transactions of the American Mathematical Society 366(12):6591–6628]. With the help of these formulas, some new Karush–Kuhn–Tucker optimality conditions are established.

On the roots of polynomials with log-convex coefficients

AbstractIn this paper, we consider the family of nth degree polynomials whose coefficients form a log-convex sequence (up to binomial weights), and investigate their roots. We study, among others, the structure of the set of roots of such polynomials, showing that it is a closed convex cone in the upper half-plane, which covers its interior when n tends to infinity, and giving its precise description for every $n\in \mathbb {N}$ , $n\geq 2$ . Dual Steiner polynomials of star bodies are a particular case of them, and so we derive, as a consequence, further properties for their roots.

Radial symmetry of solutions to anisotropic and weighted diffusion equations with discontinuous nonlinearities

We prove radial symmetry for bounded nonnegative solutions of a weighted anisotropic problem. Given the anisotropic setting that we deal with, the term "radial" is understood in the Finsler framework. In the whole space, J. Serra obtained the symmetry result in the isotropic unweighted setting. In this case we provide the extension of his result to the anisotropic setting. This provides a generalization to the anisotropic setting of a celebrated result due to Gidas-Ni-Nirenberg and such a generalization is new even for in the case of linear operators whenever the dimension is greater than 2. In proper cones, the results presented are new even in the isotropic and unweighted setting for suitable nonlinear cases. Even for the previously known case of unweighted isotropic setting, the present paper provides an approach to the problem by exploiting integral (in)equalities which is new for $N>2$: this complements the corresponding symmetry result obtained via the moving planes method by Berestycki-Pacella.

On the Extreme Rays of the Cone of $$3\times 3$$ Quasiconvex Quadratic Forms: Extremal Determinants Versus Extremal and Polyconvex Forms

This work is concerned with the study of the extreme rays of the convex cone of \(3\times 3\) quasiconvex quadratic forms (denoted by \(\mathcal{C}_3\)). We characterize quadratic forms \(f\in \mathcal{C}_3,\) the determinant of the acoustic tensor of which is an extremal polynomial, and conjecture/discuss about other cases. We prove that in the case when the determinant of the acoustic tensor of a form \(f\in \mathcal{C}_3\) is an extremal polynomial other than a perfect square, then the form must itself be an extreme ray of \(\mathcal{C}_3;\) when the determinant is a perfect square, then the form is either an extreme ray of \(\mathcal{C}_3\) or polyconvex; finally, when the determinant is identically zero, then the form f must be polyconvex. The zero determinant case plays an important role in the proofs of the other two cases. We also make a conjecture on the extreme rays of \(\mathcal{C}_3,\) and discuss about weak and strong extremals of \(\mathcal{C}_d\) for \(d\ge 3,\) where it turns out that several properties of \(\mathcal{C}_3\) do not hold for \(\mathcal{C}_d\) for \(d>3,\) and thus case \(d=3\) is special. These results recover all previously known results (to our best knowledge) on examples of extreme points of \(\mathcal{C}_3\) that were proved to be such. Our results also improve the ones proven by Harutyunyan and Milton (Commun Pure Appl Math 70(11):2164–2190, 2017) on weak extremals in \(\mathcal{C}_3\) (or extremals in the sense of Milton) introduced in (Commun Pure Appl Math XLIII:63–125, 1990). In the language of positive biquadratic forms, quasiconvex quadratic forms correspond to nonnegative biquadratic forms and the results read as follows: if the determinant of the \({\varvec{y}}\) (or \({\varvec{x}}\)) matrix of a \(3\times 3\) nonnegative biquadratic form in \({\varvec{x}},{\varvec{y}}\in \mathbb R^3\) is an extremal polynomial that is not a perfect square, then the form must be an extreme ray of the convex cone of \(3\times 3\) nonnegative biquadratic forms \((\mathcal{C}_3);\) if the determinant is identically zero, then the form must be a sum of squares; if the determinant is a nonzero perfect square, then the form is either an extreme ray of \(\mathcal{C}_3,\) or is a sum of squares. The proofs are all established by means of several classical results from linear algebra, convex analysis (geometry), real algebraic geometry, and the calculus of variations.

Comparison of Image Endmember- and Object-Based Classification of Very-High-Spatial-Resolution Unmanned Aircraft System (UAS) Narrow-Band Images for Mapping Riparian Forests and Other Land Covers

Riparian forests are critical for carbon storage, biodiversity, and river water quality. There has been an increasing use of very-high-spatial-resolution (VHR) unmanned aircraft systems (UAS)-based remote sensing for riparian forest mapping. However, for improved riparian forest/zone monitoring, restoration, and management, an enhanced understanding of the accuracy of different classification methods for mapping riparian forests and other land covers at high thematic resolution is necessary. Research that compares classification efficacies of endmember- and object-based methods applied to VHR (e.g., UAS) images is limited. Using the Sequential Maximum Angle Convex Cone (SMACC) endmember extraction algorithm (EEA) jointly with the Spectral Angle Mapper (SAM) classifier, and a separate multiresolution segmentation/object-based classification method, we map riparian forests/land covers and compare the classification accuracies accrued via the application of these two approaches to narrow-band, VHR UAS orthoimages collected over two river reaches/riparian areas in Austria. We assess the effect of pixel size on classification accuracy, with 7 and 20 cm pixels, and evaluate performance across multiple dates. Our findings show that the object-based classification accuracies are markedly higher than those of the endmember-based approach, where the former generally have overall accuracies of >85%. Poor endmember-based classification accuracies are likely due to the very small pixel sizes, as well as the large number of classes, and the relatively small number of bands used. Object-based classification in this context provides for effective riparian forest/zone monitoring and management.

Positive hulls of random walks and bridges

We study random convex cones defined as positive hulls of d-dimensional random walks and bridges. We compute expectations of various geometric functionals of these cones such as the number of k-dimensional faces and the sums of conic quermassintegrals of their k-dimensional faces. These expectations are expressed in terms of Stirling numbers of both kinds and their B-analogues.

The multidimensional truncated moment problem: The moment cone

Let A={a1,…,am}, m∈N, be measurable functions on a measurable space (X,A). If μ is a positive measure on (X,A) such that ∫aidμ<∞ for all i, then the sequence (∫a1dμ,…,∫amdμ) is called a truncated moment sequence. By Richter's Theorem each truncated moment sequence has a k-atomic representing measure with k≤m. The set SA of all truncated moment sequences is the truncated moment cone. The aim of this paper is to analyze the various structures of the truncated moment cone. The main results concern the facial structure (exposed faces, facial dimensions) and lower and upper bounds of the Carathéodory number (that is, the smallest number of atoms which suffices for all truncated moment sequences) of the convex cone SA. In the case when X⊆Rn and ai∈C1(X,R), the differential structure of the moment map and regularity/singularity properties of moment sequences are analyzed. The maximal mass problem is considered and some applications to other problems are sketched.

The Appropriate of Cone Depth in Loop Electrical Excision Procedure (LEEP) for Negative Pathological Margin from High Grade Precancerous Lesion of Cervix, Retrospective Study.

Objective:To determine the appropriate cone depth for treating high grade precancerous lesions to achieve negative pathological margins of cones from LEEPs. Other factors associated with positive pathological margin were also investigated. Methods:A Retrospective study recruited 170 patients who received indications for LEEP during January 2015 to July 2020 were enrolled. The participants were operated by a single cut of LEEP and not had previously conization before. All patient data were collected into two groups, including negative and positive cone margin groups. Then, we used the cone depth by calculating from cone tissue after formalin fixation to eliminate shrinkage effect. The appropriate cut-off points for cone depth were calculated by ROC and analyzed factors that influence positive cone margin. Results:The depth of cone (mm ±SD) of negative margin group was 8.70 (±3.36) and 6.13 (±2.28) mm in positive margin group. The appropriate cut-off points for cone depth were calculated by ROC presented at resection depth of 7.21 mm, which displayed proper cone depth with a sensitivity of 63.53% and specificity of 71.76%. Elderly age (adjusted OR 1.061, 95%CI 1.008-1.117, p=0.002), number of quadrants of lesion involvement (adjusted OR 1.182, 95%CI 1.312-2.513, p=<0.001) and glandular involvement (adjusted OR 3.648, 95%CI 1.605-8.292, p=0.002) were the significant risk factors for positive margin. Conclusion:The appropriate cone depth for treating high grade precancerous lesions was at least 7.21 mm to achieve a negative cone margin from LEEP. The significant factors associated with positive cone margin include elderly age, more quadrants of lesion involvement and glandular involvement.

Fractional mean curvature flow of Lipschitz graphs

We consider the fractional mean curvature flow of entire Lipschitz graphs. We provide regularity results, and we study the long time asymptotics of the flow. In particular we show that in a suitable rescaled framework, if the initial graph is a sublinear perturbation of a cone, the evolution asymptotically approaches an expanding self-similar solution. We also prove stability of hyperplanes and of convex cones in the unrescaled setting.