Convex Cone Research Articles

Self-associated three-dimensional cones

For every proper convex cone \(K \subset {\mathbb {R}}^3\) there exists a unique complete hyperbolic affine 2-sphere with mean curvature \(-1\) which is asymptotic to the boundary of the cone. Two cones are associated if the corresponding affine spheres can be mapped to each other by an orientation-preserving isometry. This equivalence relation is generated by the groups \(SL(3,{\mathbb {R}})\) and \(S^1\), where the former acts by linear transformations of the ambient space, and the latter by multiplication of the cubic holomorphic differential of the affine sphere by unimodular complex constants. The action of \(S^1\) generalizes conic duality, which acts by multiplication of the cubic differential by \(-1\). We call a cone self-associated if it is linearly isomorphic to all its associated cones, in which case the action of \(S^1\) induces (nonlinear) isometries of the corresponding affine sphere. We give a complete classification of the self-associated cones and compute isothermal parameterizations of the corresponding affine spheres. Their metrics can be expressed in terms of degenerate Painlevé III transcendents. The boundaries of generic self-associated cones can be represented as conic hulls of vector-valued solutions of a certain third-order linear ordinary differential equation with periodic coefficients, but there exist also self-associated cones with polyhedral boundary parts. The self-associated cones are the second family of non-trivial 3-dimensional cones for which the affine spheres can be computed explicitly, the first being the semi-homogeneous cones.

Nonlinear Cone Model for Investigation of Runaway Electron Synchrotron Radiation Spot Shape

The runaway electron event is the fundamental physical phenomenon and tokamak is the most advanced conception of the plasma magnetic confinement. The energy of disruption generated runaway electrons can reach as high as tens of mega-electron-volt and they can cause a catastrophic damage of plasma-facing-component surfaces in large tokamaks and International Thermonuclear Experimental Reactor (ITER). Due to its importance, this phenomenon is being actively studied both theoretically and experimentally in leading thermonuclear fusion centers. Thus, effective monitoring of the runaway electrons is an important task. The synchrotron radiation diagnostic allows direct observation of such runaway electrons and an analysis of their parameters and promotes the safety operation of present-day large tokamaks and future ITER. In 1990 such diagnostic had demonstrated its effectiveness on the TEXTOR (Tokamak Experiment for Technology Oriented Research, Germany) tokamak for investigation of runaway electrons beam size, position, number, and maximum energy. Now this diagnostic is installed practically on all the present-day’s tokamaks. The parameter v┴/|v||| strongly influences on the runaway electron synchrotron radiation behavior (v|| is the longitudinal velocity, v┴ is the transverse velocity with respect to the magnetic field B). The paper is devoted to the theoretical investigation of runaway electron synchrotron radiation spot shape when this parameter is not small that corresponds to present-day tokamak experiments. The features of the relativistic electron motion in a tokamak are taken into account. The influence of the detector position on runaway electron synchrotron radiation data is discussed. Analysis carried out in the frame of the nonlinear cone model. In this model, the ultrarelativistic electrons emit radiation in the direction of their velocity v→ and the velocity vector runs along the surface of a cone whose axis is parallel to the magnetic field B. The case of the small parameter v┴/|v||| (v┴/|v|||<<1, linear cone model) was considered in the paper: Plasma Phys. Rep. 22, 535 (1996) and these theoretical results are used for experimental data analysis.

On minimal positive basis for polyhedral cones in n?>

For polyhedral cones in the Euclidean space, we present its conic dimension, which is invariant under linear isomorphisms that is sensitive to the number of generators of this cone, and the related notion of conic basis. We may interpret these two notions as versions of the definitions of linear dimension and linear basis for linear subspaces in the setting of polyhedral cones. We establish a conic version of the rank-nullity theorem that, in this case, is an inequality involving the conic dimensions of both the cone and its image under a linear map. We use this conic rank-nullity inequality to establish both a decomposition and a union of conic basis, involving the lineality space of the cone. We introduce the signature of a polyhedral cone and establish some results on the injectivity of a linear map and the preservation of the signature of a polyhedral cone under linear maps. In particular, we show that a linear map that acts injectively on the linear span of a polyhedral cone preserves its signature.

On the tensor semigroup of affine Kac-Moody lie algebras

The support of the tensor product decomposition of integrable irreducible highest weight representations of a symmetrizable Kac-Moody Lie algebra g \mathfrak {g} defines a semigroup of triples of weights. Namely, given λ \lambda in the set P + P_+ of dominant integral weights, V ( λ ) V(\lambda ) denotes the irreducible representation of g \mathfrak {g} with highest weight λ \lambda . We are interested in the tensor semigroup Γ N ( g ) ≔ { ( λ 1 , λ 2 , μ ) ∈ P + 3 | V ( μ ) ⊂ V ( λ 1 ) ⊗ V ( λ 2 ) } , \begin{equation*} \Gamma _{\mathbb {N}}(\mathfrak {g})≔\{(\lambda _1,\lambda _2,\mu )\in P_{+}^3\,|\, V(\mu )\subset V(\lambda _1)\otimes V(\lambda _2)\}, \end{equation*} and in the tensor cone Γ ( g ) \Gamma (\mathfrak {g}) it generates: Γ ( g ) ≔ { ( λ 1 , λ 2 , μ ) ∈ P + , Q 3 | ∃ N ≥ 1 V ( N μ ) ⊂ V ( N λ 1 ) ⊗ V ( N λ 2 ) } . \begin{equation*} \Gamma (\mathfrak {g})≔\{(\lambda _1,\lambda _2,\mu )\in P_{+,{\mathbb {Q}}}^3\,|\,\exists N\geq 1 \quad V(N\mu )\subset V(N\lambda _1)\otimes V(N\lambda _2)\}. \end{equation*} Here, P + , Q P_{+,{\mathbb {Q}}} denotes the rational convex cone generated by P + P_+ . In the special case when g \mathfrak {g} is a finite-dimensional semisimple Lie algebra, the tensor semigroup is known to be finitely generated and hence the tensor cone to be convex polyhedral. Moreover, the cone Γ ( g ) \Gamma (\mathfrak {g}) is described in Belkale and Kumar [Invent. Math. 166 (2006), pp. 185–228] by an explicit finite list of inequalities. In general, Γ ( g ) \Gamma (\mathfrak {g}) is neither polyhedral, nor closed. In this article we describe the closure of Γ ( g ) \Gamma (\mathfrak {g}) by an explicit countable family of linear inequalities for any untwisted affine Lie algebra, which is the most important class of infinite-dimensional Kac-Moody algebra. This solves a Brown-Kumar’s conjecture in this case (see Brown and Kumar [Math. Ann. 360 (2014), pp. 901–936]). The difference between the tensor cone and the tensor semigroup is measured by the saturation factors. Namely, a positive integer d d is called a saturation factor, if V ( N λ 1 ) ⊗ V ( N λ 2 ) V(N\lambda _1)\otimes V(N\lambda _2) contains V ( N μ ) V(N\mu ) for some positive integer N N then V ( d λ 1 ) ⊗ V ( d λ 2 ) V(d\lambda _1)\otimes V(d\lambda _2) contains V ( d μ ) V(d\mu ) , assuming that μ − λ 1 − λ 2 \mu -\lambda _1-\lambda _2 belongs to the root lattice. For g = s l n \mathfrak {g}={\mathfrak {sl}}_n , the famous Knutson-Tao theorem asserts that d = 1 d=1 is a saturation factor (see Knutson and Tao [J. Amer. Math. Soc. 12 (1999), pp. 1055–1090]). More generally, for any simple Lie algebra, explicit saturation factors are known. In the Kac-Moody case, Γ N ( g ) \Gamma _{\mathbb {N}}(\mathfrak {g}) is not necessarily finitely generated and hence the existence of such a factor is unclear a priori. Here, we obtain explicit saturation factors for any affine Kac-Moody Lie algebra. For example, in type A ~ n \tilde A_n , we prove that any integer d ≥ 2 d\geq 2 is a saturation factor, generalizing the case A ~ 1 \tilde A_1 shown in Brown and Kumar [Math. Ann. 360 (2014), pp. 901–936].

Alfonso: Matlab Package for Nonsymmetric Conic Optimization

We present alfonso, an open-source Matlab package for solving conic optimization problems over nonsymmetric convex cones. The implementation is based on the authors’ corrected analysis of a method of Skajaa and Ye. It enables optimization over any convex cone as long as a logarithmically homogeneous self-concordant barrier is available for the cone or its dual. This includes many nonsymmetric cones, for example, hyperbolicity cones and their duals (such as sum-of-squares cones), semidefinite and second-order cone representable cones, power cones, and the exponential cone. Besides enabling the solution of problems that cannot be cast as optimization problems over a symmetric cone, algorithms for nonsymmetric conic optimization also offer performance advantages for problems whose symmetric cone programming representation requires a large number of auxiliary variables or has a special structure that can be exploited in the barrier computation. The worst-case iteration complexity of alfonso is the best known for nonsymmetric cone optimization: [Formula: see text] iterations to reach an ε-optimal solution, where ν is the barrier parameter of the barrier function used in the optimization. Alfonso can be interfaced with a Matlab function (supplied by the user) that computes the Hessian of a barrier function for the cone. A simplified interface is also available to optimize over the direct product of cones for which a barrier function has already been built into the software. This interface can be easily extended to include new cones. Both interfaces are illustrated by solving linear programs. The oracle interface and the efficiency of alfonso are also demonstrated using an optimal design of experiments problem in which the tailored barrier computation greatly decreases the solution time compared with using state-of-the-art, off-the-shelf conic optimization software. Summary of Contribution: The paper describes an open-source Matlab package for optimization over nonsymmetric cones. A particularly important feature of this software is that, unlike other conic optimization software, it enables optimization over any convex cone as long as a suitable barrier function is available for the cone or its dual, not limiting the user to a small number of specific cones. Nonsymmetric cones for which such barriers are already known include, for example, hyperbolicity cones and their duals (such as sum-of-squares cones), semidefinite and second-order cone representable cones, power cones, and the exponential cone. Thus, the scope of this software is far larger than most current conic optimization software. This does not come at the price of efficiency, as the worst-case iteration complexity of our algorithm matches the iteration complexity of the most successful interior-point methods for symmetric cones. Besides enabling the solution of problems that cannot be cast as optimization problems over a symmetric cone, our software can also offer performance advantages for problems whose symmetric cone programming representation requires a large number of auxiliary variables or has a special structure that can be exploited in the barrier computation. This is also demonstrated in this paper via an example in which our code significantly outperforms Mosek 9 and SCS 2.

Generation of Chow Parameters and Reduced Variables Through Nearest Neighbor Relations in Threshold Networks.

Generation of useful variables and features is an important issue throughout the machine learning, artificial intelligence, and applied fields for their efficient computations. In this paper, the nearest neighbor relations are proposed for the minimal generation and the reduced variables of the functions in the threshold networks. First, the nearest neighbor relations are shown to be minimal and inherited for threshold functions and they play an important role in the iterative generation of the Chow parameters. Further, they give a solution for the Chow parameters problem. Second, convex cones are made of the nearest neighbor relations for the generation of the reduced variables. Then the edges of convex cones are compared for the discrimination of variables. Finally, the reduced variables based on the nearest neighbor relations are shown to be useful for documents classification.

Threshold Phenomena for Random Cones

We consider an even probability distribution on the d-dimensional Euclidean space with the property that it assigns measure zero to any hyperplane through the origin. Given N independent random vectors with this distribution, under the condition that they do not positively span the whole space, the positive hull of these vectors is a random polyhedral cone (and its intersection with the unit sphere is a random spherical polytope). It was first studied by Cover and Efron. We consider the expected face numbers of these random cones and describe a threshold phenomenon when the dimension d and the number N of random vectors tend to infinity. In a similar way we treat the solid angle, and more generally the Grassmann angles. We further consider the expected numbers of k-faces and of Grassmann angles of index d-k when also k tends to infinity.

Spin-Gapless States in Two-Dimensional Molecular Ferromagnet Fe2(TCNQ)2.

Two-dimensional van der Waals magnetic atomic crystals have provided unprecedented access to magnetic ground states due to a quantum confinement effect. Here, using first-principles calculations, we demonstrate a spin-gapless molecular ferromagnet, namely, Fe2(TCNQ)2, with superior mechanical stability and a remarkable linear Dirac cone, which can be exfoliated from its already-synthesized van der Waals crystal. Especially, Young's modulus has values of 175.28 GPa·nm along the x- and y-directions with a Poisson's ratio of 0.29, while the Curie temperature within the Ising model is considerably higher than room temperature. Furthermore, spin-orbit coupling can open a band gap at the Dirac point, leading to topologically nontrivial electronic states characterized by an integer value of the Chern number and the edge states of its nanoribbon. Our results offer versatile platforms for achieving plastic spin filtering or a quantum anomalous Hall effect with promising applications in spintronics devices.

Some counterexamples related to sectorial matrices and matrix phases

A sectorial matrix is an n×n matrix whose numerical range is contained in an open half-plane, and such matrices have many nice properties. In particular, the subset of strictly accretive matrices is a convex cone in the space of n×n matrices, and results related to positive definite matrices have recently been generalized to this cone. Moreover, sectorial matrices have recently been used to define phases of a matrix, and these phases can be used to angularly bound the eigenvalues by majorization-type inequalities similar to the ones for the singular values and the absolute value of the eigenvalues. Nevertheless, many traits that would be desirable are not true for sectorial matrices and matrix phases, and in this note we present a number of counterexamples for such traits. More precisely, the counterexamples are related to sectorial polar decompositions, majorization inequalities for phases of products, the spectral and numerical radius, and Schur complements.

Some Applications of the Open Mapping Theorem in Locally Convex Cones

Some Applications of the Open Mapping Theorem in Locally Convex Cones

Constrained mutual convex cone method for image set based recognition

In this paper, we propose convex cone-based frameworks for image-set classification. Image-set classification aims to classify a set of images, usually obtained from video frames or multi-view cameras, into a target object. To accurately and stably classify a set, it is essential to accurately represent structural information of the set. There are various image features, such as histogram-based features and convolutional neural network features. We should note that most of them have non-negativity and thus can be effectively represented by a convex cone. This leads us to introduce the convex cone representation to image-set classification. To establish a convex cone-based framework, we mathematically define multiple angles between two convex cones, and then use the angles to define the geometric similarity between them. Moreover, to enhance the framework, we introduce two discriminant spaces. We first propose a discriminant space that maximizes gaps between cones and minimizes the within-class variance. We then extend it to a weighted discriminant space by introducing weights on the gaps to deal with complicated data distribution. In addition, to reduce the computational cost of the proposed methods, we develop a novel strategy for fast implementation. The effectiveness of the proposed methods is demonstrated experimentally by using five databases.

Mouse Models of Achromatopsia in Addressing Temporal "Point of No Return" in Gene-Therapy.

Achromatopsia is characterized by amblyopia, photophobia, nystagmus, and color blindness. Previous animal models of achromatopsia have shown promising results using gene augmentation to restore cone function. However, the optimal therapeutic window to elicit recovery remains unknown. Here, we attempted two rounds of gene augmentation to generate recoverable mouse models of achromatopsia including a Cnga3 model with a knock-in stop cassette in intron 5 using Easi-CRISPR (Efficient additions with ssDNA inserts-CRISPR) and targeted embryonic stem (ES) cells. This model demonstrated that only 20% of CNGA3 levels in homozygotes derived from target ES cells remained, as compared to normal CNGA3 levels. Despite the low percentage of remaining protein, the knock-in mouse model continued to generate normal cone phototransduction. Our results showed that a small amount of normal CNGA3 protein is sufficient to form “functional” CNG channels and achieve physiological demand for proper cone phototransduction. Thus, it can be concluded that mutating the Cnga3 locus to disrupt the functional tetrameric CNG channels may ultimately require more potent STOP cassettes to generate a reversible achromatopsia mouse model. Our data also possess implications for future CNGA3-associated achromatopsia clinical trials, whereby restoration of only 20% functional CNGA3 protein may be sufficient to form functional CNG channels and thus rescue cone response.

Error bound analysis for vector equilibrium problems with partial order provided by a polyhedral cone

The aim of this paper is to establish new results on the error bounds for a class of vector equilibrium problems with partial order provided by a polyhedral cone generated by some matrix. We first propose some regularized gap functions of this problem using the concept of mathcal {G}_{A}-convexity of a vector-valued function. Then, we derive error bounds for vector equilibrium problems with partial order given by a polyhedral cone in terms of regularized gap functions under some suitable conditions. Finally, a real-world application to a vector network equilibrium problem is given to illustrate the derived theoretical results.

Correction to: Duality for quasiconvex minimization over closed convex cones

Correction to: Duality for quasiconvex minimization over closed convex cones

Convergence of two-stage iterative scheme for [formula omitted]-weak regular splittings of type II

Monotone matrices play a key role in the convergence theory of regular splittings and different types of weak regular splittings. If monotonicity fails, then it is difficult to guarantee the convergence of the above-mentioned classes of matrices. In such a case, K-monotonicity is sufficient for the convergence of K-regular and K-weak regular splittings, where K is a proper cone in Rn. However, the convergence theory of a two-stage iteration scheme in a general proper cone setting is a gap in the literature. Especially, the same study for weak regular splittings of type II (even if in standard proper cone setting, i.e., K=R+n), is open. To this end, we propose convergence theory of two-stage iterative scheme for K-weak regular splittings of both types in the proper cone setting. We provide some sufficient conditions which guarantee that the induced splitting from a two-stage iterative scheme is a K-regular splitting and then establish some comparison theorems. We also study K-monotone convergence theory of the stationary two-stage iterative method in case of a K-weak regular splitting of type II. The most interesting and important part of this work is on M-matrices appearing in the Covid-19 pandemic model. Finally, numerical computations are performed using the proposed technique to compute the next generation matrix involved in the pandemic model. The computation of large PageRank matrices shows that the two-stage Gauss-Seidel method performs better than the Gauss-Seidel methods.

On Iteration Complexity of a First-Order Primal-Dual Method for Nonlinear Convex Cone Programming

On Iteration Complexity of a First-Order Primal-Dual Method for Nonlinear Convex Cone Programming

No-arbitrage concepts in topological vector lattices

We provide a general framework for no-arbitrage concepts in topological vector lattices, which covers many of the well-known no-arbitrage concepts as particular cases. The main structural condition we impose is that the outcomes of trading strategies with initial wealth zero and those with positive initial wealth have the structure of a convex cone. As one consequence of our approach, the concepts NUPBR, NAA_1 and NA_1 may fail to be equivalent in our general setting. Furthermore, we derive abstract versions of the fundamental theorem of asset pricing (FTAP), including an abstract FTAP on Banach function spaces, and investigate when the FTAP is warranted in its classical form with a separating measure. We also consider a financial market with semimartingales which does not need to have a numéraire, and derive results which show the links between the no-arbitrage concepts by only using the theory of topological vector lattices and well-known results from stochastic analysis in a sequence of short proofs.

Cost efficiency analysis in data envelopment analysis framework: An application to sugar industries

This paper proposes an approach to maintaining unit efficiency (cost efficiency) under any fluctuation in input costs. It can accommodate situations where the level of influence over costs ranges from minimal to considerable. For this purpose, a two-step procedure is applied: First, finding the convex cone. It is formed by the intersection of convex cones themselves created from comparing the new input costs of the efficient unit and the extreme efficient units which are adjacent to it in the first quadrant. Second, finding the subset that maintains the unit efficient in the presence of annual inflation. The procedure is demonstrated by an algorithm and a working example. Finally, it is illustrated via application to a real case in sugar industries.

An abstract inf-sup problem inspired by limit analysis in perfect plasticity and related applications

This paper is concerned with an abstract inf-sup problem generated by a bilinear Lagrangian and convex constraints. We study the conditions that guarantee no gap between the inf-sup and related sup-inf problems. The key assumption introduced in the paper generalizes the well-known Babuška–Brezzi condition. It is based on an inf-sup condition defined for convex cones in function spaces. We also apply a regularization method convenient for solving the inf-sup problem and derive a computable majorant of the critical (inf-sup) value, which can be used in a posteriori error analysis of numerical results. Results obtained for the abstract problem are applied to continuum mechanics. In particular, examples of limit load problems and similar ones arising in classical plasticity, gradient plasticity and delamination are introduced.

Buckling behavior of AA6061 circular tube under axial compression by considering contact condition of tube end

When a circular tube is under axial compression, it can be developed into specific forming technologies (e.g., tube inversion). However buckling failure is prone to occurring and seriously restricts the successful realization of the forming process or service performance. In this paper the conical convex or concave dies with different semi-cone angles were adopted to change tube end contact conditions and the buckling process, critical buckling force and the buckling waves of AA6061 tube under axial compression were investigated by using finite element simulation combined with experiments. The results show that the buckling deformation, the position where initial buckling wave appears, and the distribution of axial stress in tube blank are quite different when axially compressed on a flat die or a conical convex die (semi-cone angle β = 60°). For the conical convex dies as semi-cone angle increases from 75° to 87.5°, or for concave dies as semi-cone angle decreases from 110° to 92.5°, the axial compressive stress, critical buckling force and critical reduction increase, while the peak of the buckled wave decreases, i.e., occurrence of buckling delays and the buckling becomes more difficult to occur. When β is 90° (flat die), the critical buckling force and reduction reach the peak values.