Convex Cone Research Articles

Conic relaxations for conic minimax convex polynomial programs with extensions and applications

Abstract In this paper, we analyze conic minimax convex polynomial optimization problems. Under a suitable regularity condition, an exact conic programming relaxation is established based on a positivity characterization of a max function over a conic convex system. Further, we consider a general conic minimax $$\rho $$ ρ -convex polynomial optimization problem, which is defined by appropriately extending the notion of conic convexity of a vector-valued mapping. For this problem, it is shown that a Karush-Kuhn-Tucker condition at a global minimizer is necessary and sufficient for ensuring an exact relaxation with attainment of the conic programming relaxation. The exact conic programming relaxations are applied to SOS-convex polynomial programs, where appropriate choices of the data allow the associated conic programming relaxation to be reformulated as a semidefinite programming problem. In this way, we can further elaborate the obtained results for other special settings including conic robust SOS-convex polynomial problems and difference of SOS-convex polynomial programs.

Study of the cone of sums of squares plus sums of nonnegative circuit forms

Abstract In this article, we combine sums of squares (SOS) and sums of nonnegative circuit (SONC) forms, two independent nonnegativity certificates for real homogeneous polynomials. We consider the convex cone SOS+SONC of forms that decompose into a sum of an SOS and a SONC form and study it from a geometric point of view. We show that the SOS+SONC cone is proper and neither closed under multiplication nor under linear transformation of variables. Moreover, we present an alternative proof of an analog of Hilbert’s 1888 Theorem for the SOS+SONC cone and prove that in the non-Hilbert cases it provides a proper superset of the union of the SOS and SONC cones. This follows by exploiting a new necessary condition for membership in the SONC cone.

Quasistatic Tensile and Fatigue Performances of Titanium Alloy Open Hole Structures Strengthened by Different Cold Expansion Mandrel Front Cone Angles

ABSTRACTIn this paper, the cold expansion test of titanium alloy open hole structures was conducted, and the fatigue performance was also assessed. A parametric study of the mandrel front cone angle revealed a correlation with the strengthening quality and fatigue performance evolution, providing valuable insight for optimization of open hole structure designs. The results showed that the strengthening resistance decreased and then increased with the increase of the front cone angle. The microhardness of the hole edge increased and exhibited obvious sensitivity to the mandrel front cone angle. As expected, the fatigue life of specimens increased after strengthening. The best fatigue life strengthening front cone angle was 15°. Although the front cone angle exhibited little influence on the location of the fatigue crack initiation zone and crack propagating direction, the crack extension rate and the stiffness degradation rate could be reduced by the proper mandrel front cone angle.

An example of fractional ODE loss of maximum principle and Hopf’s lemma

It has been known that the fractional ODEs involving Riemann–Liouville fractional derivatives maintain many “analogies” of the classic elliptic-type properties, including maximum principles, Hopf’s Lemma, the existence of real eigenvalues, etc. However, in this short work, we construct a counterexample demonstrating that Caputo-type fractional operators (and their compositions) fail to uphold the maximum principle and Hopf’s lemma. This finding is somewhat unexpected and serves as a benchmark problem or counterexample, reminding us to remain cautious when developing parallel theories for fractional differential equations and applying them in modeling. Since the maximum principle and Hopf’s lemma are often used to construct convex cones in Banach spaces and guarantee the existence of a principal eigenvalue in classical elliptic theory, this result leads us to conjecture that fractional Sturm–Liouville problems involving the Caputo fractional derivative may lack a conventional principal eigenvalue. This conjecture is partially proved in a following-up work.

Cooperative navigation using constrained convex generators

In this article, we suggest a technique of set-based cooperative navigation for a fleet of vehicles under communication constraints. In the suggested approach, only the distances and bearings to other vehicles or known locations can be measured. Each vehicle in the fleet must determine its location and the positions of the other vehicles based on the measurements as well as information shared among the vehicles. Since the measurement set may be nonconvex, it must be approximated by a convex set. To address this issue, we employ constrained convex generators, which is a generalization of the definition for constrained zonotopes that allows ℓ2 unit norm balls and convex cones. To keep the amount of exchanged information low the vehicles transmit an ellipse containing the state estimate to the other vehicles. The obtained estimates are worst-case bounds for the true position, which is important in certain applications such as collision avoidance. The computed sets are then applied to vehicle control algorithms, taking into account the positions of all the agents. Through numerical simulations, we illustrate the application of this method to the problem of cooperative navigation of autonomous underwater vehicles and the problem of fire detection with multiple UAVs.

A capillary problem for spacelike mean curvature flow in a cone of Minkowski space

Consider a convex cone in three-dimensional Minkowski space which either contains the light cone or is contained in it. This work considers mean curvature flow of a proper spacelike strictly mean convex disc in the cone which is graphical with respect to its rays. Its boundary is required to have constant intersection angle with the boundary of the cone. We prove that the corresponding parabolic boundary value problem for the graph admits a solution for all time which rescales to a self-similarly expanding solution.

Spatial-temporal imaging electronic dynamics in few-layer graphene

Abstract Revealing the ultrafast carrier dynamics with spatial, energy and time resolution is critical for achieving a comprehensive understanding of the ultrafast electronic dynamics of two-dimensional (2D) materials and heterostructures. Here, by time-resolved photoemission electron microscopy (Tr-PEEM) measurements, we reveal the layer-dependent ultrafast electronic dynamics of monolayer (1 ML), bilayer (2 ML), 4 ML graphene as well as bulk graphite. In contrast to 2 ML and thicker graphene, photo-excited electrons in 1 ML graphene relax much faster, which is attributed to the unique massless linear Dirac cone dispersion, allowing electrons to relax by continuously emitting phonons. Moreover, transient photoelectron redistribution shows an energy shift between 1 ML and 2 ML graphene, highlighting the critical role of the bandgap in 2 ML graphene on the carrier relaxation dynamics. Our work demonstrates the power of Tr-PEEM in revealing the ultrafast dynamics of 2D materials and heterostructures with spatial-temporal information, and provides information for the layer-dependent ultrafast relaxation dynamics in few-layer graphene.

ϕp amplitudes from the positive tropical Grassmannian: Triangulations of extended diagrams

The global Schwinger formula, introduced by Cachazo and Early as a single integral over the positive tropical Grassmannian, provides a way to uncover properties of scattering amplitudes which are hard to see in their standard Feynman diagram formulation. In a recent work, Cachazo and one of the authors extended the global Schwinger formula to general ϕp theories. When p=4, it was conjectured that the integral decomposes as a sum over cones which are in bijection with noncrossing chord diagrams, and further that these can be obtained by finding the zeroes of a piecewise linear function, H(x). In this note we give a proof of this conjecture. We also present a purely combinatorial way of computing ϕp amplitudes by triangulating a trivial extended version of noncrossing (p−2)-chord diagrams, called extended diagrams, and present a proof of the bijection between triangulated extended diagrams and Feynman diagrams when p=4. This is reminiscent of recent constructions using Stokes polytopes and accordiohedra. However, the ϕp amplitude is now partitioned by a new collection of objects, each of which characterizes a polyhedral cone in the positive tropical Grassmannian in the form of an associahedron or of an intersection of two associahedra. Moreover, we comment on the bijection between extended diagrams and double-ordered biadjoint scalar amplitudes. We also conjecture the form of the general piecewise linear function, Hϕp(x), whose zeroes generate the regions in which the ϕp global Schwinger formula decomposes into. Published by the American Physical Society 2024

Towards an Improved Approach of Clay Minerals Mapping in the Northwestern Region of Algeria

Abstract The availability of hyperspectral images has significantly facilitated clay minerals identification and mapping. Based on the Hyperion L1R hyperspectral dataset, this research aims to improve previous scientific work related to identifying and mapping clay minerals in the Djebel Meni region (northwestern of Algeria). Firstly, we enhance the dataset quality through pre-processing techniques like the removal of bad bands, radiometric calibration, and atmospheric correction. Secondly, the Spectral Information Divergence (SID) algorithm was employed by introducing endmembers derived with the Sequential Maximum Angle Convex Cone (SMACC) algorithm initially and then using Jet Propulsion Laboratory (JPL) spectral signatures of Illite, Kaolinite, and Montmorillonite. The classification results show a correspondence between areas occupied by endmembers and JPL spectral signatures, which helps in matching endmembers with specific minerals. Finally, we conducted a comparative analysis of the two classifications outcomes against a reference map. This last is generated using SID algorithm, which takes ground truth spectral signatures as input. Our proposed approach using the SID classifier has given an overall accuracy score of 97.13% and 84.17% using endmembers image and JPL library respectively. The Kappa coefficient is respectively, with endmembers image and JPL, 0.93 and 0.57. These results show that the SID classification with the endmembers image is better than the SID classification with the JPL library because extracting the endmembers from the image is more accurate than doing that from the pure minerals analyzed in the laboratory. These promising results suggest that our approach could be extended to the identification of clay minerals in diverse hyperspectral datasets.

An MPC-DCM Control Method for a Forward-Bending Biped Robot Based on Force and Moment Control

For a forward-bending biped robot with 10 degrees of freedom on its legs, a new control framework of MPC-DCM based on force and moment is proposed in this paper. Specifically, the Diverging Component of Motion (DCM) is a stability criterion for biped robots based on linear inverted pendulum, and Model Predictive Control (MPC) is an optimization solution strategy using rolling optimization. In this paper, DCM theory is applied to the state transition matrix of the system, combined with simplified rigid body dynamics, the mathematical description of the biped robot system is established, the classical MPC method is used to optimize the control input, and DCM constraints are added to the constraints of MPC, making the real-time DCM approximate to a straight line in the walking single gait. At the same time, the linear angle and friction cone constraints are considered to enhance the stability of the robot during walking. In this paper, MATLAB/Simulink is used to simulate the robot. Under the control of this algorithm, the robot can reach a walking speed of 0.75 m/s and has a certain anti-disturbance ability and ground adaptability. In this paper, the Model-H16 robot is used to deploy the physical algorithm, and the linear walking and obstacle walking of the physical robot are realized.

Exact moment representation in polynomial optimization

We investigate the problem of representing moment sequences by measures in the context of Polynomial Optimization Problems, that consist in finding the infimum of a real polynomial on a real semialgebraic set defined by polynomial inequalities. We analyze the exactness of Moment Matrix (MoM) hierarchies, dual to the Sum of Squares (SoS) hierarchies, which are sequences of convex cones introduced by Lasserre to approximate measures and positive polynomials. We investigate in particular flat truncation properties, which allow testing effectively when MoM exactness holds and recovering the minimizers.We show that the dual of the MoM hierarchy coincides with the SoS hierarchy extended with the real radical of the support of the defining quadratic module Q. We deduce that flat truncation happens if and only if the support of the quadratic module associated with the minimizers is of dimension zero. We also bound the order of the hierarchy at which flat truncation holds.As corollaries, we show that flat truncation and MoM exactness hold when regularity conditions, known as Boundary Hessian Conditions, hold (and thus that MoM exactness holds generically); and when the support of the quadratic module Q is zero-dimensional. Effective numerical computations illustrate these flat truncation properties.

The Orlicz–Pettis theorem for locally convex cones

The Orlicz–Pettis theorem for locally convex cones

Shape and stability of rotation sets for incompressible flows on the torus

AbstractIn this paper we introduce the concept of rotation cones, determined by rotation sets, and use these to describe the stability of rotation sets of incompressible and fixed-point free continuous flows on the torus $$\mathbb{T}^{d}$$ T d , d ⩾ 2. The previous concept is equivalent and extends the stability of rotation numbers (and rotation vectors) in the special case of fixed-point free continuous flows on $$\mathbb{T}^{2}$$ T 2 . We prove that incompressible Lipschitz vector fields with stable rotation sets have convex rotation cones with non-empty interior, and that all extremal edges are collinear with vectors in ℤd. If, in addition, the rotation cones are proper, then these are polyhedral with finitely many edges. Finally we prove that the set of vector fields with stable rotation sets is C0-open and dense among those having proper rotation cones.

On the Continuous-Time Complementarity Problem

This work deals with solving continuous-time nonlinear complementarity problems defined on two types of nonempty closed convex cones: a polyhedral cone (positive octant) and a second-order cone. Theoretical results that establish a relationship between such problems and the variational inequalities problem are presented. We show that global minimizers of an unconstrained continuous-time programming problem are solutions to the continuous-time nonlinear complementarity problem. Moreover, a relation is set upso that a stationary point of an unconstrained continuous-time programming problem, in which the objective function involves the Fischer-Burmeister function, is a solution for the continuous-time complementarity problem. To guarantee the validity of the K.K.T. conditions for some auxiliary continuous-time problems which appear during the theoretical development, we use the linear independence constraint qualification. These constraint qualification are posed in the continuous-time context and appeared in the literature recently. In order to exemplify the developed theory, some simple examples are presented throughout the text.

Characterization of continuous additive set-valued maps “modulo K” on finite dimensional linear spaces

Abstract Let Y be a real vector metric space and K ⊂ Y be a closed convex cone such that K ∩ (− K) = {0}. We prove that a convex compact-valued map F : ℝ → 2 Y ∖ {∅} is K-continuous and K-additive if and only if there are non-empty convex compact sets A, B ⊂ Y such that 0 ∈ A − B ⊂ K and F is equal “modulo K” to the continuous set-valued map G ( t ) = t A , t ≥ 0 , t B , t < 0. $$\begin{array}{} \displaystyle G(t)=\begin{cases} tA,&t\geq0,\\ tB,& t \lt 0. \end{cases} \end{array}$$ Next, we use this result to characterize convex compact-valued maps F : ℝ N → 2 Y ∖ {∅}.

Invariant cones for jump-diffusions in infinite dimensions

In this paper we provide sufficient conditions for stochastic invariance of closed convex cones for stochastic partial differential equations (SPDEs) of jump-diffusion type, and clarify when these conditions are necessary. We emphasize that the diffusion coefficient of the SPDE does not need to be smooth, an assumption which is frequently imposed when dealing with stochastic invariance problems. Our results apply to the positive cone of abstract L2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^2$$\\end{document}-spaces. Furthermore, we present a series of applications, where we investigate SPDEs arising in natural sciences and economics.

Vector optimization problems in near vector space

ABSTRACT The conventional vector optimization problems intend to optimize the objective functions, which are defined on a vector space into another vector space. In this paper, we are going to study the vector optimization problems by minimizing the objective functions, which are defined on a vector space into another so-called near vector space. The concept of near vector space is axiomatically defined, and it does not necessarily contain the additive inverse element. A so-called pointed-like convex cone is introduced to propose the concept of minimizer of this new type vector optimization problem. Its corresponding scalar optimization problem is also formulated. Finally, we show that the optimal solution of the corresponding scalar optimization problem is also a minimizer of the original vector optimization problem based on the near vector space.

Early detection and lesion visualization of pear leaf anthracnose based on multi-source feature fusion of hyperspectral imaging.

Pear anthracnose, caused by Colletotrichum bacteria, is a severe infectious disease that significantly impacts the growth, development, and fruit yield of pear trees. Early detection of pear anthracnose before symptoms manifest is of great importance in preventing its spread and minimizing economic losses. This study utilized hyperspectral imaging (HSI) technology to investigate early detection of pear anthracnose through spectral features, vegetation indices (VIs), and texture features (TFs). Healthy and diseased pear leaves aged 1 to 5 days were selected as subjects for capturing hyperspectral images at various stages of health and disease. Characteristic wavelengths (OWs1 and OWs2) were extracted using the Successive Projection Algorithm (SPA) and Competitive Adaptive Reweighted Sampling (CARS) algorithm. Significant VIs were identified using the Random Forest (RF) algorithm, while effective TFs were derived from the Gray Level Co-occurrence Matrix (GLCM). A classification model for pear leaf early anthracnose disease was constructed by integrating different features using three machine learning algorithms: Support Vector Machine (SVM), Extreme Learning Machine (ELM), and Back Propagation Neural Network (BPNN). The results showed that: the classification identification model constructed based on the feature fusion performed better than that of single feature, with the OWs2-VIs-TFs-BPNN model achieving a highest accuracy of 98.61% in detection and identification of pear leaf early anthracnose disease. Additionally, to intuitively and effectively monitor the progression and severity of anthracnose in pear leaves, the visualization of anthracnose lesions was achieved using Successive Maximum Angle Convex Cone (SMACC) and Spectral Information Divergence (SID) techniques. According to our research results, the fusion of multi-source features based on hyperspectral imaging can be a reliable method to detect early asymptomatic infection of pear leaf anthracnose, and provide scientific theoretical support for early warning and prevention of pear leaf diseases.

Bionic Design of High-Performance Joints: Differences in Failure Mechanisms Caused by the Different Structures of Beetle Femur-Tibial Joints.

Beetle femur-tibial joints can bear large loads, and the joint structure plays a crucial role. Differences in living habits will lead to differences in femur-tibial joint structure, resulting in different mechanical properties. Here, we determined the structural characteristics of the femur-tibial joints of three species of beetles with different living habits. The tibia of Scarabaeidae Protaetia brevitarsis and Cetoniidae Torynorrhina fulvopilosa slide through cashew-shaped bumps on both sides of the femur in a guide rail consisting of a ring and a cone bump. The femur-tibial joint of Buprestidae Chrysodema radians is composed of a conical convex tibia and a circular concave femur. A bionic structure design was developed out based on the characteristics of the structure of the femur-tibial joints. Differences in the failure of different joint models were obtained through experiments and finite element analysis. The experimental results show that although the spherical connection model can bear low loads, it can maintain partial integrity of the structure and avoid complete failure. The cuboid connection model shows a higher load-bearing capacity, but its failure mode is irreversible deformation. As key parts of rotatable mechanisms, the bionic models have the potential for wide application in the high-load engineering field.

Discrete dynamics and supergeometry

We formulate a geometric measurement theory of dynamical classical systems possessing both continuous and discrete degrees of freedom. The approach is covariant with respect to choices of clocks and naturally incorporates laboratories. The latter are embedded symplectic submanifolds of an odd-dimensional symplectic structure. When suitably defined, symplectic geometry in odd dimensions is exactly the structure needed for covariance. A fundamentally probabilistic viewpoint allows classical supergeometries to describe discrete dynamics. We solve the problem of how to construct probabilistic measures on supermanifolds given a (possibly odd dimensional) supersymplectic structure. This relies on a superanalog of the Hodge star for differential forms and a description of probabilities by convex cones. We also show how stochastic processes such as Markov chains can be described by supergeometry.