Convexity Conditions Research Articles

Generalized Nonlinear Variational Inequality Problems with Random Variation

This text deals with exploring random solutions for generalized nonlinear variational inequality problems. By leveraging the Fan-KKM theorem and Aumann’s measurable selection theorem, we can establish the existence and uniqueness of random solution sets, given the conditions of monotonicity and convexity. Additionally, we use Minty’s lemma to demonstrate the compactness and convexity of the random solution sets.

Kolmogorov’s theorem for degenerate Hamiltonian systems with Hölder continuous parameters

Abstract In this paper, we investigate Kolmogorov-type theorems for small perturbations of degenerate Hamiltonian systems. These systems are index by a parameter ξ as $ H(y,x,\xi) = \langle\omega(\xi),y\rangle {+ \bar h(y,\xi)}+\varepsilon P(y,x,\xi,\varepsilon) $ , where ɛ > 0. We assume that the frequency mapping $\omega(\cdot)$ , $\bar h(y,\cdot)=O(|y|^2)$ and the perturbation $\varepsilon P(y,x,\cdot, \varepsilon)$ maintain Hölder continuity about ξ. We prove that the persistent invariant tori retain the same frequency as those of the unperturbed tori, under a certain topological degree condition and a weak convexity condition for the frequency mapping. Notably, this paper presents, to our understanding, pioneering results on the KAM theorem under such conditions with only assumption of Hölder continuous dependence of frequency mapping ω on the parameter.

Weak set-equilibrium problems: existence and stability conditions

We formulate four types of set-equilibrium problems based on some well-known set-order relations. Then, we study the existence of solutions for these problems by imposing standard generalized convexity and Lipschitz conditions. A comparison with some widely studied vector equilibrium problems is performed. Finally, we are interested in an issue concerning the stability of solutions of some related approximate set-optimization problems.

Existence Results of Some Nonlinear Minimization Problems with Application to a System of PDE

A new class of non-self mappings, called proximal condensing operators, is introduced and used to investigate the existence of best proximity points for non-self mappings in reflexive Banach spaces by applying an appropriate measure of noncompactness. In this way, we present generalizations of well-known Schauder and Darbo fixed point problems. We then define the notion of best proximity points for multiplication of two non-self mappings in the setting of Banach algebras and present some sufficient conditions to ensure the existence of such points. Finally, we renorm a Banach space consists of all real valued and continuous functions with two variables to obtain the strict convexity condition and then we survey the existence of an optimal solution for a system of partial differential equations.

Polynomial convexity with degree bounds

We introduce different notions of polynomial convexity with bounds on degrees of polynomials in C n . We provide some examples in higher dimensions and show necessary and sufficient conditions for polynomial convexity with degree bounds for certain sets of points in C and for certain arcs in the unit circle.

Starlikeness, Convexity, Close-to-Convexity, and Quasi-Convexity for Functions with Fixed Initial Coefficients

In this paper, we employ the theory of differential subordination to establish a theorem that delineates certain sufficient conditions for starlikeness, convexity, close-to-convexity, and quasi-convexity in relation to functions with fixed initial coefficients. Furthermore, we introduce some results derived from these conditions. Building upon this framework, we derive an extension of Nunokawa’s lemma for analytic functions with fixed initial coefficients.

Analysis of regularized federated learning

Federated learning is an efficient machine learning tool for dealing with heterogeneous big data and privacy protection. Federated learning methods with regularization can control the level of communications between the central and local machines. Stochastic gradient descent is often used for implementing such methods on heterogeneous big data, to reduce the communication costs. In this paper, we consider such an algorithm called Loopless Local Gradient Descent which has advantages in reducing the expected communications by controlling a probability level. We improve the method by allowing flexible step sizes and carry out novel analysis for the convergence of the algorithm in a non-convex setting in addition to the standard strongly convex setting. In the non-convex setting, we derive rates of convergence when the smooth objective function satisfies a Polyak-Łojasiewicz condition. When the objective function is strongly convex, a sufficient and necessary condition for the convergence in expectation is presented.

Towards sharper excess risk bounds for differentially private pairwise learning

Pairwise learning is a vital part of machine learning. It depends on pairs of training instances, and is naturally fit for modeling relationships between samples. However, as a data driven paradigm, it faces huge privacy issues. Differential privacy (DP) is a useful tool to protect the privacy of machine learning, but corresponding excess population risk bounds are loose in existing DP pairwise learning analysis. In this paper, we propose a gradient perturbation algorithm for pairwise learning to get better risk bounds under Polyak–Łojasiewicz condition, including both convex and non-convex cases. Specifically, for the theoretical risk bound in expectation, previous best results are of rates O(pn2ϵ2+1n) and O(pnϵ+1n) under strongly convex condition and convex conditions, respectively. In this paper, we use the on-average stability and achieve an O(min{pn1.5ϵ+p1.5n2.5ϵ3,pn2ϵ2+1n}) bound, significantly improving previous bounds. For the high probability risk bound, previous best results are analyzed by the uniform stability, and O(βnU+pnϵ) excess population risk bounds are achieved under strongly convex or convex conditions, where βnU is the traditional pairwise uniform stability parameter, it is large since it considers the worst case of the loss sensitivity. In this paper, we propose the pairwise locally elastic stability and improve the high probability bound to O(βEn+pnϵ), in which the pairwise locally elastic stability parameter βE≪βnU because it considers the average sensitivity of the pairwise loss function.

Towards improved description of plastic anisotropy in sheet metals under biaxial loading: A novel generalization of Hill48 yield criterion

Lightweight design and development of automobiles are strongly dependent on lightweight high-strength sheet metals. An accurate prediction of anisotropy and yield surface evolution of these materials under different sampling directions of biaxial tension (BT) is a necessary condition for the numerical analyses of stamping forming. In this work, the Hill48 model considering the principal stress direction is generalized to describe the anisotropic evolution and the yield loci on the normal and diagonal planes. The material parameters of the generalized M-Hill48 model (GM-Hill48) have analytical form and can be calibrated by the yield stresses and r-values expressed by the continuous equivalent plastic strain (EPS). A quadratic function interpolation method based on the near-plane strain (NPS) stress is established to control the yield locus under different plane stress states. In addition, a method for predicting the yield loci in arbitrary sampling directions under BT is proposed by tensor coordinate transformation. Different from the traditional model which can only determine material parameters by using equi-biaxial tension (EBT) or NPS stress along the rolling direction (RD) and transverse direction (TD), the yield stresses of BT in different sampling directions can be incorporated into the generalized model. The predictive capability of the GM-Hill48 model is verified by capturing the continuous evolution of yield stresses and strain for DP590 and AA6016-T4 materials. The convexity condition of the established modified model is given and discussed. Finally, the advantages of new model are further expounded in terms of the prediction accuracy and identified parameters compared with the advanced yield models.

Convexity conditions for fiber-reinforced elastic shells

Convexity conditions of the Legendre–Hadamard and quasiconvexity type are derived in the context of a theory for fiber-reinforced shells based on Cosserat elasticity. These furnish two-dimensional shell-theoretic versions of their three-dimensional counterparts.

Linear convergence of event‐triggered distributed optimization with metric subregularity condition

AbstractThis paper designs a continuous‐time algorithm with event‐triggered communication (ETC) for solving a class of distributed convex optimization problems with a metric subregularity condition. First, we develop an event‐triggered continuous‐time optimization algorithm to overcome the bandwidth limitation of multi‐agent systems. Besides, with the aid of Lyapunov theory, we prove that the distributed event‐triggered algorithm converges to the optimum set with an exact linear convergence rate, without the strongly convex condition. Moreover, we provide the discrete version of the continuous‐time algorithm and show its exact linear convergence rate. Finally, we give a comparison example to validate the effectiveness of the designed algorithm in communication resource saving.

Generalized convergence of solutions for nonlinear Hamilton–Jacobi equations with state-constraint

For a continuous Hamiltonian H:(x,p,u)∈T⁎Rn×R→R, we consider the asymptotic behavior of associated Hamilton–Jacobi equations with state-constraint{H(x,Du,λu)≤Cλ,x∈Ωλ⊂Rn,H(x,Du,λu)≥Cλ,x∈Ω‾λ⊂Rn, as λ→0+. When H satisfies certain convex, coercive and monotone conditions, the domain Ωλ:=(1+r(λ))Ω keeps bounded, star-shaped for all λ>0 with limλ→0+⁡r(λ)=0, and limλ→0+⁡Cλ=c(H) equals the ergodic constant of H(⋅,⋅,0), we prove the convergence of solutions uλ to a specific solution of the critical equation{H(x,Du,0)≤c(H),x∈Ω⊂Rn,H(x,Du,0)≥c(H),x∈Ω‾⊂Rn. We also discuss the generalization of such a convergence for equations with more general Cλ and Ωλ.

Regularized mean curvature flow for invariant hypersurfaces in a Hilbert space and its application to gauge theory

In this paper, we investigate a regularized mean curvature flow starting from an invariant hypersurface in a Hilbert space equipped with an isometric and almost free action of a Hilbert Lie group whose orbits are minimal regularizable submanifolds. We prove that, if the initial invariant hypersurface satisfies a certain kind of horizontally convexity condition and some additional conditions, then it collapses to an orbit of the Hilbert Lie group action along the regularized mean curvature flow. In the final section, we state a vision for applying the study of the regularized mean curvature flow to the gauge theory.

An infinite dimensional KAM theorem with normal degeneracy

In this paper, we consider a classical Hamiltonian normal form with degeneracy in the normal direction. In previous results, one needs to assume that the perturbation satisfies certain non-degenerate conditions in order to remove the degeneracy in the normal form. Instead of that, we introduce a topological degree condition and a weak convexity condition, which are easy to verify, and we prove the persistence of lower dimensional tori without any restriction on perturbation but only smallness and analyticity.

Beam pointing optimization for spatial diversity MIMO free space optical communications

In this paper, we propose a beam pointing algorithm to find optimal beam coordinates maximizing link performance in multiple-input multiple-output (MIMO) free space optical communications (FSOC) with spatial diversity. We first derive the received signal-to-noise ratio (SNR) with spatial diversity and analyze the convexity condition of SNR function with respect to transmit beam coordinates. Then, we obtain the beam coordinates maximizing the received SNR through an alternating optimization (AO) algorithm and Newton’s method. However, the proposed algorithm requires instantaneous channel state information (CSI), which requires a high signaling overhead in MIMO FSOC. Thus, instead of exploiting instantaneous CSI, we utilize statistical CSI and propose a robust beam pointing algorithm with respect to atmospheric turbulence and pointing error. Numerical results show that our proposed algorithm outperforms conventional schemes in a dynamic FSOC network. Furthermore, we confirm that the optimal coordinates which are robust to the channel conditions approach to the centroid of receiver lenses as the strength of turbulence and pointing error increases.

Extending and improving conical bicombings

We study metric spaces that admit a conical bicombing and thus obey a weak form of non-positive curvature. Prime examples of such spaces are injective metric spaces. In this article, we give a complete characterization of complete metric spaces admitting a conical bicombing by showing that every such space is isometric to a \sigma -convex subset of some injective metric space. In addition, we show that every proper metric space that admits a conical bicombing also admits a consistent bicombing that satisfies certain convexity conditions. This can be seen as a strong indication that a question from Descombes and Lang about improving conical bicombings might have a positive answer. As an application, we prove that any group acting geometrically on a proper metric space with a conical bicombing admits a \mathcal{Z} -structure.

Extrapolation of Compactness on Banach Function Spaces

We prove an extrapolation of compactness theorem for operators on Banach function spaces satisfying certain convexity and concavity conditions. In particular, we show that the boundedness of an operator T in the weighted Lebesgue scale and the compactness of T in the unweighted Lebesgue scale yields compactness of T on a very general class of Banach function spaces. As our main new tool, we prove various characterizations of the boundedness of the Hardy-Littlewood maximal operator on such spaces and their associate spaces, using a novel sparse self-improvement technique. We apply our main results to prove compactness of the commutators of singular integral operators and pointwise multiplication by functions of vanishing mean oscillation on, for example, weighted variable Lebesgue spaces.

Distributed optimization via dynamic event-triggered scheme with metric subregularity condition

In this paper, we present a continuous-time algorithm with a dynamic event-triggered communication (DETC) mechanism for solving a class of distributed convex optimization problems that satisfy a metric subregularity condition. The proposed algorithm addresses the challenge of limited bandwidth in multi-agent systems by utilizing a continuous-time optimization approach with DETC. Furthermore, we prove that the distributed event-triggered algorithm converges exponentially to the optimal set, even without strong convexity conditions. Finally, we provide a comparison example to demonstrate the efficiency of our algorithm in communication resource-saving.

Nonlinear response modelling of material systems using constrained Gaussian processes

AbstractThis article investigates the suitability of constrained Gaussian process regression in predicting nonlinear mechanical responses of material systems with notably reduced uncertainties. This study reinforces the conventional Gaussian processes with mechanics‐informed prior knowledge observed in various kinematic responses. Stiffening and softening responses of material systems mostly demonstrate at least one of the boundedness, monotonicity and convexity conditions with respect to some kinematic variables. These relationships or impositions in turn are encoded into a constrained Gaussian process for prediction, uncertainty quantification and extrapolation. Using numerous examples and comparative studies, this article evidently proves that the use of constrained Gaussian processes is data‐efficient, highly accurate, yields low uncertainties, recovers model overfitting and extrapolates very well compared to unconstrained or conventional Gaussian processes. Moreover, the usability of the proposed numerical method across various engineering modelling domains such as multiscale homogenisation, experimentation, structural optimisation, material constitutive modelling and structural idealisation is demonstrated.

Multi-objective Variational Curves

AbstractRiemannian cubics in tension are critical points of the linear combination of two objective functionals, namely the squared $$L^2$$ L 2 norms of the velocity and acceleration of a curve on a Riemannian manifold. We view this variational problem of finding a curve as a multi-objective optimization problem and construct the Pareto fronts for some given instances where the manifold is a sphere and where the manifold is a torus. The Pareto front for the curves on the torus turns out to be particularly interesting: the front is disconnected and it reveals two distinct Riemannian cubics with the same boundary data, which is the first known nontrivial instance of this kind. We also discuss some convexity conditions involving the Pareto fronts for curves on general Riemannian manifolds.