Semiconvex Function Research Articles

A novel semi-convex function for simultaneous identification of moving vehicle loads and bridge damage

For the simultaneous identification (SI) problem on the moving vehicle loads and bridge damage, the existing methods fail to appropriately consider the sparse characteristics of structural damage and ignore the prior information of vehicle loads, which make them suffering from challenges such as prolonged computation cost, poor noise robustness, and limited identification accuracy. To address this issue, a novel semi-convex function is proposed to construct a theoretical framework for the SI problem using the bridge dynamic responses subjected to moving vehicles in this study. Firstly, the bridge damage-induced virtual forces are conceptualized as a new form of moving residual forces characterized by block sparsity properties, and the damage location is then determined by the frequency characteristics with the largest fundamental frequency amplitude of moving residual forces. Secondly, a semi-convex function model encompassing moving forces and moving residual forces is defined. Combined with sparse regularization strategy, an improved alternating direction method of multipliers algorithm is proposed and applied to simultaneously solve for both moving vehicle loads and damage degrees. To assess the effectiveness and feasibility of the proposed method, extensive numerical validations are conducted in various bridge damage scenarios. Additionally, a series of truss bridge experiments are performed in laboratory under multiple damage and vehicle axle-weight conditions. The results show that compared with the existing outstanding methods, the proposed semi-convex function method can significantly reduce the identification cost, enhance robustness to noise, and demonstrate notable improvements in accuracy for the SI problem, which provides an innovative and more efficient solution to the SI problem in bridge engineering.

Maximum time step for high order BDF methods applied to gradient flows

For a backward differentiation formula (BDF) applied to the gradient flow of a semiconvex function, quadratic stability implies gradient stability. Namely, it is possible to build a Lyapunov functional for the discrete-in-time dynamical system, with a restriction on the time step. The maximum time step which can be derived from quadratic stability has previously been obtained for the BDF1, BDF2 and BDF3 schemes. Here, we compute it for the BDF4 and BDF5 methods. We also prove that the BDF6 scheme is not quadratically stable. Our results are based on the tools developed by Dahlquist and other authors to show the equivalence of A-stability and G-stability. We give several applications of gradient stability to the asymptotic behaviour of sequences generated by BDF schemes.

A semi-convex function for both constant and time-varying moving force identification

In recent years, many moving force identification (MFI) methods have been proposed to monitor moving forces indirectly from structural responses. However, the prior knowledge about moving forces has not been reasonably considered in the existing methods, so the identified moving forces are not accurate enough. To improve the accuracy of MFI results, a novel MFI method is proposed based on a semi-convex function in this study. Firstly, a model of a simply-supported beam subjected to moving forces is taken as an example to establish the relationship between moving forces and structural responses. Then, the prior knowledge that time-varying force component of a moving force usually fluctuates around its static weight in the time domain is adopted. Thus, the moving force is decomposed into constant and time-varying force components. The constant force component is unchanged over time and the time-varying force component is changed over time. Based on this prior knowledge, a semi-convex function is defined for MFI, which is inspired by an alternating iterative method (AIM) and regularization techniques. As a result, the semi-convex function is solved by the AIM, so the constant and time-varying force components can be respectively obtained through iterations, and the moving forces are identified from structural responses. Finally, to evaluate the effectiveness of the proposed method, both numerical simulations and experimental verifications are carried out. The identified results show that the proposed method can further improve the MFI results by comparing with the L2 regularization and the moving average Tikhonov regularization. The characteristics of the identified moving forces are consistent with that described in the given prior knowledge. Moreover, the proposed method can provide a good ability to estimate the total weight of the model car.

Compensated convex transforms and geometric singularity\\ extraction from semiconvex functions

The upper and lower compensated convex transforms are `tight' one-sided approximations for a given function. We apply these transforms to the extraction of fine geometric singularities from general semiconvex/semiconcave functions and DC-functions in $\mathbb{R}^n$ (difference of convex functions). Well-known geometric examples of (locally) semiconcave functions include the Euclidean distance function and the Euclidean squared-distance function. For a locally semiconvex function $f$ with general modulus, we show that `locally' a point is singular (a non-differentiable point) if and only if it is a scale $1$-valley point, hence by using our method we can extract all fine singular points from a given semiconvex function. More precisely, if $f$ is a semiconvex function with general modulus and $x$ is a singular point, then locally the limit of the scaled valley transform exists at every point $x$ and can be calculated as $ \lim_{\lambda\to+\infty}\lambda V_\lambda (f)(x)=r_x^2/4$, where $r_x$ is the radius of the minimal bounding sphere of the (Frechet) subdifferential $\partial_- f(x)$ of the locally semiconvex $f$ and $V_\lambda (f)(x)$ is the valley transform at $x$. Thus the limit function $ \mathcal{V}_\infty(f)(x):=\lim_{\lambda\to+\infty}\lambda V_\lambda (f)(x)=r_x^2/4$ provides a `scale $1$-valley landscape function' of the singular set for a locally semiconvex function $f$. At the same time, the limit also provides an asymptotic expansion of the upper transform $C^u_\lambda(f)(x)$ when $\lambda$ approaches $+\infty$. For a locally semiconvex function $f$ with linear modulus we show further that the limit of the gradient of the upper compensated convex transform $ \lim_{\lambda\to+\infty}\nabla C^u_\lambda(f)(x)$ exists and equals the centre of the minimal bounding sphere of $\partial_- f(x)$. We also show that for a DC-function $f=g-h$, the scale $1$-edge transform, when $\lambda\to+\infty$, satisfies $ \liminf_{\lambda\to+\infty}\lambda E_\lambda (f)(x)\geq (r_{g,x}-r_{h,x})^2/4$, where $r_{g,x}$ and $r_{h,x}$ are the radii of the minimal bounding spheres of the subdifferentials $\partial_- g$ and $\partial_- h$ of the two convex functions $g$ and $h$ at $x$, respectively.

A Liouville property for gradient graphs and a Bernstein problem for Hamiltonian stationary equations

Using a rotation of Yuan, we observe that the gradient graph of any semi-convex function is a Liouville manifold, that is, does not admit non-constant bounded harmonic functions. As a corollary, we find that any solution of the fourth order Hamiltonian stationary equation satisfying $$\theta \geq \left( n - 2\right) \frac{\pi}{2} + \delta$$ for some \({\delta > 0}\) must be a quadratic.

A non-convex setup for multivalued differential equations driven by oblique subgradients

The paper proves the existence and uniqueness of the solution for the following multivalued deterministic variational inequality with oblique subgradients, considered in a non-convex domain: {x′(t)+H(t,x(t))∂−φ(x(t))∋g(t,x(t)),t≥0,x(0)=x0, where ∂−φ stands for the Fréchet subdifferential of the semiconvex function φ and the matrix application x↦H(⋅,x) is a Lipschitz mapping. The presence of the oblique reflection brought by the term H∂−φ leads to the use of different techniques comparing to the cases of standard reflection in non-convex domains or oblique reflection in convex domains. The last section of the article is focused on the qualitative analysis of a non-convex Skorohod problem, with generalized reflection and, as applications, stochastic variational inequalities driven by oblique Fréchet subgradients are addressed.

On some inequalities for relative semi-convex functions

We consider and study a new class of convex functions that are called relative semi-convex functions. Some Hermite-Hadamard inequalities for the relative semi-convex function and its variant forms are derived. Several special cases are also discussed. Results proved in this paper may stimulate further research in this area. MSC:26D15, 26A51, 49J40.

Long-time behavior of solutions to Hamilton–Jacobi equations with quadratic gradient term

We study a rate of convergence appearing in the long-time behavior of viscosity solutions of the Cauchy problem for the Hamilton–Jacobi equation $$u_t(x,t)+\alpha x \cdot Du(x,t)+\beta|Du(x,t)|^2=f(x)\quad{\rm{in}}\,{{\mathbb R}^n}\times(0,\infty),$$where α, β > 0 are constants and f is a Lipschitz and semiconvex function on \({{{\mathbb R}^n}}\) . Our goal of this paper is to show that the semiconvexity property of f is an important factor which determines this rate of convergence. We also establish existence, uniqueness and Lipschitz continuity of viscosity solutions of the Cauchy problem and the corresponding ergodic problem for Hamilton–Jacobi equations in \({{\mathbb R}^n}\) .

An alternating iterative method and its application in statistical inference

This paper studies non-convex programming problems. It is known that, in statistical inference, many constrained estimation problems may be expressed as convex programming problems. However, in many practical problems, the objective functions are not convex. In this paper, we give a definition of a semi-convex objective function and discuss the corresponding non-convex programming problems. A two-step iterative algorithm called the alternating iterative method is proposed for finding solutions for such problems. The method is illustrated by three examples in constrained estimation problems given in Sasabuchi et al. (Biometrika, 72, 465–472 (1983)), Shi N. Z. (J. Multivariate Anal., 50, 282–293 (1994)) and El Barmi H. and Dykstra R. (Ann. Statist., 26, 1878–1893 (1998)).

Dirichlet boundary conditions for elliptic operators with unbounded drift

We study the realisation A of the operator A = Δ-(DΦ,D.) in L 2 (Ω,μ) with Dirichlet boundary condition, where Ω is a possibly unbounded open set in R N , Φ is a semi-convex function and the measure dμ(x) = exp(-Φ(x))dx lets A be formally self-adjoint. The main result is that A: D(A) = {u ∈ H 2 (Ω,μ): (DΦ,D u ) ∈ L 2 (Ω,μ), u = 0 at ∂Ω} is a dissipative self-adjoint operator in L 2 (Ω,μ).

Semismooth and Semiconvex Functions in Constrained Optimization

We introduce semismooth and semiconvex functions and discuss their properties with respect to nonsmooth nonconvex constrained optimization problems. These functions are locally Lipschitz, and hence have generalized gradients. The author has given an optimization algorithm that uses generalized gradients of the problem functions and converges to stationary points if the functions are semismooth. If the functions are semiconvex and a constraint qualification is satisfied, then we show that a stationary point is an optimal point. We show that the pointwise maximum or minimum over a compact family of continuously differentiable functions is a semismooth function and that the pointwise maximum over a compact family of semiconvex functions is a semiconvex function. Furthermore, we show that a semismooth composition of semismooth functions is semismooth and give a type of chain rule for generalized gradients.

Maximum Principle for Semiconvex Performance Functionals

An optimal control problem is considered where the performance functional is a semiconvex function of integral functionals, i.e., a function which can be approximated locally by a convex cone. A discussion of various types of conical approximations and a variant of the basic separation lemma are presented. A maximum principle is formulated and an extension of the basic proof of the maximum principle is given. An example shows the possible applications of the variant of the maximum principle.