拟线性半变分不等式解的存在性

考虑拟线性半变分不等式解的存在性.我们的方法是变分法及局部Lipschitz函数的非光滑临界点理论.     

鳞片形非光滑表面的仿生设计

对典型土壤动物鳞片形体表进行了研究,并建立了鳞片形非光滑表面的数学模型。根据此模型设计的仿生鳞片形非光滑推土板在进行土壤切削试验时不发生粘附,与光滑推土板相比,可平均降低推土阻力约１５％。     

A nonconvex nonsmooth regularization method for compressed sensing and low rank matrix completion

In this paper, a nonconvex and nonsmooth method for compressed sensing and low-rank matrix completion is studied. The proposed model is formulated as nonconvex regularized least square optimization problem. At first, an alternating minimization scheme is developed in which the problem can be decomposed into three subproblems, two of them are convex and the remaining one is smooth. Then, the convergence of the sequence which is generated by the alternating minimization algorithm is proved. In addition, some recovery guarantees are also analyzed. Finally, various numerical simulations are performed to test the efficiency of the method.     

一类非光滑非凸优化问题的神经网络方法

提出了解决一类带等式与不等式约束的非光滑非凸优化问题的神经网络模型。证明了当目标函数有下界时,神经网络的解轨迹在有限时间收敛到可行域。同时,神经网络的平衡点集与优化问题的关键点集一致,且神经网络最终收敛于优化问题的关键点集。与传统基于罚函数的神经网络模型不同,提出的模型无须计算罚因子。最后,通过仿真实验验证了所提出模型的有效性。     

Distributed continuous‐time algorithm for a general nonsmooth monotropic optimization problem

This paper investigates a general monotropic optimization problem for continuous‐time networks, where the global objective function is a sum of local objective functions that are only known to individual agent, and general constraints are taken into account, including local inequality constraints, global equality constraint, and local feasible constraints. In addition, all functions involved in the objective functions and inequality constraints are not necessarily differentiable. To solve the problem, a distributed continuous‐time algorithm is designed using subgradient projections, and it is shown that the proposed algorithm is well defined in the sense that the existence of its solutions can be guaranteed. Furthermore, it is proved that the algorithm converges to an optimal solution for the general monotropic optimization problem. Finally, a simulation example is provided for validating the theoretical result.     

The Exponential Stability for a Class of Hybrid Systems

The exponential stability with a nonsmooth Lyapunov function for a class of hybrid systems is studied in this paper. First, a sufficient condition is derived that has to be satisfied by the feedback control for the hybrid systems. Then, for the special case where the Lyapunov function involved is a kind of nonsmooth function, the maximum of finitely many smooth functions (for short, max‐type function), the stability of a hybrid system is considered, and a convenient criterion to determine the stability of the system is established. Finally, a numerical method of determining the control input value is developed.     

带干摩擦的非光滑动力系统Lyapunov指数的数值计算

对带干摩擦的非光滑动力系统,在不连续点利用闭上链(cocycle)得到一个传递矩阵,由此给出一种非光滑动力系统Lyapunov指数的数值计算方法.该方法是根据计算光滑动力系统Lyapunov指数的数值方法而提出的.通过对带干摩擦的动力系统进行Lyapunov指数的数值计算,结合时间历程图和Poincare映射对其动力学特性进行了分析.数值结果表明该方法的可行性和有效性,与一般的计算非光滑动力系统Lyapunov指数的数值算法相比,该方法更便于数值计算.     

Research on the drag reduction mechanism of antlion (Myrmeleon sagax) larvae nonsmooth structural surface

Antlion (Myrmeleon sagax) larvae live in sandy soil and possess the ability to enter soil quickly. In this article, the hierarchical structure of the nonsmooth surface of antlion larvae was obtained using a scanning electron microscope (SEM). Based on the results, a bionic nonsmooth structure model was established to investigate the friction and movement of soil particles above it. Then the relationship between drag reduction characteristics and the antlion larvae's nonsmooth structural surface was discussed, which would be helpful to design soil‐engaging components. When the height of each nonsmooth structure is proportional to the square of their interval distance, and is proportional to the velocity of movement, it is shown that a nonsmooth structural surface contributes to improving the bulk coefficient of granular materials, which leads to substantial drag reduction.     

Computation of maximal turning points to nonlinear equations by nonsmooth optimization

We develop a new approach to finding bifurcations of solutions of nonlinear equation systems, which is based on the detection of extreme values of a new type of variational functional associated with the considered problems. The variational functional is obtained constructively by the extended functional method which can be applied to a wide class of parametric problems including nonlinear partial differential equations. Sufficient and necessary conditions for the existence of a maximal turning point by the approach are proved. Based on these, an algorithm of the quasi-direction of steepest ascent is introduced. Simulation experiments are used to illustrate the behaviour of the method and to discuss its advantages and disadvantages in comparison with the alternatives.     

A Multifunction Approach to Extremum Problems Having Infinite-Dimensional Images. Necessary Conditions for Unilateral Constraints

In [1], extremum problems having an infinite-dimensional image were considered and some preliminary properties were established. In this article, an optimality condition of such problems is investigated for the case of unilateral constraints, which partially extends the results of [2, 3]. This is done by associating a special multifunction with the feasible set of solutions. It turns out that any classic Lagrangian multitplier functions can be factorized into a constant term and a variable one; the former term is the gradient of the separating hyperplane introduced in [2, 3], and the latter one plays the role of the selector of the above-mentioned multifunction. Finally, the need of enlarging the class of Lagrangian multiplier functions is discussed.