Second-Order Smoothing Objective Penalty Function for Constrained Optimization Problems

In this article, a novel objective penalty function as well as its second-order smoothing is introduced for constrained optimization problems (COP). It is shown that an optimal solution to the second-order smoothing objective penalty optimization problem is an optimal solution to the original optimization problem under some mild conditions. Based on the second-order smoothing objective penalty function, an algorithm that has better convergence is introduced. Numerical examples illustrate that this algorithm is efficient in solving COP.     

一类椭圆问题正解的存在性和唯一性

该文讨论了二阶拟线性椭圆型问题u|\-\{Ω=0: -div［(d+|u|\+2)\+\{〖SX(〗p〖〗2〖SX)〗-1u］ =λ\-1u\+\{p-1+g(x,u),〓 x∈Ω正解的存在性和唯一性，其中 Ω是 R\+N 中的有界区域, λ\-1 是-△\-p 在 Ω上对应于零Dirichlet边界条件的第一特征根, g(x, t) 满足增长条件lim[DD(X]t→+∞[DD)]〖SX(〗g(x,t)〖〗t\+\{p-1〖SX)〗=0, p>1, 0≤d<+∞〖HT5”H〗关键词:〖HT5”SS〗拟线性椭圆问题； 鞍点； 正解.     

Fast and Stable Multiple Smoothing Parameter Selection in Smoothing Spline Analysis of Variance Models With Large Samples

The current parameterization and algorithm used to fit a smoothing spline analysis of variance (SSANOVA) model are computationally expensive, making a generalized additive model (GAM) the preferred method for multivariate smoothing. In this article, we propose an efficient reparameterization of the smoothing parameters in SSANOVA models, and a scalable algorithm for estimating multiple smoothing parameters in SSANOVAs. To validate our approach, we present two simulation studies comparing our reparameterization and algorithm to implementations of SSANOVAs and GAMs that are currently available in R. Our simulation results demonstrate that (a) our scalable SSANOVA algorithm outperforms the currently used SSANOVA algorithm, and (b) SSANOVAs can be a fast and reliable alternative to GAMs. We also provide an example with oceanographic data that demonstrates the practical advantage of our SSANOVA framework. Supplementary materials that are available online can be used to replicate the analyses in this article.     

A method of adding–removing knots for solving smoothing problems with obstacles

We study a method of adding–removing knots that has been proposed in the literature for solving the smoothing problem with obstacles. The method uses the coefficients of natural splines in the expansion by radial basis functions. We present examples of cycling and counterexamples to possible use of some ideas. We also give some sufficient conditions for finiteness of the method.     

Numerical Algorithm Based on Quintic Nonpolynomial Spline for Solving Third-Order Boundary Value Problems Associated with Draining and Coating Flows

A numerical algorithm is developed for the approximation of the solution to certain boundary value problems involving the third-order ordinary differential equation associated with draining and coating flows. The authors show that the approximate solutions obtained by the numerical algorithm developed by using nonpolynomial quintic spline functions are better than those produced by other spline and domain decomposition methods. The algorithm is tested on two problems associated with draining and coating flows to demonstrate the practical usefulness of the approach.     

Generalized Nonparametric Mixed-Effect Models: Computation and Smoothing Parameter Selection

Generalized linear mixed-effect models are widely used for the analysis of correlated non-Gaussian data such as those found in longitudinal studies. In this article, we consider extensions with nonparametric fixed effects and parametric random effects. The estimation is through the penalized likelihood method, and our focus is on the efficient computation and the effective smoothing parameter selection. To assist efficient computation, the joint likelihood of the observations and the latent variables of the random effects is used instead of the marginal likelihood of the observations. For the selection of smoothing parameters and correlation parameters, direct cross-validation techniques are employed; the effectiveness of cross-validation with respect to a few loss functions are evaluated through simulation studies. Real data examples are presented to illustrate potential applications of the methodology. Open-source R code is demonstrated in the Appendix.     

Robust Control Problems Associated with the Multilayer Quasi-Geostrophic Equations of the Ocean

In this article we study some robust control problems associated with the multilayer quasi-geostrophic equations of the ocean and related to data assimilation in oceanography. We prove the existence and uniqueness of solutions using a general framework given in [6].     

On Finite and Strong Convergence of a Proximal Method for Equilibrium Problems

We introduce the notions of conditioning and well-posedness for equilibrium problems. Using these concepts, we obtain finite and strong convergence results for the proximal method that improve, develop, and unify several theorems in optimization and nonlinear analysis.     

Distributionally robust optimization with correlated data from vector autoregressive processes

We present a distributionally robust formulation of a stochastic optimization problem for non-i.i.d vector autoregressive data. We use the Wasserstein distance to define robustness in the space of distributions and we show, using duality theory, that the problem is equivalent to a finite convex–concave saddle point problem. The performance of the method is demonstrated on both synthetic and real data.     

偏序线性空间中向量集值映射最优化问题解的鞍点条件和Lagrange对偶

本文在没有任何拓扑结构的条件下，给出了向量集值映射最优化问题解的鞍点充分和必要条件以及Ｌａｇｒａｎｇｅ对偶，从而将文献（１）中的有关结果推广到更一般的偏序线性空间，并进一步给出了逆对偶定理。     

Polynomial-Time Algorithms for Multivariate Linear Problems with Finite-Order Weights: Worst Case Setting

We consider approximation of linear multivariate problems defined over weighted tensor product Hilbert spaces with finite-order weights. This means we consider functions of d variables that can be represented as sums of functions of at most q^* variables. Here, q^* is fixed (and presumably small) and d may be arbitrarily large. For the univariate problem, d = 1, we assume we know algorithms A1,ε that use O(ε−p) function or linear functional evaluations to achieve an error ε in the worst case setting. Based on these algorithms A_1,ε, we provide a construction of polynomial-time algorithms A_d,ε for the general d-variate problem with the number of evaluations bounded roughly by ε^−pd^q* to achieve an error ε in the worst case setting.     

Degenerate Elliptic Eigenvalue Problems with Indefinite Weights

The purpose of this paper is to provide a careful and accessible exposition of the Kreĭn and Rutman Theory of degenerate elliptic eigenvalue problems with indefinite weights that model population dynamics in environments with spatial heterogeneity. We prove that the first eigenvalue of our problem is algebraically simple and its corresponding eigenfunction may be chosen to be positive everywhere. Here the approach is distinguished by the extensive use of the ideas and techniques characteristic of the recent developments in the theory of partial differential equations. The results extend an earlier theorem due to Manes and Micheletti to the degenerate case. Dedicated to the memory of Professor Sigeru Mizohata (1924–2002)     

Smoothing Newton and Quasi-Newton Methods for Mixed Complementarity Problems

The mixed complementarity problem can be reformulated as a nonsmooth equation by using the median operator. In this paper, we first study some useful properties of this reformulation and then derive the Chen-Harker-Kanzow-Smale smoothing function for the mixed complementarity problem. On the basis of this smoothing function, we present a smoothing Newton method for solving the mixed complementarity problem. Under suitable conditions, the method exhibits global and quadratic convergence properties. We also present a smoothing Broyden-like method based on the same smoothing function. Under appropriate conditions, the method converges globally and superlinearly.     

Smoothing Trust Region Methods for Nonlinear Complementarity Problems with P 0-Functions

By using the Fischer–Burmeister function to reformulate the nonlinear complementarity problem (NCP) as a system of semismooth equations and using Kanzow’s smooth approximation function to construct the smooth operator, we propose a smoothing trust region algorithm for solving the NCP with P ₀ functions. We prove that every accumulation point of the sequence generated by the algorithm is a solution of the NCP. Under a nonsingularity condition, local Q-superlinear/Q-quadratic convergence of the algorithm is established without the strict complementarity condition. This work was partially supported by the Research Grant Council of Hong Kong and the National Natural Science Foundation of China (Grant 10171030).     

Properties of a Class of Nonlinear Transformations Over Euclidean Jordan Algebras with Applications to Complementarity Problems

For any function φ from ?^r to ?^r, Tao and Gowda [Math. Oper. Res., 30 (2005), pp. 985–1004] introduced a corresponding nonlinear transformation R_φ over a Euclidean Jordan algebra (which is called a relaxation transformation) and established some useful relations between φ and R_φ. In this paper, we further investigate some interconnections between properties of φ and properties of R_φ, including the properties of continuity, (local) Lipschitz continuity, directional differentiability, (continuous) differentiability, semismoothness, monotonicity, the P₀-property, and the uniform P-property. As an application, we investigate the symmetric cone complementarity problem with a relaxation transformation. A property of the solution set of this class of problems is given. We also investigate a smoothing algorithm for solving this class of problems and show that the algorithm is globally convergent under an assumption that the solution set of the problem concerned is nonempty.     

An extension of gap functions for a system of vector equilibrium problems with applications to optimization problems

In this paper, the notion of gap functions is extended from scalar case to vector one. Then, gap functions and generalized functions for several kinds of vector equilibrium problems are shown. As an application, the dual problem of a class of optimization problems with a system of vector equilibrium constraints (in short, OP) is established, the concavity of the dual function, the weak duality of (OP) and the saddle point sufficient condition are derived by using generalized gap functions. This work was supported by the National Natural Science Foundation of China (10671135) and the Applied Research Project of Sichuan Province (05JY029-009-1).     

带扰动映像的广义平衡问题和不动点问题的混合粘性逼近问题

引入一个用于寻求带扰动映像的广义平衡问题解集以及可数无穷多非扩张映像之族公共不动点集的公共解的新的迭代算法. 证明了由此算法生成的序列的强收敛性. 所得的结果推广改进了先前许多作者的结果.     

Extragradient and extrapolation methods with generalized Bregman distances for saddle point problems

In this paper, we introduce two Bregman-type algorithmic frameworks to generalize the extragradient and extrapolation methods. With the help of relative Lipschitzness and the Bregman distance tool, the iteration properties of the proposed frameworks are analyzed. As applied to smooth convex-concave saddle point problems, our theory rediscovers the main results in Mokhtari et al. (2020) [14] for wider frameworks under weaker assumptions via a conceptually different approach.     

强向量均衡问题与不动点问题的粘性逼近算法

讨论了强向量均衡问题与非扩张映射不动点问题的公共解.首先,给出了强向量均衡问题的辅助问题,并在适当的条件下,证明了其解的存在性和唯一性结果.然后,利用这些结果,提出了强向量均衡问题与非扩张映射不动点问题公共解的粘性逼近算法,并进一步证明了,在适当的条件下,由该算法产生的迭代序列强收敛于强向量均衡问题和非扩张映射不动点问题的公共解.     

A dimensional split preconditioner for Stokes and linearized Navier-Stokes equations

In this paper we introduce a new preconditioner for linear systems of saddle point type arising from the numerical solution of the Navier-Stokes equations. Our approach is based on a dimensional splitting of the problem along the components of the velocity field, resulting in a convergent fixed-point iteration. The basic iteration is accelerated by a Krylov subspace method like restarted GMRES. The corresponding preconditioner requires at each iteration the solution of a set of discrete scalar elliptic equations, one for each component of the velocity field. Numerical experiments illustrating the convergence behavior for different finite element discretizations of Stokes and Oseen problems are included.