Stochastic Variational Inequality Approaches to the Stochastic Generalized Nash Equilibrium with Shared Constraints

In this paper, we consider the generalized Nash equilibrium with shared constraints in the stochastic environment, and we call it the stochastic generalized Nash equilibrium. The stochastic variational inequalities are employed to solve this kind of problems, and the expected residual minimization model and the conditional value-at-risk formulations defined by the residual function for the stochastic variational inequalities are discussed. We show the risk for different kinds of solutions for the stochastic generalized Nash equilibrium by the conditional value-at-risk formulations. The properties of the stochastic quadratic generalized Nash equilibrium are shown. The smoothing approximations for the expected residual minimization formulation and the conditional value-at-risk formulation are employed. Moreover, we establish the gradient consistency for the measurable smoothing functions and the integrable functions under some suitable conditions, and we also analyze the properties of the formulations. Numerical results for the applications arising from the electricity market model illustrate that the solutions for the stochastic generalized Nash equilibrium given by the ERM model have good properties, such as robustness, low risk and so on.     

Combination of steepest descent and BFGS methods for nonconvex nonsmooth optimization

In this paper, a method is developed for solving nonsmooth nonconvex minimization problems. This method extends the classical BFGS framework. First, we generalize the Wolfe conditions for locally Lipschitz functions and prove that this generalization is well defined. Then, a line search algorithm is presented to find a step length satisfying the generalized Wolfe conditions. Next, the Goldstein e-subgradient is approximated by an iterative method and a descent direction is computed using a positive definite matrix. This matrix is updated using the BFGS method. Finally, a minimization algorithm based on the BFGS method is described. The algorithm is implemented in MATLAB and numerical results using it are reported.     

Random variables,monotone relations,and convex analysis

Random variables can be described by their cumulative distribution functions, a class of nondecreasing functions on the real line. Those functions can in turn be identified, after the possible vertical gaps in their graphs are filled in, with maximal monotone relations. Such relations are known to be the subdifferentials of convex functions. Analysis of these connections yields new insights. The generalized inversion operation between distribution functions and quantile functions corresponds to graphical inversion of monotone relations. In subdifferential terms, it corresponds to passing to conjugate convex functions under the Legendre–Fenchel transform. Among other things, this shows that convergence in distribution for sequences of random variables is equivalent to graphical convergence of the monotone relations and epigraphical convergence of the associated convex functions. Measures of risk that employ quantiles (VaR) and superquantiles (CVaR), either individually or in mixtures, are illuminated in this way. Formulas for their calculation are seen from a perspective that reveals how they were discovered. The approach leads further to developments in which the superquantiles for a given distribution are interpreted as the quantiles for an overlying “superdistribution.” In this way a generalization of Koenker–Basset error is derived which lays a foundation for superquantile regression as a higher-order extension of quantile regression.     

A parametrization method for the numerical solution of singular differential equations

The paper explains the numerical parametrization method (PM), originally created for optimal control problems, for classical calculus of variation problems that arise in connection with singular implicit (IDEs) and differential-algebraic equations (DAEs) in frame of their regularization. The PM for IDEs is based on representation of the required solution as a spline with moving knots and on minimization of the discrepancy functional with respect to the spline parameters. Such splines are named variational splines. For DAEs only finite entering functions can be represented by splines, and the functional under minimization is the discrepancy of the algebraic subsystem. The first and the second derivatives of the functionals are calculated in two ways – for DAEs with the help of adjoint variables, and for IDE directly. The PM does not use the notion of differentiation index, and it is applicable to any singular equation having a solution.     

? 1-regularized linear regression: persistence and oracle inequalities

We study the predictive performance of ? ₁-regularized linear regression in a model-free setting, including the case where the number of covariates is substantially larger than the sample size. We introduce a new analysis method that avoids the boundedness problems that typically arise in model-free empirical minimization. Our technique provides an answer to a conjecture of Greenshtein and Ritov (Bernoulli 10(6):971–988, 2004) regarding the “persistence” rate for linear regression and allows us to prove an oracle inequality for the error of the regularized minimizer. It also demonstrates that empirical risk minimization gives optimal rates (up to log factors) of convex aggregation of a set of estimators of a regression function.     

Generalized equations and the generalized Newton method

We give some convergence results on the generalized Newton method (referred to by some authors as Newton's method) and the chord method when applied to generalized equations. The main results of the paper extend the classical Kantorovich results on Newton's method to (nonsmooth) generalized equations. Our results also extend earlier results on nonsmooth equations due to Eaves, Robinson, Josephy, Pang and Chan. We also propose inner-iterative schemes for the computation of the generalized Newton iterates. These schemes generalize popular iterative methods (Richardson's method, Jacobi's method and the Gauss-Seidel method) for the solution of linear equations and linear complementarity problems and are shown to be convergent under natural generalizations of classical convergence criteria. Our results are applicable to equations involving single-valued functions and also to a class of generalized equations which includes variational inequalities, nonlinear complementarity problems and some nonsmooth convex minimization problems.     

Ekeland's variational principle, minimax theorems and existence of nonconvex equilibria in complete metric spaces

In this paper, we introduce the concept of τ-function which generalizes the concept of w-distance studied in the literature. We establish a generalized Ekeland's variational principle in the setting of lower semicontinuous from above and τ-functions. As applications of our Ekeland's variational principle, we derive generalized Caristi's (common) fixed point theorems, a generalized Takahashi's nonconvex minimization theorem, a nonconvex minimax theorem, a nonconvex equilibrium theorem and a generalized flower petal theorem for lower semicontinuous from above functions or lower semicontinuous functions in the complete metric spaces. We also prove that these theorems also imply our Ekeland's variational principle.     

Uniform Distribution,Discrepancy, and Reproducing Kernel Hilbert Spaces

In this paper we define a notion of uniform distribution and discrepancy of sequences in an abstract set E through reproducing kernel Hilbert spaces of functions on E. In the case of the finite-dimensional unit cube these discrepancies are very closely related to the worst case error obtained for numerical integration of functions in a reproducing kernel Hilbert space. In the compact case we show that the discrepancy tends to zero if and only if the sequence is uniformly distributed in our sense. Next we prove an existence theorem for such uniformly distributed sequences and investigate the relation to the classical notion of uniform distribution. Some examples conclude this paper.     

New variants of the generalized level method for minimization of convex nondifferentiable functions taking infinite values

New variants of the generalized level method for minimization of convex Lipschitz functions on a compact set with a nonempty interior are proposed. These variants include the well-known generalized and classical level methods. For the new variants, an estimate of the convergence rate is found, including the variants in which the auxiliary problems are solved approximately.     

The role of data transfer on the selection of a single vs. multiple mesh architecture for tightly coupled multiphysics applications

Data transfer from one mesh to another may be necessary in a number of situations including spatial adaptation, remeshing, arbitrary Lagrangian-Eulerian (ALE), and multiphysics simulation. Data transfer has the potential to introduce error into a simulation; the magnitude and impact of which depends on the application, transfer scenario, and the algorithm used to perform the data transfer. During a transient simulation, data transfer may occur many times, with the potential of error accumulation at each transfer. This paper examines data transfer error and its impact on a set of simple multiphysics problems. Data transfer error is examined using analytical functions to compare schemes based on interpolation, area-weighted averaging, and L² minimization. An example error analysis is performed to illustrate data transfer error and behavior for a simple problem. Data transfer error is also investigated for a one-dimensional time-dependent system of partial differential equations. This study concludes that data transfer error can be significant in coupled multiphysics systems. These numerical experiments suggest that error is a function of data transfer scheme, and characteristics of the field data and mesh. If there are significant differences in the meshes in a multiple mesh simulation, this study suggests that data transfer may lead to error and instability if care is not taken. Further, this work motivates that data transfer error should be included in the estimation of numerical error when data transfer is employed in a simulation.     

A generalized conditional gradient method and its connection to an iterative shrinkage method

This article combines techniques from two fields of applied mathematics: optimization theory and inverse problems. We investigate a generalized conditional gradient method and its connection to an iterative shrinkage method, which has been recently proposed for solving inverse problems. The iterative shrinkage method aims at the solution of non-quadratic minimization problems where the solution is expected to have a sparse representation in a known basis. We show that it can be interpreted as a generalized conditional gradient method. We prove the convergence of this generalized method for general class of functionals, which includes non-convex functionals. This also gives a deeper understanding of the iterative shrinkage method.     

Numerical methods for A-optimal designs with a sparsity constraint for ill-posed inverse problems

We consider the problem of experimental design for linear ill-posed inverse problems. The minimization of the objective function in the classic A-optimal design is generalized to a Bayes risk minimization with a sparsity constraint. We present efficient algorithms for applications of such designs to large-scale problems. This is done by employing Krylov subspace methods for the solution of a subproblem required to obtain the experiment weights. The performance of the designs and algorithms is illustrated with a one-dimensional magnetotelluric example and an application to two-dimensional super-resolution reconstruction with MRI data.     

Nonasymptotic analysis of robust regression with modified Huber's loss

To achieve robustness against the outliers or heavy-tailed sampling distribution, we consider an Ivanov regularized empirical risk minimization scheme associated with a modified Huber's loss for nonparametric regression in reproducing kernel Hilbert space. By tuning the scaling and regularization parameters in accordance with the sample size, we develop nonasymptotic concentration results for such an adaptive estimator. Specifically, we establish the best convergence rates for prediction error when the conditional distribution satisfies a weak moment condition.     

一个四边形非协调新模式及其收敛性研究

本文给出了一个新的四边形非协调元,利用广义分片检查,对其收敛性进行了研究并给出了应力和位移的误差估计.最后,对弹性力学平面问题做了数值计算.     

On vector and matrix median computation

The aim of this paper is to gain more insight into vector and matrix medians and to investigate algorithms to compute them. We prove relations between vector and matrix means and medians, particularly regarding the classical structure tensor. Moreover, we examine matrix medians corresponding to different unitarily invariant matrix norms for the case of symmetric 2×2 matrices, which frequently arise in image processing. Our findings are explained and illustrated by numerical examples. To solve the corresponding minimization problems, we propose several algorithms. Existing approaches include Weiszfeld’s algorithm for the computation of ?₂ vector medians and semi-definite programming, in particular, second order cone programming, which has been used for matrix median computation. In this paper, we adapt Weiszfeld’s algorithm for our setting and show that also two splitting methods, namely the alternating direction method of multipliers and the parallel proximal algorithm, can be applied for generalized vector and matrix median computations. Besides, we compare the performance of these algorithms numerically and apply them within local median filters.     

Global minimization of a generalized convex multiplicative function

This paper discusses an algorithm for generalized convex multiplicative programming problems, a special class of nonconvex minimization problems in which the objective function is expressed as a sum ofp products of two convex functions. It is shown that this problem can be reduced to a concave minimization problem with only 2p variables. An outer approximation algorithm is proposed for solving the resulting problem.     

Non‐Monotone Space Decomposition Methods for Minimization Problems

Parallel space decomposition methods for the numerical treatment of unconstrained minimization problems are presented. For a special case of these methods described in [1] we extend classical line search methods for subspace optimization by non‐monotone strategies of [2]. For the convergence theory the concept of a generalized minimizing sequence is introduced in extension of a concept in [3].     

Decomposition Techniques for Bilinear Saddle Point Problems and Variational Inequalities with Affine Monotone Operators

The majority of first-order methods for large-scale convex–concave saddle point problems and variational inequalities with monotone operators are proximal algorithms. To make such an algorithm practical, the problem’s domain should be proximal-friendly—admit a strongly convex function with easy to minimize linear perturbations. As a by-product, this domain admits a computationally cheap linear minimization oracle (LMO) capable to minimize linear forms. There are, however, important situations where a cheap LMO indeed is available, but the problem domain is not proximal-friendly, which motivates search for algorithms based solely on LMO. For smooth convex minimization, there exists a classical algorithm using LMO—conditional gradient. In contrast, known to us similar techniques for other problems with convex structure (nonsmooth convex minimization, convex–concave saddle point problems, even as simple as bilinear ones, and variational inequalities with monotone operators, even as simple as affine) are quite recent and utilize common approach based on Fenchel-type representations of the associated objectives/vector fields. The goal of this paper was to develop alternative (and seemingly much simpler) decomposition techniques based on LMO for bilinear saddle point problems and for variational inequalities with affine monotone operators.     

Nonconservative Estimating Functions and Approximate Quasi-Likelihoods

The estimating function approach unifies two dominant methodologies in statistical inferences: Gauss's least square and Fisher's maximum likelihood. However, a parallel likelihood inference is lacking because estimating functions are in general not integrable, or nonconservative. In this paper, nonconservative estimating functions are studied from vector analysis perspective. We derive a generalized version of the Helmholtz decomposition theorem for estimating functions of any dimension. Based on this theorem we propose locally quadratic potentials as approximate quasi-likelihoods. Quasi-likelihood ratio tests are studied. The ideas are illustrated by two examples: (a) logistic regression with measurement error model and (b) probability estimation conditional on marginal frequencies.     

Newsvendor solutions via conditional value-at-risk minimization

In this paper, we consider the minimization of the conditional value-at-risk (CVaR), a most preferable risk measure in financial risk management, in the context of the well-known single-period newsvendor problem, which is originally formulated as the maximization of the expected profit or the minimization of the expected cost. We show that downside risk measures including the CVaR are tractable in the problem due to their convexity, and consequently, under mild assumptions on the probability distribution of products’ demand, we provide analytical solutions or linear programming (LP) formulation of the minimization of the CVaR measures defined with two different loss functions. Numerical examples are also exhibited, clarifying the difference among the models analyzed in this paper, and demonstrating the efficiency of the LP solutions.