若干基于模糊随机样本的SLT定理

基于模糊随机样本空间,提出了模糊随机期望风险泛函,模糊随机经验风险泛函和模糊随机经验风险泛函最小化原则等概念;基于模糊随机变量及其期望,讨论了相关概率不等式;最后证明了基于模糊随机样本统计学习理论的关键定理并研究了学习过程一致收敛速度的界等问题.     

Soft Randomized Machine Learning

Doklady Mathematics - A new method for entropy-randomized machine learning is proposed based on empirical risk minimization instead of the exact fulfillment of empirical balance conditions. The...     

Robust bi-level optimization of relief logistics operations

In this paper, a robust bi-level optimization model is developed for a supply–distribution relief network under uncertainty in demand and supply parameters. It optimizes the relief operating costs as well as considering a penalty term for unsatisfied victims’ demands. Moreover, the proposed framework optimizes the relief commodity flow in a relief chain along with the supply risk minimization by identifying the suppliers with a lower risk. This paper proposes an integrated optimization method in which the supply risk value for each supplier is obtained via the TOPSIS method. Next, these values are utilized in a robust bi-level model to select appropriate suppliers and allocate orders. Finally, the robustness and effectiveness of the proposed model are demonstrated by a case of flood disaster.     

Inverse source problem for a diffusion equation involving the fractional spectral Laplacian

In this paper, we consider an inverse problem related to a fractional diffusion equation. The model problem is governed by a nonlinear partial differential equation involving the fractional spectral Laplacian. This study is focused on the reconstruction of an unknown source term from a partial internal measured data. The considered ill‐posed inverse problem is formulated as a minimization one. The existence, uniqueness, and stability of the solution are discussed. Some theoretical results are established. The numerical reconstruction of the unknown source term is investigated using an iterative process. The proposed method involves a denoising procedure at each iteration step and provides a sequence of source term approximations converging in norm to the actual solution of the minimization problem. Some numerical results are presented to show the efficiency and the accuracy of the proposed approach.     

Portfolio selection based on fuzzy cross-entropy

In this paper, the Kapur cross-entropy minimization model for portfolio selection problem is discussed under fuzzy environment, which minimizes the divergence of the fuzzy investment return from a priori one. First, three mathematical models are proposed by defining divergence as cross-entropy, average return as expected value and risk as variance, semivariance and chance of bad outcome, respectively. In order to solve these models under fuzzy environment, a hybrid intelligent algorithm is designed by integrating numerical integration, fuzzy simulation and genetic algorithm. Finally, several numerical examples are given to illustrate the modeling idea and the effectiveness of the proposed algorithm.     

A trust region algorithm with adaptive cubic regularization methods for nonsmooth convex minimization

By using the Moreau-Yosida regularization and proximal method, a new trust region algorithm is proposed for nonsmooth convex minimization. A cubic subproblem with adaptive parameter is solved at each iteration. The global convergence and Q-superlinear convergence are established under some suitable conditions. The overall iteration bound of the proposed algorithm is discussed. Preliminary numerical experience is reported.     

A convexification method for a class of global optimization problems with applications to reliability optimization

A convexification method is proposed for solving a class of global optimization problems with certain monotone properties. It is shown that this class of problems can be transformed into equivalent concave minimization problems using the proposed convexification schemes. An outer approximation method can then be used to find the global solution of the transformed problem. Applications to mixed-integer nonlinear programming problems arising in reliability optimization of complex systems are discussed and satisfactory numerical results are presented.     

On multiplicative and Volterra minimization methods

Theory and applications of multiplicative and Volterra calculi have been evolving rapidly over the recent years. As numerical minimization methods have a wide range of applications in science and engineering, the idea of the design of minimization methods based on multiplicative and Volterra calculi is self-evident. In this paper, the well-known Newton minimization method for one and two variables is developed in the frameworks of multiplicative and Volterra calculi. The efficiency of these proposed minimization methods is exposed by examples, and the results are compared with the original minimization method. One of the striking results of the proposed method is that the rate of convergence and the range of initial values are considerably larger compared to the original method.     

NONDESCENT SUBGRADIENT METHOD FOR NONSMOOTH CONSTRAINED MINIMIZATION

A kind of nondecreasing subgradient algorithm with appropriate stopping rule has been proposed for nonsmooth constrained minimization problem. The dual theory is invoked in dealing with the stopping rule and general global minimiizing algorithm is employed as a subroutine of the algorithm. The method is expected to tackle a large class of nonsmooth constrained minimization problem.     

最小风险组合证券非负投资比例系数的确定

研究非负投资比例系数约束条件下,实现风险最小化的组合证券投资问题.应用罚函数法,对最小风险组合证券的非负投资比例系数进行研究.实例表明:这一方法是可行的、有效的.     

Aggregation via empirical risk minimization

Given a finite set F of estimators, the problem of aggregation is to construct a new estimator whose risk is as close as possible to the risk of the best estimator in F. It was conjectured that empirical minimization performed in the convex hull of F is an optimal aggregation method, but we show that this conjecture is false. Despite that, we prove that empirical minimization in the convex hull of a well chosen, empirically determined subset of F is an optimal aggregation method.     

A multiplicative Gauss-Newton minimization algorithm:Theory and application to exponential functions

Multiplicative calculus(MUC) measures the rate of change of function in terms of ratios, which makes the exponential functions significantly linear in the framework of MUC.Therefore, a generally non-linear optimization problem containing exponential functions becomes a linear problem in MUC. Taking this as motivation, this paper lays mathematical foundation of well-known classical Gauss-Newton minimization(CGNM) algorithm in the framework of MUC. This paper formulates the mathematical derivation of proposed method named as multiplicative Gauss-Newton minimization(MGNM) method along with its convergence properties.The proposed method is generalized for n number of variables, and all its theoretical concepts are authenticated by simulation results. Two case studies have been conducted incorporating multiplicatively-linear and non-linear exponential functions. From simulation results, it has been observed that proposed MGNM method converges for 12972 points, out of 19600 points considered while optimizing multiplicatively-linear exponential function, whereas CGNM and multiplicative Newton minimization methods converge for only 2111 and 9922 points, respectively. Furthermore, for a given set of initial value, the proposed MGNM converges only after 2 iterations as compared to 5 iterations taken by other methods. A similar pattern is observed for multiplicatively-non-linear exponential function. Therefore, it can be said that proposed method converges faster and for large range of initial values as compared to conventional methods.     

基于模糊粗糙样本的统计学习理论的理论基础

介绍模糊粗糙理论的基本内容;提出模糊粗糙经验风险泛函,模糊粗糙期望风险泛函,模糊粗糙经验风险最小化原则等概念;最后证明基于模糊粗糙样本的统计学习理论的关键定理并构建学习过程一致收敛速度的界.     

On an inverse linear programming problem

A method for solving the following inverse linear programming (LP) problem is proposed. For a given LP problem and one of its feasible vectors, it is required to adjust the objective function vector as little as possible so that the given vector becomes optimal. The closeness of vectors is estimated by means of the Euclidean vector norm. The inverse LP problem is reduced to a problem of unconstrained minimization for a convex piecewise quadratic function. This minimization problem is solved by means of the generalized Newton method.     

Dynamic conditional value-at-risk model for routing and scheduling of hazardous material transportation networks

This paper illustrates a dynamic model of conditional value-at-risk (CVaR) measure for risk assessment and mitigation of hazardous material transportation in supply chain networks. The well-established market risk measure, CVaR, which is commonly used by financial institutions for portfolio optimizations, is investigated. In contrast to previous works, we consider CVaR as the main objective in the optimization of hazardous material (hazmat) transportation network. In addition to CVaR minimization and route planning of a supply chain network, the time scheduling of hazmat shipments is imposed and considered in the present study. Pertaining to the general dynamic risk model, we analyzed several scenarios involving a variety of hazmats and time schedules with respect to optimal route selection and CVaR minimization. A solution algorithm is then proposed for solving the model, with verifications made using numerical examples and sensitivity analysis.     

A discrepancy principle for the Landweber iteration based on risk minimization

In this paper we propose a criterion based on risk minimization to stop the Landweber algorithm for estimating the solution of a linear system with noisy data. Under the hypothesis of white Gaussian noise, we provide an unbiased estimator of the risk and we use it for defining a variant of the classical discrepancy principle. Moreover, we prove that the proposed variant satisfies the regularization property in expectation. Finally, we perform some numerical simulations when the signal formation model is given by a convolution or a Radon transform, to show that the proposed method is numerically reliable and furnishes slightly better solutions than classical estimators based on the predictive risk, namely the Unbiased Predictive Risk Estimator and the Generalized Cross Validation.     

An alternating direction method for linear‐constrained matrix nuclear norm minimization

The aim of the nuclear norm minimization problem is to find a matrix that minimizes the sum of its singular values and satisfies some constraints simultaneously. Such a problem has received more attention largely because it is closely related to the affine rank minimization problem, which appears in many control applications including controller design, realization theory, and model reduction. In this paper, we first propose an exact version alternating direction method for solving the nuclear norm minimization problem with linear equality constraints. At each iteration, the method involves a singular value thresholding and linear matrix equations which are solved exactly. Convergence of the proposed algorithm is followed directly. To broaden the capacity of solving larger problems, we solve approximately the subproblem by an iterative method with the Barzilai–Borwein steplength. Some extensions to the noisy problems and nuclear norm regularized least‐square problems are also discussed. Numerical experiments and comparisons with the state‐of‐the‐art method FPCA show that the proposed method is effective and promising. Copyright © 2011 John Wiley & Sons, Ltd.     

导数振荡与应变能极小的平面分段三次参数Hermite插值

由于分段三次参数Hermite插值的切矢往往被作为变量,故可对其进行优化以使得构造的插值曲线满足特定的要求.为了构造兼具保形性与光顺性的平面分段三次参数Hermite插值曲线,给出了一种通过同时极小化导数振荡和应变能来确定切矢的方法.首先以导数振荡函数和应变能函数为双目标建立了切矢满足的方程系统;然后证明了方程系统存在唯一解,并给出了解的具体表达式;最后给出了误差分析,并通过数值算例表明方法的有效性.结果表明,相对于导数振荡极小化方法和应变能极小化方法,所提出的导数振荡和应变能极小化方法同时兼顾了平面分段三次参数Hermite插值曲线的保形性和光顺性.     

Support vector regression based metamodeling for seismic reliability analysis of structures

The present study deals with support vector regression-based metamodeling approach for efficient seismic reliability analysis of structure. Various metamodeling approaches e.g. response surface method, Kriging interpolation, artificial neural network, etc. are usually adopted to overcome computational challenge of simulation based seismic reliability analysis. However, the approximation capability of such empirical risk minimization principal-based metamodeling approach is largely affected by number of training samples. The support vector regression based on the principle of structural risk minimization has revealed improved response approximation ability using small sample learning. The approach is explored here for improved estimate of seismic reliability of structure in the framework of Monte Carlo Simulation technique. The parameters necessary to construct the metamodel are obtained by a simple effective search algorithm by solving an optimization sub-problem to minimize the mean square error obtained by cross-validation method. The simulation technique is readily applied by random selection of metamodel to implicitly consider record to record variations of earthquake. Without additional computational burden, the approach avoids a prior distribution assumption about approximated structural response unlike commonly used dual response surface method. The effectiveness of the proposed approach compared to the usual polynomial response surface and neural network based metamodels is numerically demonstrated.     

New reformulation and feasible semismooth Newton method for a class of stochastic linear complementarity problems

A class of stochastic linear complementarity problems (SLCPs) with finitely many realizations is considered. We first formulate the problem as a new constrained minimization problem. Then, we propose a feasible semismooth Newton method which yields a stationary point of the constrained minimization problem. We study the condition for the level set of the objective function to be bounded. As a result, the condition for the solution set of the constrained minimization problem is obtained. The global and quadratic convergence of the proposed method is proved under certain assumptions. Preliminary numerical results show that this method yields a reasonable solution with high safety and within a small number of iterations.