Least trimmed squares regression,least median squares regression,and mathematical programming

In this paper, we study LTS and LMS regression, two high breakdown regression estimators, from an optimization point of view. We show that LTS regression is a nonlinear optimization problem that can be treated as a concave minimization problem over a polytope. We derive several important properties of the corresponding objective function that can be used to obtain algorithms for the exact solution of LTS regression problems, i.e., to find a global optimum to the problem. Because of today's limited problem-solving capabilities in exact concave minimization, we give an easy-to-implement pivoting algorithm to determine regression parameters corresponding to local optima of the LTS regression problem. For the LMS regression problem, we briefly survey the existing solution methods which are all based on enumeration. We formulate the LMS regression problem as a mixed zero-one linear programming problem which we analyze in depth to obtain theoretical insights required for future algorithmic and computational work.     

CVaR风险度量模型在投资组合中的运用

风险价值(VaR)是近年来金融机构广泛运用的风险度量指标，条件风险价值(CVaR)是VaR的修正模型，也称为平均超额损失或尾部VaR，它比VaR具有更好的性质。在本中，我们将运用风险度量指标VaR和CVaR，提出一个新的最优投资组合模型。介绍了模型的算法，而且利用我国的股票市场进行了实证分析，验证了新模型的有效性，为制定合理的投资组合提供了一种新思路。     

Computing S Estimators for Regression and Multivariate Location/Dispersion

Abstract

An improved resampling algorithm for S estimators reduces the number of times the objective function is evaluated and increases the speed of convergence. With this algorithm, S estimates can be computed in less time than least median squares (LMS) for regression and minimum volume ellipsoid (MVE) for location/scatter estimates with the same accuracy. Here accuracy refers to the randomness due to the algorithm. S estimators are also more statistically efficient than the LMS and MVE estimators, that is, they have less variability due to the randomness of the data.     

Estimating value at risk with semiparametric support vector quantile regression

Value at Risk (VaR) has been used as an important tool to measure the market risk under normal market. Usually the VaR of log returns is calculated by assuming a normal distribution. However, log returns are frequently found not normally distributed. This paper proposes the estimation approach of VaR using semiparametric support vector quantile regression (SSVQR) models which are functions of the one-step-ahead volatility forecast and the length of the holding period, and can be used regardless of the distribution. We find that the proposed models perform better overall than the variance-covariance and linear quantile regression approaches for return data on S&P 500, NIKEI 225 and KOSPI 200 indices.     

求解线性模型稳健参数估计的整数编码遗传算法

线性模型回归系数的一些稳健估计如LMS、LQS、LTS、LTA的应用越来越广泛,然而它们的精确计算依赖于NP难题,在遇到高维大规模数据集时不可能在较短时间内得到精确解.为尽快得到较高精度的近似解,提出了求解线性模型的稳健参数估计的整数编码遗传算法,通过计算机模拟试验验证了算法可以更快地找出全局最优解.     

均值-VaR模型的一种新解法：鞍点近似、遗传算法

以均值度量收益,方差度量风险的均值.方差模型,广泛应用于资产组合优化.随着对金融风险度量方法研究的不断深入,VaR作为一种简便、易于理解的风险度量方法,在金融企业中得到日益广泛的应用.本文用VaR代替均值-方差模型中的方差,构建了均值-VaR模型应用干投资组合优化.均值-VaR模型是非线性规划,仅当VaR满足凸性和可微性的前提下,满足库恩-塔克条件的解才是全局最优解.本文在CreditRisk+框架下,提出一个在不允许卖空条件下,不需对VaR的性质做出前提假定的新解法:将鞍点近似法用于计算VaR,在资产头寸与VaR之间建立起函数关系,采用遗传算法寻找模型的近似最优解.并用一个债券组合说明该方法的有效性。     

Credit risk optimization with Conditional Value-at-Risk criterion

This paper examines a new approach for credit risk optimization. The model is based on the Conditional Value-at-Risk (CVaR) risk measure, the expected loss exceeding Value-at-Risk. CVaR is also known as Mean Excess, Mean Shortfall, or Tail VaR. This model can simultaneously adjust all positions in a portfolio of financial instruments in order to minimize CVaR subject to trading and return constraints. The credit risk distribution is generated by Monte Carlo simulations and the optimization problem is solved effectively by linear programming. The algorithm is very efficient; it can handle hundreds of instruments and thousands of scenarios in reasonable computer time. The approach is demonstrated with a portfolio of emerging market bonds. Received: November 1, 1999 / Accepted: October 1, 2000?Published online December 15, 2000     

Nonsmooth nonconvex optimization approach to clusterwise linear regression problems

Clusterwise regression consists of finding a number of regression functions each approximating a subset of the data. In this paper, a new approach for solving the clusterwise linear regression problems is proposed based on a nonsmooth nonconvex formulation. We present an algorithm for minimizing this nonsmooth nonconvex function. This algorithm incrementally divides the whole data set into groups which can be easily approximated by one linear regression function. A special procedure is introduced to generate a good starting point for solving global optimization problems at each iteration of the incremental algorithm. Such an approach allows one to find global or near global solution to the problem when the data sets are sufficiently dense. The algorithm is compared with the multistart Späth algorithm on several publicly available data sets for regression analysis.     

基于有效约束识别技术的一个SSLE算法及其收敛性分析

基于一个有效约束识别技术, 给出了具有不等式约束的非线性最优化问题的一个可行SSLE算法. 为获得搜索方向算法的每步迭代只需解两个或三个具有相同系数矩阵的线性方程组. 在一定的条件下, 算法全局收敛到问题的一个KKT点. 没有严格互补条件, 在比强二阶充分条件弱的条件下算法具有超线性收敛速度.     

Static and dynamic VaR constrained portfolios with application to delegated portfolio management

We give a closed-form solution to the single-period portfolio selection problem with a Value-at-Risk (VaR) constraint in the presence of a set of risky assets with multivariate normally distributed returns and the risk-less account, without short sales restrictions. The result allows to obtain a very simple, myopic dynamic portfolio policy in the multiple period version of the problem. We also consider mean-variance portfolios under a probabilistic chance (VaR) constraint and give an explicit solution. We use this solution to calculate explicitly the bonus of a portfolio manager to include a VaR constraint in his/her portfolio optimization, which we refer to as the price of a VaR constraint.     

Bayesian variable selection in generalized linear models using a combination of stochastic optimization methods

In this paper the usage of a stochastic optimization algorithm as a model search tool is proposed for the Bayesian variable selection problem in generalized linear models. Combining aspects of three well known stochastic optimization algorithms, namely, simulated annealing, genetic algorithm and tabu search, a powerful model search algorithm is produced. After choosing suitable priors, the posterior model probability is used as a criterion function for the algorithm; in cases when it is not analytically tractable Laplace approximation is used. The proposed algorithm is illustrated on normal linear and logistic regression models, for simulated and real-life examples, and it is shown that, with a very low computational cost, it achieves improved performance when compared with popular MCMC algorithms, such as the MCMC model composition, as well as with “vanilla” versions of simulated annealing, genetic algorithm and tabu search.     

On the convergence of asynchronous parallel algorithm for large-scale linearly constrained minimization problem

As a synchronization parallel framework, the parallel variable transformation (PVT) algorithm is effective to solve unconstrained optimization problems. In this paper, based on the idea that a constrained optimization problem is equivalent to a differentiable unconstrained optimization problem by introducing the Fischer Function, we propose an asynchronous PVT algorithm for solving large-scale linearly constrained convex minimization problems. This new algorithm can terminate when some processor satisfies terminal condition without waiting for other processors. Meanwhile, it can enhances practical efficiency for large-scale optimization problem. Global convergence of the new algorithm is established under suitable assumptions. And in particular, the linear rate of convergence does not depend on the number of processors.     

An Algorithm for Solving Optimization Problems with One Linear Objective Function and Finitely Many Constraints of Fuzzy Relation Inequalities

An optimization model with one linear objective function and fuzzy relation equation constraints was presented by Fang and Li (1999) as well as an efficient solution procedure was designed by them for solving such a problem. A more general case of the problem, an optimization model with one linear objective function and finitely many constraints of fuzzy relation inequalities, is investigated in this paper. A new approach for solving this problem is proposed based on a necessary condition of optimality given in the paper. Compared with the known methods, the proposed algorithm shrinks the searching region and hence obtains an optimal solution fast. For some special cases, the proposed algorithm reaches an optimal solution very fast since there is only one minimum solution in the shrunk searching region. At the end of the paper, two numerical examples are given to illustrate this difference between the proposed algorithm and the known ones.     

一种求解分类问题的新算法

本文提出了基于支持向量回归机(SVR)的一种新分类算法.它和标准的支持向量机(SVM)不同:标准的支持向量机(SVM)采用固定的模度量间隔且最优化问题与参数有关.本文中我们可以用任意模度量间隔,得到的最优化问题是无参数的线性规划问题,避免了参数选择.数值试验表明了该算法的有效性.     

基于动力系统的线性不等式组的解法

本文提出了一种新的求解线性不等式组可行解的方法-基于动力系统的方法．假设线性不等式组的可行域为非空,在可行域的相对内域上建立一个非线性关系表达式,进而得到一个结构简单的动力系统模型．同时,定义了穿越方向。文章最后的数值实验结果表明此算法是有效的．     

求解线性不等式组的方法

本提出了一个新的求解线性不等式组可行解的方法－－无约束极值方法。通过在线性不等式组的非空可行域的相对内域上建立一个非线性极值问题，根据对偶关系，得到了一个对偶空间的无约束极值及原始，对偶变量之间的简单线性映射关系，这样将原来线性不等式组问题的求解转化为一个无约束极值问题。中主要讨论了求解无约束极值问题的共轭梯度算法。同时，在寻找不等式组可行解的过程中，定义了穿越方向，这样大大减少计算量。中最后数值实验结果表明此算法是有效的。     

金融风险的最坏VaR方法与投资组合优化

在股价及其走势均不确定的情况下,采用最坏VaR方法,对投资的潜在损失进行最保守的度量,并得到其等价的优化形式为一个二阶锥优化问题.接着考虑相应的投资组合优化问题:如何选择合适的头寸,使得当股票组合的期望收益达到给定水平的情况下,风险最低,即最坏VaR值最小,最后对模型进行实证分析.     

Primal and dual multi-objective linear programming algorithms for linear multiplicative programmes

Multiplicative programming problems (MPPs) are global optimization problems known to be NP-hard. In this paper, we employ algorithms developed to compute the entire set of nondominated points of multi-objective linear programmes (MOLPs) to solve linear MPPs. First, we improve our own objective space cut and bound algorithm for convex MPPs in the special case of linear MPPs by only solving one linear programme in each iteration, instead of two as the previous version indicates. We call this algorithm, which is based on Benson’s outer approximation algorithm for MOLPs, the primal objective space algorithm. Then, based on the dual variant of Benson’s algorithm, we propose a dual objective space algorithm for solving linear MPPs. The dual algorithm also requires solving only one linear programme in each iteration. We prove the correctness of the dual algorithm and use computational experiments comparing our algorithms to a recent global optimization algorithm for linear MPPs from the literature as well as two general global optimization solvers to demonstrate the superiority of the new algorithms in terms of computation time. Thus, we demonstrate that the use of multi-objective optimization techniques can be beneficial to solve difficult single objective global optimization problems.     

Box-constrained multi-objective optimization: A gradient-like method without “a priori” scalarization

The aim of this paper is the development of an algorithm to find the critical points of a box-constrained multi-objective optimization problem. The proposed algorithm is an interior point method based on suitable directions that play the role of gradient-like directions for the vector objective function. The method does not rely on an “a priori” scalarization and is based on a dynamic system defined by a vector field of descent directions in the considered box. The key tool to define the mentioned vector field is the notion of vector pseudogradient. We prove that the limit points of the solutions of the system satisfy the Karush–Kuhn–Tucker (KKT) first order necessary condition for the box-constrained multi-objective optimization problem. These results allow us to develop an algorithm to solve box-constrained multi-objective optimization problems. Finally, we consider some test problems where we apply the proposed computational method. The numerical experience shows that the algorithm generates an approximation of the local optimal Pareto front representative of all parts of optimal front.     

TV‐like regularization for backward parabolic problems

This paper investigates an inverse problem for parabolic equations backward in time, which is solved by total‐variation‐like (TV‐like, in abbreviation) regularization method with cost function ∥u_x∥². The existence, uniqueness and stability estimate for the regularization problem are deduced in the linear case. For numerical illustration, the variational adjoint method, which presents a simple method to derive the gradient of the optimization functional, is introduced to reconstruct the unknown initial condition for both linear and nonlinear parabolic equations. The conjugate gradient method is used to iteratively search for the optimal approximation. Numerical results validate the feasibility and effectiveness of the proposed algorithm. Copyright © 2016 John Wiley & Sons, Ltd.