Global Optimization of Econometric Functions

Estimating the values of the parameter estimates of econometric functions (maximum likelihood functions or nonlinear least squares functions) are often challenging global optimization problems. Determining the global optimum for these functions is necessary to understand economic behavior and to develop effective economic policies. These functions often have flat surfaces or surfaces characterized by many local optima. Classical deterministic optimization methods often do not yield successful results. For that reason, stochastic optimization methods are becoming widely used in econometrics. Selected stochastic methods are applied to two difficult econometric functions to determine if they might be useful in estimating the parameters of these functions.     

Extended Global Convergence Framework for Unconstrained Optimization

An extension of the global convergence framework for unconstrained derivative-free optimization methods is presented. The extension makes it possible for the framework to include optimization methods with varying cardinality of the ordered direction set. Grid-based search methods are shown to be a special case of the more general extended global convergence framework. Furthermore,the required properties of the sequence of ordered direction sets listed in the definition of grid-based methods are relaxed and simplified by removing the requirement of structural equivalence.     

A New Filled Function Method for Global Optimization

A novel filled function is suggested in this paper for identifying a global minimum point for a general class of nonlinear programming problems with a closed bounded domain. Theoretical and numerical properties of the proposed filled function are investigated and a solution algorithm is proposed. The implementation of the algorithm on several test problems is reported with satisfactory numerical results.     

An Analysis of the Role of Positivity and Mixture Model Constraints in Poisson Deconvolution Problems

We consider a class of estimation problems in which data of a Poisson character are related by a linear model to a target function that satisfies certain physical constraints. The classic example of this situation is the reconstruction problem of positron emission tomography (PET). There the function of interest satisfies positivity constraints. This article examines the impact of such constraints by comparing simple unconstrained reconstruction methods with constrained alternatives based on maximum likelihood (ML) and least squares (LS) formulations. Data from a series of numerical experiments are presented to quantify the significance of constraints. Although these experiments show that constraints are important, the differences between ML and LS based implementations of constraints are quite small. Thus, in order to evaluate the impact of constraints, it appears to be sufficient to focus on comparing constrained versus unconstrained implementations of LS. This simplifies the analysis of constraints considerably. A perturbation analysis technique is proposed to summarize the impact of constraints in terms of a single relative efficiency measure. The predictions obtained by this analysis are found to be in good agreement with experimental data.     

一种无约束全局优化的水平值下降算法

本文研究无约束全局优化问题,建立了一种新的水平值下降算法(Level-value Descent Method,LDM).讨论并建立了概率意义下取全局最小值的一个充分必要条件,证明了算法LDM是依概率测度收敛的.这种LDM算法是基于重点度取样(Improtance Sampling)和Markov链Monte-Carlo随机模拟实现的,并利用相对熵方法(TheCross-Entropy Method)自动更新取样密度,算例表明LDM算法具有较高的数值精度和较好的全局收敛性.     

Maximum likelihood least squares identification for systems with autoregressive moving average noise

Maximum likelihood methods are important for system modeling and parameter estimation. This paper derives a recursive maximum likelihood least squares identification algorithm for systems with autoregressive moving average noises, based on the maximum likelihood principle. In this derivation, we prove that the maximum of the likelihood function is equivalent to minimizing the least squares cost function. The proposed algorithm is different from the corresponding generalized extended least squares algorithm. The simulation test shows that the proposed algorithm has a higher estimation accuracy than the recursive generalized extended least squares algorithm.     

无约束全局优化的一个可调参数的填充函数

A filled function with adjustable parameters is suggested in this paper for finding a global minimum point of a general class of nonlinear programming problems with a bounded and closed domain. This function has two adjustable parameters. We will discuss the properties of the proposed filled function. Conditions on this function and on the values of parameters are given so that the constructed function has the desired properties of traditional filled function.     

Global Convergence of a Memory Gradient Method for Unconstrained Optimization

Memory gradient methods are used for unconstrained optimization, especially large scale problems. The first idea of memory gradient methods was proposed by Miele and Cantrell (1969) and Cragg and Levy (1969). In this paper, we present a new memory gradient method which generates a descent search direction for the objective function at every iteration. We show that our method converges globally to the solution if the Wolfe conditions are satisfied within the framework of the line search strategy. Our numerical results show that the proposed method is efficient for given standard test problems if we choose a good parameter included in the method.     

求无约束连续全局优化问题的单参数填充函数法

填充函数法是求解全局优化问题的一个重要的确定性算法,这种方法的关键是构造具有良好性质的填充函数.构造了一个新的求解无约束全局优化问题的填充函数.函数连续可微且只包含一个参数.通过分析该函数的相关性质,设计了相应的算法.数值实验表明该算法简单有效.     

New Subinterval Selection Criteria for Interval Global Optimization

The theoretical convergence properties of interval global optimization algorithms that select the next subinterval to be subdivided according to a new class of interval selection criteria are investigated. The latter are based on variants of the RejectIndex: , a recently thoroughly studied indicator, that can quite reliably show which subinterval is close to a global minimizer point. Extensive numerical tests on 40 problems confirm that substantial improvements can be achieved both on simple and sophisticated algorithms by the new method (utilizing the known minimum value), and that these improvements are larger when hard problems are to be solved.     

基于非拟牛顿方法无约束最优化问题的全局收敛性

In this paper, the non-quasi-Newton's family with inexact line search applied to unconstrained optimization problems is studied. A new update formula for non-quasi-Newton's family is proposed. It is proved that the constituted algorithm with either Wolfe-type or Armijotype line search converges globally and Q-superlinearly if the function to be minimized has Lipschitz continuous gradient.     

Global Optimization of Nonlinear Bilevel Programming Problems

A novel technique that addresses the solution of the general nonlinear bilevel programming problem to global optimality is presented. Global optimality is guaranteed for problems that involve twice differentiable nonlinear functions as long as the linear independence constraint qualification condition holds for the inner problem constraints. The approach is based on the relaxation of the feasible region by convex underestimation, embedded in a branch and bound framework utilizing the basic principles of the deterministic global optimization algorithm, BB [2, 4, 5, 11]. Epsilon global optimality in a finite number of iterations is theoretically guaranteed. Computational studies on several literature problems are reported.     

Target-Oriented Branch and Bound Method for Global Optimization

We introduce a very simple but efficient idea for branch and bound (&) algorithms in global optimization (GO). As input for our generic algorithm, we need an upper bound algorithm for the GO maximization problem and a branching rule. The latter reduces the problem into several smaller subproblems of the same type. The new & approach delivers one global optimizer or, if stopped before finished, improved upper and lower bounds for the problem. Its main difference to commonly used & techniques is its ability to approximate the problem from above and from below while traversing the problem tree. It needs no supplementary information about the system optimized and does not consume more time than classical & techniques. Experimental results with the maximum clique problem illustrate the benefit of this new method.     

A new linesearch algorithm for nonlinear least squares problems

A linesearch (steplength) algorithm for unconstrained nonlinear least squares problems is described. To estimate the steplength inside the linesearch algorithm a new method that interpolates the residual vector is used together with a standards method that interpolates the sums of squares. Numerical results are reported that point out the advantage with the new steplength estimation method.     

Global Optimization of Nonconvex Polynomial Programming Problems Having Rational Exponents

This paper considers the solution of nonconvex polynomial programming problems that arise in various engineering design, network distribution, and location-allocation contexts. These problems generally have nonconvex polynomial objective functions and constraints, involving terms of mixed-sign coefficients (as in signomial geometric programs) that have rational exponents on variables. For such problems, we develop an extension of the Reformulation-Linearization Technique (RLT) to generate linear programming relaxations that are embedded within a branch-and-bound algorithm. Suitable branching or partitioning strategies are designed for which convergence to a global optimal solution is established. The procedure is illustrated using a numerical example, and several possible extensions and algorithmic enhancements are discussed.     

Metric-Based Parameterizations for Multi-Step Unconstrained Optimization

We consider multi-step quasi-Newton methods for unconstrained optimization. These methods were introduced by Ford and Moghrabi (Appl. Math., vol. 50, pp. 305–323, 1994; Optimization Methods and Software, vol. 2, pp. 357–370, 1993), who showed how interpolating curves could be used to derive a generalization of the Secant Equation (the relation normally employed in the construction of quasi-Newton methods). One of the most successful of these multi-step methods makes use of the current approximation to the Hessian to determine the parameterization of the interpolating curve in the variable-space and, hence, the generalized updating formula. In this paper, we investigate new parameterization techniques to the approximate Hessian, in an attempt to determine a better Hessian approximation at each iteration and, thus, improve the numerical performance of such algorithms.     

无约束最优化的Polak—Ribiere和Hestenes—Stiefel共轭梯度法的全局收敛性

本文在很弱的条件下得到了无约束最优化的Polak-Ribiere和Hestenes-Stiefel共轭梯度法的全局收敛性的新结果,这里PR方法和HS方法中的参数β^PRk和β^HSk可以在某个负的区域内取值,这一负的区域与k有关,这些新的收敛性结果改进了文献中已有的结果。数值检验的结果表明了本文中新的PR方法和HS方法是相当有效的。     

Global Optimization Versus Integer Programming in Portfolio Optimization under Nonconvex Transaction Costs

This paper is concerned with a portfolio optimization problem under concave and piecewise constant transaction cost. We formulate the problem as nonconcave maximization problem under linear constraints using absolute deviation as a measure of risk and solve it by a branch and bound algorithm developed in the field of global optimization. Also, we compare it with a more standard 0–1 integer programming approach. We will show that a branch and bound method elaborating the special structure of the problem can solve the problem much faster than the state-of-the integer programming code.     

解无约束最优化问题的块共轭方向法

在给出块共轭概念的基础上，提出了适合并行计算的向量组的块共轭化方法，进而得到解无约束最优化问题的并行块共轭方向法．有大量数值结果表明块共轭方向法具有工作量少．适用函数范围广等特点，是一种比较有效的无约束最优化方法．     

Frame Based Methods for Unconstrained Optimization

This paper describes a wide class of direct search methods for unconstrained optimization, which make use of fragments of grids called frames. Convergence is shown under mild conditions which allow successive frames to be rotated, translated, and scaled relative to one another.