全局优化及其在金融中的应用
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摘要
本文研究全局优化及其在金融中的应用.并研究含有奇异解的凸函数极小化问题的数值算法。
首先研究求解无约束全局优化问题的算法.我们提出一种新的单纯型反射搜索机制用于全局优化的局部搜索阶段.这种单纯型反射的一个重要性质是当目标函数为二次凹函数时,可以保证沿着下降方向进行反射；若目标函数是一般的凹函数也能保证每次迭代总是由一个较“差”的点关于一个较“好”的点做反射.为使得算法不陷入局部极值,我们将单纯型反射机制与模拟退火算法进行杂交,并考虑将非单调的思想引入,提出一种新的单纯型模型退火算法.最后通过数值试验对算法的有效性进行了验证,数值试验结果表明这种单纯型机制在全局搜索中是合理有效的.
Markowtiz于1952年提出的的投资组合理论是现代金融学的起点,其重要观点是用资产收益率的方差或标准差作为资产收益风险的度量.经过了50余年的发展,投资组合在理论和实践上取得了巨大的成就,而最优化理论也成为研究投资组合的一种重要工具.投资组合中有两类基本策略：积极策略和消极策略.消极的策略投资者认为市场是有效的,要长期胜出市场是很难的.近来消极的基金管理模式日益受到重视,据估计仅在美国就有价值1012美元的消极管理基金[1].指数跟踪是一类消极的基金管理方式,它的目标是构造一个投资组合使得该组合在某个指定的时间段的表现与被跟踪的指数的表现尽可能的相似.在实际中构造指数基金往往有一些约束条件,例如：不允许卖空,对资产头寸的限制,跟踪资产的数目的限制以及交易费用的限制等等.本文研究带实际约束的指数跟踪问题,考虑到跟踪模型目标函数的非凸性,我们导出模型的一种近似形式.它是一个混合整数(二次)规划问题.我们证明,在一定的条件下,在最优解附近,近似形式与原目标可充分接近.为了求解混合二次规划问题,我们提出一种简单的启发式算法.数值试验表明,我们的近似技术在实际操作中运行正常,提出的启发式算法简单高效.
投资组合中最重要的问题是如何有效地分散风险,用通俗的语言说就是“不要把鸡蛋放在同一个篮子里”.边际风险用于衡量单个资产对整体投资组合的风险贡献程度,可用做风险分散化程度的衡量指标.近来边际风险控制已得到越来越高的重视,然而带有边际风险控制的投资组合问题往往是棘手的非凸规划问题,大多数带边际风险约束的投资管理策略仍然停滞在事后分析阶段.由于非系统性风险能够通过充分的分散化而消除,我们关注于系统性边际风险的控制.为使提出的投资组合模型能够易于对系统性风险进行管理,我们将因子模型的思想引入边际风险的投资模型中.提出的边际风险控制模型是一个具有一些特殊的结构非凸优化问题.我们通过利用问题的特殊结构,提出一种分枝限界算法用于全局最优策略的求解.特别的,提出的分枝限界算法的分枝变量不依赖与总资产数目n,仅与因子的个数m及受边际风险限制的资产的个数C有关,特别适用于n远大于m+C的情况.我们通过实证分析对提出的模型与均值方差模型进行了比较,并通过数值算例证实本文分枝限界算法的有效性.
目前对投资组合风险的控制大多讨论的是对单个资产或者整体组合的风险的控制,事实上对投资组合的风险因子进行控制可以让投资者或基金经理对组合的风险结构有更加清楚的了解,根据自身对各类风险因素的承受能力更有效的进行投资管理.带有因子风险约束的投资模型很多时候是一个非凸规划问题,模型的求解难度很大；我们通过对风险因子的单位风险回报率进行控制,研究均值方差框架下带因子风险控制的投资模型.得到一个简单的二次规划投资模型,并通过实证分析对因子风险控制的投资策略与均值方差模型进行比较,结果表明对因子的单位风险回报率进行控制可以均匀投资组合中各风险因子的分布,并提高投资组合的市场表现.
全局优化问题研究的是存在多个局部极值点的优化问题,是一类非常难的优化问题,目前尚未存在非常有效的算法.事实上局部优化中的一些特殊问题也是非常棘手的,例如奇异解的极小化问题.传统的牛顿算法对无约束的凸极小化问题是非常有效的,其显著优点是算法的局部二次收敛性,但若极小化问题存在奇异解,牛顿算法的收敛性将降为线性.我们研究了带奇异解的凸函数极小化问题,提出一种截断正则化牛顿算法进行求解,并在一定的条件下证明了算法的全局收敛性和局部二次收敛性.数值结果表明截断正则化牛顿法对奇异解的凸极小化问题是非常有效的,且问题规模的增大时优势更明显.
In the thesis, we study the numerical methods for global optimization and its applications in portfolio selection. We also study the numerical method for solving convex minimization with singular solutions.
First, we consider to find a global solution of minimizing a nonconvex function with box constrains. To solve the problem, we propose a new simplex mechanism for the local search stage of global optimization. We show that when the objective function is a concave quadratic function, the reflection of the simplex mechanism is done along a descent direction. If the objective function is concave but not necessary quadratic, the mechanism always reflect a "worse" vertex with respective to a "better" vertex. To avoid the algorithm being trapped in a local optimizer, we combined the simplex mechanism with simulated annealing and introduced the idea of non-monotone line search in our algorithm to propose a new SA-Simplex method. Numerical results show that as a local explorer in global optimization, our simplex mechanism is reasonable and practically effective.
The portfolio selection theory was founded by Markowtiz in 1952. It is a new starting point of modern finance. Its most important view is to use the variance or the standard variance of the return of the portfolio to measure portfolio risk. Portfolio theory has made great progress in the past 50 years both in theory and practice, and the optimization theory has becomes an important tool in the solution of those models. There are two basic strategies in portfolio management:active and passive. Most fund managers inclines to passive strategies because they think that in an efficient market, it is hardly beat the market for a long period. Recently, passive strategies have received much attention. It is estimated that there are in total over 1012 dollars passively invested in index funds in the USA market [1]. Index tracking is a popular form of passive fund management, which consists in reproducing the performance of a stock market index by investing in a subset of stocks included in the index. There are some realistic constraints that should be posed on the tracking portfolio, for example, no short sell, constraints on portfolio positions, the number of assets that allowed to track the index, transaction cost constraints and so on. In this thesis, we consider index tracking with realistic constraints. The objective function of tracking model is very complex nonconvex minimization problem. To solve the problem, we propose a convex approximation to it, which is a mix-integer quadratic programming. Under certain conditions, we prove that the objective function value of the approximation problem can be sufficiently close to the value of the original objective function near the optimal portfolio. We then propose a simple heuristic to find its global optimizer. Numeri-cal results show that our approximation works well and the heuristic is simple and efficient.
The most important thing in portfolio is how to diversify risk as well as possi-ble. In plain language, the investor should not put all eggs in one basket. Marginal risk can be used to judge the risk contribution of a single asset, and could be used as a measure of portfolio diversification. Recently, marginal risk control has at-tracted increasing attention. However, as portfolio selection with marginal risk control always leads to a non-convex optimization problem, most of the related work dwells on the stage of ex post analysis. As the non-systematic risk could be elimiated through diversification, we focus on the management of systematic risk. To get a formulation that could be useful for management of systematic risk, we introduce the idea of factor models in our portfolio selection model. Our portfolio model is a non-convex programming but with some special structures. By ex-ploiting the special structure of the problems, we propose an efficient branch and bound method for its global optimizer. Especially, our branching variables are only related to the number of assets that are restricted with marginal risk constraints (m) and the number of market factors (C). Our branch an bound method is very efficient when the number of assets (n) is much less than m+C. We compare our proposed portfolio model with mean variance model through empirical studies and test the efficient of our proposed branch and bound method through numerical experiments.
Currently, most of risk control in portfolio management refers to the risk re-striction of certain assets in portfolio. In fact, by properly examining the market risk factors that derive the return of the portfolio, we could control the portfolio risk profile more efficiently according to the tolerance level for some specific risk. In general, factor risk constrained portfolio selection problems lead to nonconvex optimization problems and are generally very hard to solve. We consider the port-folio selection problems with factor risk constraints in another way. Specifically, we control the market factors'risk-reward ratio and derive a quadratic programming model, which could be efficiently solve by some standard software. We compare our proposed portfolio model with the mean variance model through empirical analysis. The results show that by property control the market factors'risk-reward ratio, we could make the distribution of risk more evenly, and improve the performance of our managed portfolio as well.
Global optimization refers to the problem with multiplier local optimizer. Traditional optimization methods are easily trapped in a local optimizer and fails when dealing with global optimization problems. Global optimization is a kind of very hard optimization problems and there exists no efficient algorithm yet. In fact, there are some very hard problems in local optimizations too, for exam-ple, convex minimization with singular solutions. Traditional Newton's method is very efficient for unconstrained convex minimization, and has the attractive local quadratic convergence property. However, if the problem has singular solutions, the convergence rate of Newton's method may reduce to be linear. In the last part of this thesis, we propose a truncated regularized Newton method to minimize a convex function with singular solutions. Under appropriate conditions, we prove the global convergence and local quadratic convergence of the proposed method. Numerical results show that our proposed method has potential advantages for solving large scale problems with singular solutions.
This dissertation is supported by the National Natural Science Foundation of China (10771057) and the Major Project of Ministry of Education of China granted (309023).
首先研究求解无约束全局优化问题的算法.我们提出一种新的单纯型反射搜索机制用于全局优化的局部搜索阶段.这种单纯型反射的一个重要性质是当目标函数为二次凹函数时,可以保证沿着下降方向进行反射；若目标函数是一般的凹函数也能保证每次迭代总是由一个较“差”的点关于一个较“好”的点做反射.为使得算法不陷入局部极值,我们将单纯型反射机制与模拟退火算法进行杂交,并考虑将非单调的思想引入,提出一种新的单纯型模型退火算法.最后通过数值试验对算法的有效性进行了验证,数值试验结果表明这种单纯型机制在全局搜索中是合理有效的.
Markowtiz于1952年提出的的投资组合理论是现代金融学的起点,其重要观点是用资产收益率的方差或标准差作为资产收益风险的度量.经过了50余年的发展,投资组合在理论和实践上取得了巨大的成就,而最优化理论也成为研究投资组合的一种重要工具.投资组合中有两类基本策略：积极策略和消极策略.消极的策略投资者认为市场是有效的,要长期胜出市场是很难的.近来消极的基金管理模式日益受到重视,据估计仅在美国就有价值1012美元的消极管理基金[1].指数跟踪是一类消极的基金管理方式,它的目标是构造一个投资组合使得该组合在某个指定的时间段的表现与被跟踪的指数的表现尽可能的相似.在实际中构造指数基金往往有一些约束条件,例如：不允许卖空,对资产头寸的限制,跟踪资产的数目的限制以及交易费用的限制等等.本文研究带实际约束的指数跟踪问题,考虑到跟踪模型目标函数的非凸性,我们导出模型的一种近似形式.它是一个混合整数(二次)规划问题.我们证明,在一定的条件下,在最优解附近,近似形式与原目标可充分接近.为了求解混合二次规划问题,我们提出一种简单的启发式算法.数值试验表明,我们的近似技术在实际操作中运行正常,提出的启发式算法简单高效.
投资组合中最重要的问题是如何有效地分散风险,用通俗的语言说就是“不要把鸡蛋放在同一个篮子里”.边际风险用于衡量单个资产对整体投资组合的风险贡献程度,可用做风险分散化程度的衡量指标.近来边际风险控制已得到越来越高的重视,然而带有边际风险控制的投资组合问题往往是棘手的非凸规划问题,大多数带边际风险约束的投资管理策略仍然停滞在事后分析阶段.由于非系统性风险能够通过充分的分散化而消除,我们关注于系统性边际风险的控制.为使提出的投资组合模型能够易于对系统性风险进行管理,我们将因子模型的思想引入边际风险的投资模型中.提出的边际风险控制模型是一个具有一些特殊的结构非凸优化问题.我们通过利用问题的特殊结构,提出一种分枝限界算法用于全局最优策略的求解.特别的,提出的分枝限界算法的分枝变量不依赖与总资产数目n,仅与因子的个数m及受边际风险限制的资产的个数C有关,特别适用于n远大于m+C的情况.我们通过实证分析对提出的模型与均值方差模型进行了比较,并通过数值算例证实本文分枝限界算法的有效性.
目前对投资组合风险的控制大多讨论的是对单个资产或者整体组合的风险的控制,事实上对投资组合的风险因子进行控制可以让投资者或基金经理对组合的风险结构有更加清楚的了解,根据自身对各类风险因素的承受能力更有效的进行投资管理.带有因子风险约束的投资模型很多时候是一个非凸规划问题,模型的求解难度很大；我们通过对风险因子的单位风险回报率进行控制,研究均值方差框架下带因子风险控制的投资模型.得到一个简单的二次规划投资模型,并通过实证分析对因子风险控制的投资策略与均值方差模型进行比较,结果表明对因子的单位风险回报率进行控制可以均匀投资组合中各风险因子的分布,并提高投资组合的市场表现.
全局优化问题研究的是存在多个局部极值点的优化问题,是一类非常难的优化问题,目前尚未存在非常有效的算法.事实上局部优化中的一些特殊问题也是非常棘手的,例如奇异解的极小化问题.传统的牛顿算法对无约束的凸极小化问题是非常有效的,其显著优点是算法的局部二次收敛性,但若极小化问题存在奇异解,牛顿算法的收敛性将降为线性.我们研究了带奇异解的凸函数极小化问题,提出一种截断正则化牛顿算法进行求解,并在一定的条件下证明了算法的全局收敛性和局部二次收敛性.数值结果表明截断正则化牛顿法对奇异解的凸极小化问题是非常有效的,且问题规模的增大时优势更明显.
In the thesis, we study the numerical methods for global optimization and its applications in portfolio selection. We also study the numerical method for solving convex minimization with singular solutions.
First, we consider to find a global solution of minimizing a nonconvex function with box constrains. To solve the problem, we propose a new simplex mechanism for the local search stage of global optimization. We show that when the objective function is a concave quadratic function, the reflection of the simplex mechanism is done along a descent direction. If the objective function is concave but not necessary quadratic, the mechanism always reflect a "worse" vertex with respective to a "better" vertex. To avoid the algorithm being trapped in a local optimizer, we combined the simplex mechanism with simulated annealing and introduced the idea of non-monotone line search in our algorithm to propose a new SA-Simplex method. Numerical results show that as a local explorer in global optimization, our simplex mechanism is reasonable and practically effective.
The portfolio selection theory was founded by Markowtiz in 1952. It is a new starting point of modern finance. Its most important view is to use the variance or the standard variance of the return of the portfolio to measure portfolio risk. Portfolio theory has made great progress in the past 50 years both in theory and practice, and the optimization theory has becomes an important tool in the solution of those models. There are two basic strategies in portfolio management:active and passive. Most fund managers inclines to passive strategies because they think that in an efficient market, it is hardly beat the market for a long period. Recently, passive strategies have received much attention. It is estimated that there are in total over 1012 dollars passively invested in index funds in the USA market [1]. Index tracking is a popular form of passive fund management, which consists in reproducing the performance of a stock market index by investing in a subset of stocks included in the index. There are some realistic constraints that should be posed on the tracking portfolio, for example, no short sell, constraints on portfolio positions, the number of assets that allowed to track the index, transaction cost constraints and so on. In this thesis, we consider index tracking with realistic constraints. The objective function of tracking model is very complex nonconvex minimization problem. To solve the problem, we propose a convex approximation to it, which is a mix-integer quadratic programming. Under certain conditions, we prove that the objective function value of the approximation problem can be sufficiently close to the value of the original objective function near the optimal portfolio. We then propose a simple heuristic to find its global optimizer. Numeri-cal results show that our approximation works well and the heuristic is simple and efficient.
The most important thing in portfolio is how to diversify risk as well as possi-ble. In plain language, the investor should not put all eggs in one basket. Marginal risk can be used to judge the risk contribution of a single asset, and could be used as a measure of portfolio diversification. Recently, marginal risk control has at-tracted increasing attention. However, as portfolio selection with marginal risk control always leads to a non-convex optimization problem, most of the related work dwells on the stage of ex post analysis. As the non-systematic risk could be elimiated through diversification, we focus on the management of systematic risk. To get a formulation that could be useful for management of systematic risk, we introduce the idea of factor models in our portfolio selection model. Our portfolio model is a non-convex programming but with some special structures. By ex-ploiting the special structure of the problems, we propose an efficient branch and bound method for its global optimizer. Especially, our branching variables are only related to the number of assets that are restricted with marginal risk constraints (m) and the number of market factors (C). Our branch an bound method is very efficient when the number of assets (n) is much less than m+C. We compare our proposed portfolio model with mean variance model through empirical studies and test the efficient of our proposed branch and bound method through numerical experiments.
Currently, most of risk control in portfolio management refers to the risk re-striction of certain assets in portfolio. In fact, by properly examining the market risk factors that derive the return of the portfolio, we could control the portfolio risk profile more efficiently according to the tolerance level for some specific risk. In general, factor risk constrained portfolio selection problems lead to nonconvex optimization problems and are generally very hard to solve. We consider the port-folio selection problems with factor risk constraints in another way. Specifically, we control the market factors'risk-reward ratio and derive a quadratic programming model, which could be efficiently solve by some standard software. We compare our proposed portfolio model with the mean variance model through empirical analysis. The results show that by property control the market factors'risk-reward ratio, we could make the distribution of risk more evenly, and improve the performance of our managed portfolio as well.
Global optimization refers to the problem with multiplier local optimizer. Traditional optimization methods are easily trapped in a local optimizer and fails when dealing with global optimization problems. Global optimization is a kind of very hard optimization problems and there exists no efficient algorithm yet. In fact, there are some very hard problems in local optimizations too, for exam-ple, convex minimization with singular solutions. Traditional Newton's method is very efficient for unconstrained convex minimization, and has the attractive local quadratic convergence property. However, if the problem has singular solutions, the convergence rate of Newton's method may reduce to be linear. In the last part of this thesis, we propose a truncated regularized Newton method to minimize a convex function with singular solutions. Under appropriate conditions, we prove the global convergence and local quadratic convergence of the proposed method. Numerical results show that our proposed method has potential advantages for solving large scale problems with singular solutions.
This dissertation is supported by the National Natural Science Foundation of China (10771057) and the Major Project of Ministry of Education of China granted (309023).
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[44]Dembo R S, Steihaug T. Truncated-Newton algorithms for large-scale unconstrained optimization. Mathematical Programming,1983,26:190-212
[45]Nash S G, Sofer A. Assessing a search direction within a truncated-Newton method. Operations Research Letters,1990,9:219-221
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[49]Hedar A R. Studies on Metaheuristics for Continuous Global Optimization Prob-lems:[dissertation].2004
[50]Vanderbilt D, Louie S G. A Monte Carlo simulated annealing approach to opti-mization over continuous variables. Journal of Computational Physics,1984,56: 259-271
[51]Press W H, Teukolsky S A. Simulated annealing optimization over continuous spaces. Computers in Physics,1991,5:426-429
[52]Nelder J A, Mead R. A simplex method for function minimization, The Computer Journal,1965,7:308-313
[53]Cardoso M F, Salcedo R L, Azevedo S F D, et al. A simulated annealing approach to the solution of minlp problems. Computers and Chemical Engineering,1997,21: 1349-1364
[54]Cardoso M F, Salcedo R L, Azevedo S F D. The simplex-simulated annealing ap-proach to continuous non-linear optimization. Computers and Chemical Engineer-ing,1996,20(9):1065-1080
[55]Kvasnicka V, Pospichal J. A hybrid of simplex method and simulated annealing. Chemometrics and Intelligent Laboratory Systems,1997,39:161-173
[56]Hedar A R, Fukushima M. Hybrid simulated annealing and direct search method for nonlinear unconstrained global optimization. Optimization Methods and Software, 2002,17:891-912
[57]Kelley C T. Detection and remediation of stagnation in the Nelder-Mead algorithm using a sufficient decrease condition. SIAM Journal on Optimization,1999,10:43-55
[58]Kelley C T. Iterative Methods for Optimization. SIAM Frontiers in Applied Math-ematics, Philadelphia, PA.1999,8
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[60]Dennis J E, Torczon V. Direct search methods on parallel machines. SIAM Journal on Optimization,1991,1:448-474
[61]Grippo L, Lampariello F, Lucidi S. A nonmonotone line search technique for new-ton's method. SIAM Journal on Numerical Analysis,1986,23:707-716
[62]Beasley J E, Meade N, Chang T J. An evolutionary heuristc for the index tracing problem. European Journal of Operation Research,2003,148(3):621-643
[63]Gill M, Kellezi E. The threshold heuristic for index tracking. Financil Engineering E-Commerce and Supply Chain Kluwer Applied Optimization Series,2001,1-18
[64]Torrubiano R R, Suarez A. A hybrid optimization approach to index tracking. Anneals of Operation Research,2009,166:57-71
[65]Canakgoz N A, Beasley J E. Mixed-integer programming approaches for index track-ing and enhanced indexation. European Journal of Operational Research,2009,196: 384-399
[66]Shapcott J. Index tracking:genetic algorithms for investment portfolio selection. Edinburgh, Parallel Computing Centre,1992
[67]Mansini R, Speranza M G. Heuristic algorithms for the portfolio selection problem with minimum transaction lots. European Journal of Operation Reasearch,1999, 114:219-233
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[69]Laarhoven P J, Aarts E H. Simulated Annealing:Theory and Applications. Dor-drecht:D. Reidel Publishing Company,1987
[70]Perold A F, Sharpe W F. Dynamic strategies for asset allocation. Financial analysis Journal,1998,17-27
[71]Chang T J, Meade N, Beasley J E, et al. Heuristics for cardinality constrained portfolio optimisation. Computers and Operations Research,2000,27:1271-1302
[72]Browne S. Beating a moving target:Optimal portfolio strategies for o utperforming a stochastic benchmark. Finance and Stochastics,1999,3:275-294
[73]Fletcher R. Ageneral quadratic programming algorithm. Journal of the Institute ofMathematics and its Applications,1971,7:76-91
[74]Gill P E, Murray W, Saunders M A, et al. User's guide for SOL/QPSOL. Technical Report SOL, Department of Operations Research, StanfordUniversity, Stanford, California,1984,84-86
[75]Dantzig G B. Linear Programming and Extensions. Princeton:Princeton University Press,1963.
[76]Beasley J E. OR-Library:distributing test problems by electronic mail. Journal of the Operational Research Society,1990,41(11):1069-1072
[77]Zhu S S, Li D, Sun X L. Portfolio selection with marginal risk control, Journal of Computational Finance,2009 (accepted)
[78]Philippe D. Value at risk:The New Benchmark for Controlling Market Risk. Chicago:Irwin Professional Publishing,1996
[79]Kurth A, Tasche D. Credit risk contributions to value-at-risk and expected shortfall. Risk,2003,16:84-88
[80]Sahinidis N V. BARON:a general purpose global optimization software package. Journal of Global Optimization,1996,8:201-205
[81]Burmeister E, Ross S A. Using macroeconomic factors to control portfolio risk. Technical report, Duke University,2003
[82]Sharpe W F. Captial asset prices:A theory of market equilibrium under conditions of risk. Journal of Finance,1964,19:425-442
[83]Khayyal F A, Laren C, Voorhis TV. A relaxation method for nonconvex quadrati-cally constraine quadratic programs. Journal of Global optimization,1995,6:215-230
[84]Audet C, Hansen P, Jaumard B, Savard G. A branch and cut algorithm for non-convex quadratically constrained quadratica programming. Mathematical Program-ming, Seriers A,2000,87:131-152
[85]Alizadeh F, Gldfarb D. Second-order cone programming. Mathmatical Program-ming, Seriers B,2003,95:3-51
[86]Lobo M S, Vandenberghe L, Boyd S, Lebret H. Applications of second-order cone programming. Linear Algebra and its applications,1998,284:193-228
[87]Nesterov Y, Nemirovski A. Interior Point Polynomial Methods in Convex Program-ming:Theory and Application. society for industrial and applied mathematics, Philadelphia,1994
[88]Nesterov Y E, Todd M J. Self-scaled barriers and interior-point methods for convex programming. Mathmatic of Operation research,1997,22:1-42
[89]Nesterov Y E, Todd M J. Primal-Dual Interior-Point Methods for Self-Scaled Cones. SIAM Journal on Optimization,1998,8:324-364
[90]Lobo M S, Vandenberghe L, Boyd S. SOCP:Software for second-order cone pro-gramming. Information Systems Laboratory, Standford University,1997
[91]Sturm J F. Using SeDuMi 1.02, a matlab toolbox for optimization over symmetric cones. Optimization Methods and Software,1999,11:625-653
[92]Dan H, Yamashita N, Fukushima M. Convergence properties of the inexact Levenberg-Marquardt method under local error bound conditions. Optimization Methods and Software,2002,17:605-626
[93]Fasano G, Lampariello F, Sciandrone M. Truncated nonmontone Gauss-Newton method for large-scale nonlinear least-squares problems. Computational Optimoiza-tion and Applications.2006,34:343-358
[94]Luksan L, VIcek J. Sparse and Partially Separable Test Problems for Unconstrained and Equality Constrained Optimization. Academy of the Czech Republic, Institude of Computer Science, Technical Report no.767:1999
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