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Biconvex programming (or inequality constrained biconvex optimization) is an important model in solving many engineering optimization problems in areas like machine learning and signal and information processing. In this paper, the partial exactness of the partial optimum for the penalty function of biconvex programming is studied. The penalty function is partially exact if the partial Karush–Kuhn–Tucker (KKT) condition is true. The sufficient and necessary partially local stability condition used to determine whether the penalty function is partially exact for a partial optimum solution is also proven. Based on the penalty function, an algorithm is presented for finding a partial optimum solution to an inequality constrained biconvex optimization, and its convergence is proven under some conditions.
Min Jiang; Zhiqing Meng; Rui Shen. Partial Exactness for the Penalty Function of Biconvex Programming. Entropy 2021, 23, 132 .
AMA StyleMin Jiang, Zhiqing Meng, Rui Shen. Partial Exactness for the Penalty Function of Biconvex Programming. Entropy. 2021; 23 (2):132.
Chicago/Turabian StyleMin Jiang; Zhiqing Meng; Rui Shen. 2021. "Partial Exactness for the Penalty Function of Biconvex Programming." Entropy 23, no. 2: 132.
This paper introduces a health index for measuring the health level of societies during the lockdown era, i. e., for the period from March 21, 2020 to April 7, 2020. For this purpose, individual-level survey data from the Global Behaviors and Perceptions in the COVID-19 Pandemic dataset are considered. We focus on cases in the United States and the United Kingdom, and the data come from 11,270 and 11,459 respondents, respectively. We then use unit root tests with structural breaks to examine whether COVID-19-related economic shocks significantly affect the health levels of the United States and the United Kingdom. The empirical results indicate that the health levels in the United States and the United Kingdom are not significantly affected by the COVID-19-related economic shocks. The evidence shows that government directives (such as lockdowns) did not significantly change the health levels of these societies.
Baozhen Jiang; Zhaohui Liu; Rui Shen; Leping Huang; Yang Tong; Yuxin Xia. Have COVID-19-Related Economic Shocks Affected the Health Levels of Individuals in the United States and the United Kingdom? Frontiers in Public Health 2020, 8, 611325 .
AMA StyleBaozhen Jiang, Zhaohui Liu, Rui Shen, Leping Huang, Yang Tong, Yuxin Xia. Have COVID-19-Related Economic Shocks Affected the Health Levels of Individuals in the United States and the United Kingdom? Frontiers in Public Health. 2020; 8 ():611325.
Chicago/Turabian StyleBaozhen Jiang; Zhaohui Liu; Rui Shen; Leping Huang; Yang Tong; Yuxin Xia. 2020. "Have COVID-19-Related Economic Shocks Affected the Health Levels of Individuals in the United States and the United Kingdom?" Frontiers in Public Health 8, no. : 611325.
Uncertainties from retail price-fluctuation sales as well as constraints from suppliers make it difficult for retailers to place accurate orders, which have a great impact on the whole supply chain. Thus, this paper studies a supply chain ordering problem for two-level price-fluctuation sales and establishes a bilevel programming model by Copula function measuring the correlation between price and demand. The optimal order quantity is derived by transforming the bilevel programming model into a single-level model. An algorithm is given for solving the approximate optimal order quantity for the discrete model, and the convergence of the algorithm is proved. The results show that the approximate optimal order quantity decreases with the increase in the uncertainties of price and demand. Supply chain members should sell more products at the normal level, thereby increasing profits of each member in the supply chain under two-level price-fluctuation sales.
MinChao Zheng; Zhiqing Meng; Rui Shen. Research on Two-Level Price-Fluctuation Supply Chain Ordering Strategy Problem. Discrete Dynamics in Nature and Society 2020, 2020, 1 -14.
AMA StyleMinChao Zheng, Zhiqing Meng, Rui Shen. Research on Two-Level Price-Fluctuation Supply Chain Ordering Strategy Problem. Discrete Dynamics in Nature and Society. 2020; 2020 ():1-14.
Chicago/Turabian StyleMinChao Zheng; Zhiqing Meng; Rui Shen. 2020. "Research on Two-Level Price-Fluctuation Supply Chain Ordering Strategy Problem." Discrete Dynamics in Nature and Society 2020, no. : 1-14.
This paper focuses on the role of local governments in the development of tourism in China by examining 30 Chinese provinces from 2000 to 2018. The results of empirical research show that fiscal decentralization in China provides local governments with incentives for the development of high pollution industries and of large state-owned enterprises, which do not help the sustainable development of tourism. In addition, there is an “inverted U-shaped” relationship between pollution level and tourism development. Although the growth of China’s tourism industry is pollution-based currently, tourism revenue is considered to decline once a threshold is reached. The competition from local governments for foreign investment is conducive to the improvement of environmental quality and increase in tourism revenue. Based on this, we have proposed a series of sustainable tourism development measures.
Shaolong Zeng; Lingyun Gao; Rui Shen; Yingying Ma; Haiping Li. Fiscal Decentralization, Pollution and China’s Tourism Revenue. Sustainability 2020, 12, 1925 .
AMA StyleShaolong Zeng, Lingyun Gao, Rui Shen, Yingying Ma, Haiping Li. Fiscal Decentralization, Pollution and China’s Tourism Revenue. Sustainability. 2020; 12 (5):1925.
Chicago/Turabian StyleShaolong Zeng; Lingyun Gao; Rui Shen; Yingying Ma; Haiping Li. 2020. "Fiscal Decentralization, Pollution and China’s Tourism Revenue." Sustainability 12, no. 5: 1925.
To stimulate purchases from consumers, retailers nowadays use the multiple retail prices strategy (MRPS), i.e., selling the products at multiple prices simultaneously. The paper extends the current newsboy model and proposes an optimal ordering model for MRPS corresponding to uncertain consumer demands. The Lagrangian multiplier method is applied to solve the problem, and an algorithm for finding the approximate optimal total order quantity is designed. Numerical results show that MRPS is better than the single retail price strategy (SRPS). It further reveals that when there is an order quantity constraint, the retailer needs to control the number of retail prices; that is, retailer’s MRPS is affected by order quantity constraint; sensitivity analysis demonstrates that MRPS is also affected by the price discount coefficient in the case of no order quantity constraint while it is not affected by demand volatility. The research work provides some useful managerial inspirations for retailers.
Yunzhi Mu; Zhiqing Meng; Rui Shen; Gengui Zhou; Leiyan Xu; MinChao Zheng. Optimal Ordering Strategy for Goods at Multiple Retail Prices under Simultaneous Sales. Discrete Dynamics in Nature and Society 2019, 2019, 1 -14.
AMA StyleYunzhi Mu, Zhiqing Meng, Rui Shen, Gengui Zhou, Leiyan Xu, MinChao Zheng. Optimal Ordering Strategy for Goods at Multiple Retail Prices under Simultaneous Sales. Discrete Dynamics in Nature and Society. 2019; 2019 ():1-14.
Chicago/Turabian StyleYunzhi Mu; Zhiqing Meng; Rui Shen; Gengui Zhou; Leiyan Xu; MinChao Zheng. 2019. "Optimal Ordering Strategy for Goods at Multiple Retail Prices under Simultaneous Sales." Discrete Dynamics in Nature and Society 2019, no. : 1-14.
This paper introduces a concession equilibrium solution without weighted aggregation operators to multiattribute group decision-making problems (in short MGDMPs). It is of practical significance for all decision-makers to find an optimal solution to MGDMPs or to sort out all candidate solutions to MGDMPs. It is proved that under certain conditions the optimal concession equilibrium solution does exist, and on this important result the optimal concession equilibrium solution is obtained by solving a single objective optimization problem. Moreover, the optimal concession equilibrium solution is equivalent to the robust optimal solution with the group weight aggregation under the worst weight condition. Finally, it is proved that the concession equilibrium solution is equivalent to a complete order, i.e. all candidate alternatives can be sorted by concession equilibrium solution. By defining the triangular fuzzy numbers of target concession value, the optimal concession equilibrium solution or the order of the alternative solutions can be obtained in the range of objective concession ambiguity. Numerical experiment shows that the solution can balance the evaluations of multiattribute group decision makers. This paper provides a new approach to solving multiattribute group decision-making problems.
Min Jiang; Rui Shen; Zhiqing Meng. A Concession Equilibrium Solution Method without Weighted Aggregation Operators for Multiattribute Group Decision-Making Problems. Discrete Dynamics in Nature and Society 2019, 2019, 1 -10.
AMA StyleMin Jiang, Rui Shen, Zhiqing Meng. A Concession Equilibrium Solution Method without Weighted Aggregation Operators for Multiattribute Group Decision-Making Problems. Discrete Dynamics in Nature and Society. 2019; 2019 ():1-10.
Chicago/Turabian StyleMin Jiang; Rui Shen; Zhiqing Meng. 2019. "A Concession Equilibrium Solution Method without Weighted Aggregation Operators for Multiattribute Group Decision-Making Problems." Discrete Dynamics in Nature and Society 2019, no. : 1-10.
This paper introduces an approach for group decision-making problems (GDMP) without weighted aggregation operators. This approach is more suitable for scenarios with infinite number of individuals. A mathematical model approach is established based on the new concept of s⁎-optimal concession equilibrium solution without weighted aggregation operators for group decision-making problems. It is of practical significance for all decision-makers (experts) to find an optimal solution or to sort out all the candidate solutions. We prove that the s⁎-optimal concession equilibrium solution is equivalent to solving a single objective optimization problem, and, under certain conditions, the s⁎-optimal equilibrium solution always exists. Moreover, it is proven that the s⁎-optimal concession equilibrium solution is equivalent to the robust optimal solution of the group weight aggregation and the optimal solution under the worst weighted aggregation operators.
Min Jiang; Zhiqing Meng; Rui Shen. Group Decision-Making Approach without Weighted Aggregation Operators. Discrete Dynamics in Nature and Society 2018, 2018, 1 -11.
AMA StyleMin Jiang, Zhiqing Meng, Rui Shen. Group Decision-Making Approach without Weighted Aggregation Operators. Discrete Dynamics in Nature and Society. 2018; 2018 ():1-11.
Chicago/Turabian StyleMin Jiang; Zhiqing Meng; Rui Shen. 2018. "Group Decision-Making Approach without Weighted Aggregation Operators." Discrete Dynamics in Nature and Society 2018, no. : 1-11.
In this paper, an algorithm of barrier objective penalty function for inequality constrained optimization is studied and a conception–the stability of barrier objective penalty function is presented. It is proved that an approximate optimal solution may be obtained by solving a barrier objective penalty function for inequality constrained optimization problem when the barrier objective penalty function is stable. Under some conditions, the stability of barrier objective penalty function is proved for convex programming. Specially, the logarithmic barrier function of convex programming is stable. Based on the barrier objective penalty function, an algorithm is developed for finding an approximate optimal solution to an inequality constrained optimization problem and its convergence is also proved under some conditions. Finally, numerical experiments show that the barrier objective penalty function algorithm has better convergence than the classical barrier function algorithm.
Rui Shen; Zhiqing Meng; Chuangyin Dang; Min Jiang. Algorithm of Barrier Objective Penalty Function. Numerical Functional Analysis and Optimization 2017, 38, 1473 -1489.
AMA StyleRui Shen, Zhiqing Meng, Chuangyin Dang, Min Jiang. Algorithm of Barrier Objective Penalty Function. Numerical Functional Analysis and Optimization. 2017; 38 (11):1473-1489.
Chicago/Turabian StyleRui Shen; Zhiqing Meng; Chuangyin Dang; Min Jiang. 2017. "Algorithm of Barrier Objective Penalty Function." Numerical Functional Analysis and Optimization 38, no. 11: 1473-1489.
The paper develops a multi-product supply chain model where supplyproduction- sale integration is considered and the worst-case conditional value at risk (WCVaR) model is applied as the risk measure, and also provides a coordination strategy to minimize the supply chain risk. First, by analyzing the source demand of market in the supply chain, three WCVaR models consisting of three tiers—the supplier, the manufacturer and the retailer in the supply chain are proposed to measure the market risk. Then, a risk coordination model is proposed to cover the whole supply chain including producing, order, inventory and sales. Finally, the numerical results show the efficiency of the model in mitigating risks. And we make a summary of supply chain risk management strategies.
Min Jiang; Rui Shen; Zhiqing Meng. WCVaR-Based Risk Coordination Model for Multi-Product Supply. Open Journal of Business and Management 2017, 05, 641 -652.
AMA StyleMin Jiang, Rui Shen, Zhiqing Meng. WCVaR-Based Risk Coordination Model for Multi-Product Supply. Open Journal of Business and Management. 2017; 05 (04):641-652.
Chicago/Turabian StyleMin Jiang; Rui Shen; Zhiqing Meng. 2017. "WCVaR-Based Risk Coordination Model for Multi-Product Supply." Open Journal of Business and Management 05, no. 04: 641-652.
Augmented Lagrangian function is one of the most important tools used in solving some constrained optimization problems. In this article, we study an augmented Lagrangian objective penalty function and a modified augmented Lagrangian objective penalty function for inequality constrained optimization problems. First, we prove the dual properties of the augmented Lagrangian objective penalty function, which are at least as good as the traditional Lagrangian function's. Under some conditions, the saddle point of the augmented Lagrangian objective penalty function satisfies the first-order Karush-Kuhn-Tucker condition. This is especially so when the Karush-Kuhn-Tucker condition holds for convex programming of its saddle point existence. Second, we prove the dual properties of the modified augmented Lagrangian objective penalty function. For a global optimal solution, when the exactness of the modified augmented Lagrangian objective penalty function holds, its saddle point exists. The sufficient and necessary stability conditions used to determine whether the modified augmented Lagrangian objective penalty function is exact for a global solution is proved. Based on the modified augmented Lagrangian objective penalty function, an algorithm is developed to find a global solution to an inequality constrained optimization problem, and its global convergence is also proved under some conditions. Furthermore, the sufficient and necessary calmness condition on the exactness of the modified augmented Lagrangian objective penalty function is proved for a local solution. An algorithm is presented in finding a local solution, with its convergence proved under some conditions.
Zhiqing Meng; Rui Shen; Chuangyin Dang; Min Jiang. Augmented Lagrangian Objective Penalty Function. Numerical Functional Analysis and Optimization 2015, 36, 1471 -1492.
AMA StyleZhiqing Meng, Rui Shen, Chuangyin Dang, Min Jiang. Augmented Lagrangian Objective Penalty Function. Numerical Functional Analysis and Optimization. 2015; 36 (11):1471-1492.
Chicago/Turabian StyleZhiqing Meng; Rui Shen; Chuangyin Dang; Min Jiang. 2015. "Augmented Lagrangian Objective Penalty Function." Numerical Functional Analysis and Optimization 36, no. 11: 1471-1492.
Min Jiang; Rui Shen; Xinsheng Xu; Zhiqing Meng. Second-Order Smoothing Objective Penalty Function for Constrained Optimization Problems. Numerical Functional Analysis and Optimization 2014, 35, 294 -309.
AMA StyleMin Jiang, Rui Shen, Xinsheng Xu, Zhiqing Meng. Second-Order Smoothing Objective Penalty Function for Constrained Optimization Problems. Numerical Functional Analysis and Optimization. 2014; 35 (3):294-309.
Chicago/Turabian StyleMin Jiang; Rui Shen; Xinsheng Xu; Zhiqing Meng. 2014. "Second-Order Smoothing Objective Penalty Function for Constrained Optimization Problems." Numerical Functional Analysis and Optimization 35, no. 3: 294-309.
Zhiqing Meng; Rui Shen; Min Jiang. A Penalty Function Algorithm with Objective Parameters and Constraint Penalty Parameter for Multi-Objective Programming. American Journal of Operations Research 2014, 04, 331 -339.
AMA StyleZhiqing Meng, Rui Shen, Min Jiang. A Penalty Function Algorithm with Objective Parameters and Constraint Penalty Parameter for Multi-Objective Programming. American Journal of Operations Research. 2014; 04 (06):331-339.
Chicago/Turabian StyleZhiqing Meng; Rui Shen; Min Jiang. 2014. "A Penalty Function Algorithm with Objective Parameters and Constraint Penalty Parameter for Multi-Objective Programming." American Journal of Operations Research 04, no. 06: 331-339.
For the problem of supply chain management, the existing literature mainly focuses on the research of the single-stage supply chain or the two-stage supply chain that consists of a manufacturer and a retailer. To our best knowledge, little attention has been paid to the study of a more extensive supply chain that consists of a material supplier, a manufacturer and a retailer, which is a more practical and interesting case. Therefore, based on the Conditional Value-at-Risk (CVaR) measure of risk management, this paper proposes a tri-level programming model for the three-stage supply chain management. In this model, the material supplier and the manufacturer maximize their own profit while the retailer maximize his/her CVaR of expected profit. Further, we show that the proposed tri-level programming model can be transferred into a bilevel programming model, which can be solved by the existing methods. Numerical results show that the proposed model is efficient for improving the risk management of the three-stage supply chain.
Xinsheng Xu; Zhiqing Meng; Rui Shen. A tri-level programming model based on Conditional Value-at-Risk for three-stage supply chain management. Computers & Industrial Engineering 2013, 66, 470 -475.
AMA StyleXinsheng Xu, Zhiqing Meng, Rui Shen. A tri-level programming model based on Conditional Value-at-Risk for three-stage supply chain management. Computers & Industrial Engineering. 2013; 66 (2):470-475.
Chicago/Turabian StyleXinsheng Xu; Zhiqing Meng; Rui Shen. 2013. "A tri-level programming model based on Conditional Value-at-Risk for three-stage supply chain management." Computers & Industrial Engineering 66, no. 2: 470-475.
Xinsheng Xu; Zhiqing Meng; Rui Shen. A cooperation model based on CVaR measure for a two-stage supply chain. International Journal of Systems Science 2013, 46, 1865 -1873.
AMA StyleXinsheng Xu, Zhiqing Meng, Rui Shen. A cooperation model based on CVaR measure for a two-stage supply chain. International Journal of Systems Science. 2013; 46 (10):1865-1873.
Chicago/Turabian StyleXinsheng Xu; Zhiqing Meng; Rui Shen. 2013. "A cooperation model based on CVaR measure for a two-stage supply chain." International Journal of Systems Science 46, no. 10: 1865-1873.
Risk-averse suppliers optimal pricing strategies in two-stage supply chains under competitive environment are discussed. The suppliers in this paper focus more on losses as compared to profits, and they care their long-term relationship with their customers. We introduce for the suppliers a loss function, which covers both current loss and future loss. The optimal wholesale price is solved under situations of risk neutral, risk averse, and a combination of minimizing loss and controlling risk, respectively. Besides, some properties of and relations among these optimal wholesale prices are given as well. A numerical example is given to illustrate the performance of the proposed method.
Rui Shen; Zhiqing Meng; Xinsheng Xu; Min Jiang. Risk-Averse Suppliers’ Optimal Pricing Strategies in a Two-Stage Supply Chain. Discrete Dynamics in Nature and Society 2013, 2013, 1 -11.
AMA StyleRui Shen, Zhiqing Meng, Xinsheng Xu, Min Jiang. Risk-Averse Suppliers’ Optimal Pricing Strategies in a Two-Stage Supply Chain. Discrete Dynamics in Nature and Society. 2013; 2013 ():1-11.
Chicago/Turabian StyleRui Shen; Zhiqing Meng; Xinsheng Xu; Min Jiang. 2013. "Risk-Averse Suppliers’ Optimal Pricing Strategies in a Two-Stage Supply Chain." Discrete Dynamics in Nature and Society 2013, no. : 1-11.
In this article, an objective penalty function algorithm based on multi-parameters is proposed for solving bilevel programming convex to the lower level problem. Under some conditions, an optimal solution to a single-objective programming defined by the objective penalty function is proved to be an optimal solution to the original bilevel programming. We prove that the algorithm is convergent under some conditions. Numerical results show that the algorithm is efficient for solving some bilevel programming convex to the lower level problem.
Zhiqing Meng; Xinsheng Xu; Rui Shen; Min Jiang. An Objective Penalty Function Algorithm for Bilevel Programming Based on Multi-Parameters. Numerical Functional Analysis and Optimization 2013, 34, 207 -219.
AMA StyleZhiqing Meng, Xinsheng Xu, Rui Shen, Min Jiang. An Objective Penalty Function Algorithm for Bilevel Programming Based on Multi-Parameters. Numerical Functional Analysis and Optimization. 2013; 34 (2):207-219.
Chicago/Turabian StyleZhiqing Meng; Xinsheng Xu; Rui Shen; Min Jiang. 2013. "An Objective Penalty Function Algorithm for Bilevel Programming Based on Multi-Parameters." Numerical Functional Analysis and Optimization 34, no. 2: 207-219.
This paper introduces a second-order differentiability smoothing technique to the classical l 1 exact penalty function for constrained optimization problems(COP). Error estimations among the optimal objective values of the nonsmooth penalty problem, the smoothed penalty problem and the original optimization problem are obtained. Based on the smoothed problem, an algorithm for solving COP is proposed and some preliminary numerical results indicate that the algorithm is quite promising.
Xinsheng Xu; Zhiqing Meng; Jianwu Sun; Liguo Huang; Rui Shen. A second-order smooth penalty function algorithm for constrained optimization problems. Computational Optimization and Applications 2012, 55, 155 -172.
AMA StyleXinsheng Xu, Zhiqing Meng, Jianwu Sun, Liguo Huang, Rui Shen. A second-order smooth penalty function algorithm for constrained optimization problems. Computational Optimization and Applications. 2012; 55 (1):155-172.
Chicago/Turabian StyleXinsheng Xu; Zhiqing Meng; Jianwu Sun; Liguo Huang; Rui Shen. 2012. "A second-order smooth penalty function algorithm for constrained optimization problems." Computational Optimization and Applications 55, no. 1: 155-172.
This paper extends an existing cooperative multi-objective interaction programming problem with interaction constraint for two players (or two agents). First, we define an s-optimal joint solution with weight vector to multi-objective interaction programming problem with interaction constraint for two players and get some properties of it. It is proved that the s-optimal joint solution with weight vector to the multi-objective interaction programming problem can be obtained by solving a corresponding mathematical programming problem. Then, we define another s-optimal joint solution with weight value to multi-objective interaction programming problem with interaction constraint for two players and get some of its properties. It is proved that the s-optimal joint solution with weight vector to multi-objective interaction programming problem can be obtained by solving a corresponding mathematical programming problem. Finally, we build a pricing multi-objective interaction programming model for a bi-level supply chain. Numerical results show that the interaction programming pricing model is better than Stackelberg pricing model and the joint pricing model.
Min Jiang; Zhiqing Meng; Xinsheng Xu; Rui Shen; Gengui Zhou. Multiobjective Interaction Programming Problem with Interaction Constraint for Two Players. Mathematical Problems in Engineering 2012, 2012, 1 -14.
AMA StyleMin Jiang, Zhiqing Meng, Xinsheng Xu, Rui Shen, Gengui Zhou. Multiobjective Interaction Programming Problem with Interaction Constraint for Two Players. Mathematical Problems in Engineering. 2012; 2012 (4):1-14.
Chicago/Turabian StyleMin Jiang; Zhiqing Meng; Xinsheng Xu; Rui Shen; Gengui Zhou. 2012. "Multiobjective Interaction Programming Problem with Interaction Constraint for Two Players." Mathematical Problems in Engineering 2012, no. 4: 1-14.
Penalty methods are very efficient in finding an optimal solution to constrained optimization problems. In this paper, we present an objective penalty function with two penalty parameters for inequality constrained bilevel programming under the convexity assumption to the lower level problem. Under some conditions, an optimal solution to a bilevel programming defined by the objective penalty function is proved to be an optimal solution to the original bilevel programming. Moreover, based on the objective penalty function, an algorithm is developed to obtain an optimal solution to the original bilevel programming, with its convergence proved under some conditions.
Zhiqing Meng; Chuangyin Dang; Rui Shen; Ming Jiang. An Objective Penalty Function of Bilevel Programming. Journal of Optimization Theory and Applications 2011, 153, 377 -387.
AMA StyleZhiqing Meng, Chuangyin Dang, Rui Shen, Ming Jiang. An Objective Penalty Function of Bilevel Programming. Journal of Optimization Theory and Applications. 2011; 153 (2):377-387.
Chicago/Turabian StyleZhiqing Meng; Chuangyin Dang; Rui Shen; Ming Jiang. 2011. "An Objective Penalty Function of Bilevel Programming." Journal of Optimization Theory and Applications 153, no. 2: 377-387.
The paper introduces a smoothing technique for a lower order penalty function for constrained optimization problems (COP). It is proved that the optimal solution to the smoothed penalty optimization problem is a ϵ2-approximate optimal solution to the original optimization problem under some mild assumptions. Based on the smoothed penalty function, an algorithm for solving COP is proposed and some numerical examples are given.
Xinsheng Xu; Zhiqing Meng; Jianwu Sun; Rui Shen. A penalty function method based on smoothing lower order penalty function. Journal of Computational and Applied Mathematics 2011, 235, 4047 -4058.
AMA StyleXinsheng Xu, Zhiqing Meng, Jianwu Sun, Rui Shen. A penalty function method based on smoothing lower order penalty function. Journal of Computational and Applied Mathematics. 2011; 235 (14):4047-4058.
Chicago/Turabian StyleXinsheng Xu; Zhiqing Meng; Jianwu Sun; Rui Shen. 2011. "A penalty function method based on smoothing lower order penalty function." Journal of Computational and Applied Mathematics 235, no. 14: 4047-4058.