Table of Content

15 June 2012, Volume 16 Issue 2

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Original Articles

Convex function| sub-b-convex set|  sub-b-convex function| pseudo-sub-b-convex function| optimality conditions

CHAO Miantao, JIAN Jinbao , LIANG Dongying

2012, 16(2):  1-8. 

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The paper studied a new generalized convex function which called sub-b-convex function, and introduced a new concept of sub-b-convex set. The basic roperties of sub-b-convex functions were discussed in general case and differentiable case, respectively. And, obtained the sufficient conditions that the sub-b-convex function become quasi-convex function or pseudo-convex function. Furthermore, the sufficient conditions of optimality for unconstrained and inequality constrained programming were obtained under the sub-b-convexity.

 Modified lower order penalty functions   based on quadratic smoothing approximation

Bai-Fu-Sheng, LUO Xiao-Yan

2012, 16(2):  9-22. 

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In this paper, two function forms of quadratic smoothing approximation to the lower order exact penalty function are proposed to generate modified smooth penalty functions for inequality-constrained optimization problems. It is shown that under certain conditions, any global minimizer of the modified smooth penalty problem is a global minimizer to the original constrained optimization problem when the penalty parameter is sufficiently large. Two numerical examples are given to show the effectiveness of the present smoothing scheme.

Some new families of Q-integral graphs

Wang-Li-Gong, CHEN Yan-Qing

2012, 16(2):  23-31. 

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 Let G be a simple graph. The matrix Q(G)=D(G)+A(G) denotes the signless Laplacian matrix of G, where D(G) and A(G) denote the diagonal matrix and the adjacency matrix of G respectively. A graph is called Q-integral if its signless Laplacian spectrum consists entirely of integers. In this paper, we firstly construct six infinite classes of nonregular Q-integral graphs from the known six smaller Q-integral graphs identified by Stani\'{c}.  Furthermore, we obtain large families of Q-integral graphs by  the Cartesian product of graphs. Finally, we obtain some regular Q-integral graphs.

Existence and Optimality of Accessible and Approximatable Minimizers for Global Optimization with Deviation Integral

Yao-Yi-Rong, AN Liu, CHEN Xi, ZHENG Quan

2012, 16(2):  32-40. 

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The concepts of robustness of sets and functions are  proposed in view of the theory of integral global minimization. These concepts are generalized, and global minimization of quasi and pseudo upper robust function is investigated in this paper. With the deviation integral optimality condition of global minimum, the existence of accessible minimizer of quasi upper functions and approximatable minimizer of pseudo upper robust function is examined..

nvironmental efficiency evaluation based on CAR-DEA approach

BIAN Yi-Wen, Sun-Yu-Feng

2012, 16(2):  41-50. 

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The existing environmental efficiency evaluation DEA models have not considered the issue that different decision making units (DMUs) may impose different  preferences on different outputs (desirable and undesirable outputs), e.g., different DMUs may value GDP (desirable output), waste water discharge and waste gas emissions differently.  This paper aims at addressing this problem. We present a DEA model incorporating desirable outputs and undesirable outputs, which can increase desirable outputs and decrease undesirable outputs simultaneously. To restrict DMUs' different  preferences on outputs, we then extend the model by incorporating multiple sets AR constraints based on  Context-Dependent Assurance Regions DEA approach (CAR-DEA) in recent DEA literature. An application of 30  regions in China with real data set is used to illustrate the proposed approach.

On the Smoothing of the Lower Order Exact Penalty Function for Inequality Constrained Optimization

Shu-jun Lian

2012, 16(2):  51-64. 

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In this paper, we propose a method to smooth the general lower order exact penalty function for inequality constrained optimization. Error estimations are obtained among the optimal objective function values of the smoothed penalty problem, of the nonsmooth penalty problem and of the original optimization problem. It is shown that under mild assumption, an approximate global solution of the original problem can be obtained by searching a global solution of the smoothed penalty problem. We develop an algorithm for solving the original optimization problem based on the smoothed penalty function and prove the convergence of the algorithm. Some numerical examples are given to illustrate the applicability of the present smoothing method.

Equilibrium balking strategy in the Geo/Geo/1(E,SV) queueing system

Weiqi Liu Yan Ma Jihong Li

2012, 16(2):  65-76. 

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This paper considered the equilibrium balking strategy of customers in the Geo/Geo/1 queue with single vacation. To the authors' knowledge, this is the first time that the vacation policy is introduced into the economics of the discrete-time queue. The customers decide for themselves whether to enter the system or balk based on a natural  reward-cost structure. Using the theory of the quasi-birth-death process and the standard approach for solving difference equation, we obtain the stationary distribution of the system and the mean sojourn time of an arriving customer. Then by introducing appropriate functions, we provide an algorithm to identify the equilibrium balking strategy. Furthermore, the resulting stationary system behavior is explored and the equilibrium social benefit is derived. Finally, we illustrate the effects of the parameters on the equilibrium behavior via numerical experiments.

 Optimal investment strategy with stochastic  volatility and  dynamic
VaR constraint

YI Bo, LI Zhong-Fei, ZENG Yan

2012, 16(2):  77-90. 

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This paper considers an optimal  portfolio choice problem under Stein-Stein stochastic volatility model and dynamic VaR constraint. The investor aims to maximize the expected power utility of the terminal wealth, and the financial  market consists of one risk-free asset and one risky asset whose price process is described by   Stein-Stein stochastic volatility model. At the same time, the investor hopes to limit the potential   risk over investment horizon by a dynamic VaR constraint. Adopting the stochastic dynamic programming    approach and Lagrange multiple method, we derive the closed-form expressions of the optimal strategy    as well as the optimal value function in a special case. Moreover, economic implications and numerical    analysis are proposed to illustrate the impacts of stochastic volatility and dynamic VaR constraint    on the investor's optimal strategy.

A Nonmonotoe Filter Method For Minimax Problems

ZHAO Qi, ZHANG Yan

2012, 16(2):  91-104. 

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In this paper, we present a modified trust-region filter algorithm to solve  the unconstrained minimax problem. The algorithm solves an SQP subproblem to acquire the attempted step. The filter technique is used to weigh the effect of the attempted step so as to avoid the use of a penalty function.  Based on the idea of the latest reference,  we use the Lagrange function as a merit function,  also combine it with the nonmonotone technique to improve the effect of the algorithm. Under some mild conditions, we prove the global convergence and superlinear local convergence. Numerical results show the effectiveness of the algorithm.

A new level-value estimation method for  global minimization

Lou-Ye, SUN Sheng, WU Ming-Nan

2012, 16(2):  105-114. 

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In this paper, a new level-value estimation method is proposed for solving global optimization problem. For this purpose,  we introduce a deviation function and study its properties.  Based the deviation function, we give a global optimality condition, and then propose a conceptual level-value estimation algorithm, and prove the global convergence of the proposed method.  For the implementation of the proposed method, we use the Monte-Carlo method with important sampling to compute the deviation, in which the sample density is updated by the main ideas of the cross-entropy method.  Some primary numerical
results show the validity of the proposed method.

On-line scheduling on a machine of two families with lookahead

YANG Su-Fang, LI Wen-Hua

2012, 16(2):  115-120. 

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We consider on-line scheduling on a parallel batching machine of two incompatible families of unit-length jobs with lookahead where the batch capacity is unbounded. In this paper online means that jobs arrive over time. The objective is to mininize the makespan. In unbounded parallel batch scheduling, an infinite capacity machine can process many jobs simultaneously as a batch, the processing time of a batch is equal to the maximum processing time of jobs in the batch. In the lookahead model, at a time instant t, an on-line algorithm can foresee the information of all jobs arriving in the time segment (t,t+\beta]. Incompatible job families means that jobs which belong to different families cannot be assigned to the same batch. For this problem, we provide a best possible online algorithm of competitive ratio 1+\alpha for 0\leq \beta <1, where \alpha is the positive root of the equation 2\alpha^{2}+(\beta +1)\alpha +\beta -2=0.

Matrix Algorithm for LP Problem Based on Partial Basic Variables

ZHOU Kang, CHEN Jin, QIU Jiang, JIE Zhi

2012, 16(2):  121-126. 

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Matrix algorithm for LP problem based on partial basic variables is put forward, which is based on a necessary and sufficient condition of optimal basic matrix.  At first the initial matrix of algorithm is transformed into the matrix meeting the requirements of   right-constant and test number; then the matrix is transformed into basic matrix meeting the requirements   of test number; finally the matrix is transformed into optimal basic matrix. Matrix algorithm has the  advantages of wide usage, small calculation scale, simplified calculation process, easy realization and    so on. The operation of finding inverse matrix is the key operation of matrix algorithm, so finding     inverse matrix problem of matrix algorithm is put forward, and a fast algorithm finding inverse matrix is discussed and given. The fast algorithm can find inverse matrix by utilizing inverse    matrix in the last iteration. The computational efficiency of the fast algorithm is higher than   that of direct inversion method.