Table of Content

15 June 2011, Volume 15 Issue 2

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Original Articles

Duality for Minmax Fractional Problems Involving Generalized Arcwise Connected Type I

JIA Ji-Hong, LI Ze-Min

2011, 15(2):  1-10. 

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This paper deals with  a minmax fractional problems in terms of the right derivative of the function with respect to an arc.  Under arcwise connected type I and
generalized arcwise connected type I assumptions, a dual model is proposed for the minmax fractional problems. Furthermore, weak duality theorem, strong duality theorem and strict converse duality theorem are established.

Criteria for Generalized \alpha \eta -Monotonicities

TANG Wan-Mei, Rong-Wei-Dong

2011, 15(2):  11-18. 

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If  $\alpha ~{\rm and}~\eta $ are  affine in the first argument and skew instead of Condition C, the following results are obtained: (i) if the gradient of a function is (strictly) $\alpha \eta $-pseudomonotone,  the function is (strictly) pseudo $\alpha \eta $-invex;  (ii) if the gradient of a function is quasi $\alpha \eta $-monotone,  the  function is quasi $\alpha \eta $-invex.

Some New Families of Integral Trees of Diameter Four

WANG Li-Gong, Zhang-Zheng

2011, 15(2):  19-27. 

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An integral graph is a graph of which all the eigenvalues of its　adjacency matrix are integers. This paper investigates itegral trees of diameter 4.　Many new classes of such integral trees are cnstructed ifinitely by solving some　certain Diophantine equations. These results generalize some results of Wang, Li
and Zhang (see Families of integral trees with diameters 4, 6 and 8, Discrete Applied　Mathematics, 2004, 136: 349-362).

A Norm-Relaxed Algorithm with Identification Function for General Constrained Optimization

JIAN Jin-Bao, WEI Xiao-Peng, ZENG Han-Jun, PAN Hua-Qin

2011, 15(2):  28-44. 

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In this paper, based on a semi-penalty function and　an identification function used to yield a  ``working set'', as well as the norm-relaxed SQP idea, a new
algorithm for solving a kind of optimization problems with nonlinear　equality and inequality constraints is proposed. At each iteration,　to yield the search directions the algorithm solves only one reduced quadratic program (QP) subproblem and a reduced system of linear equations. The proposed algorithm possesses global convergence and superlinear convergence under some mild assumptions without the strictly complementarity. Finally, some elementary numerical experiments are reported.

Saddle-Point Equilibrium of Quadratic Differential Game for Singular Bilinear Systems

ZHU Huai-Nian, ZHANG Cheng-Ke, BIN Ning

2011, 15(2):  45-52. 

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 This paper investigates  the saddle-point equilibrium of quadratic differential game for singular bilinear systems in the finite time. As it is complex to solve the problem, a reduced-order transformation is introduced in order to obtain the fast and slow subsystems, then, the optimal control is obtained by using the maximum principle. In the end, the simulation of a numerical example is presented to illustrate the correctness and effectiveness of the proposed algorithm.

 A Newton Method for  a  Nonsmooth Nonlinear　Complementarity　Problem

GAO Yan

2011, 15(2):  53-58. 

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This paper is devoted to a nonlinear　complementarity problem with nonsmooth data. The  nonlinear　complementarity problem is reformulated as a system of nonsmooth　equations. Then, a Newton method for solving the nonsmooth equations　is proposed. In each iteration of the Newton method, an element of　the B-differential of related functions, not nonlinear　complementarity function, is required. The superlinear convergence　is shown.

Two Models of Single Machine Scheduling with General Processing Time Functions

WANG Cheng-Fei, ZHANG Yu-Zhong, MIAO Cui-Xia

2011, 15(2):  59-67. 

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Two models of single machine scheduling  with general processing time functions are considered.  Job's processing time is assumed to be the sum of  basic processing time and a function of starting time or  a function of position.  For processing over starting time model, the makespan minimization problem and total completion time minimization problem are polynomially solvable under certain conditions, which have more generality compared with previous articles. For processing  over position  model, there are optimal schedules for makespan minimization problem and total completion time minimization problem.   A property of optimal schedule and a greedy algorithm for total weighted completion time minimization problem are presented.

Branch-and-Bound  Method for Reverse Convex Programming

Buheerdun, CHEN Guo-Qing, LIU Ju-Hong

2011, 15(2):  68-76. 

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This paper considers a convex programming with an additional reverse convex constraint. A kind of conical branch and bound method is developed and onvergence conditions are obtained.

Fritz-John Condition for a Class   of Nondifferntiable Optimization

PAN Shao-Rong, ZHANG Li-Wei

2011, 15(2):  77-84. 

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Based on the properties of the space of star-shaped sets, a class of star-shaped differentiable functions,whose directional derivatives are representable as a difference of two nonnegative positively homogeneous continuous functions, is defined. The necessary optimality condition of Fritz-John type for an 
optimization problem with star-shaped differentiable inequality constraints is given.

A Nonmonotone Feasible SQP Method for Nonlinear Complementarity Problem

WANG Hua

2011, 15(2):  85-94. 

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The nonlinear complementarity problem can be reformulated as a nonlinear programming. This paper proposes a feasible SQP method with nonmonotone line search, and obtains a feasible descent direction by  full use of the K-T point pair of a QP subproblem without other additional cost. A high-order direction is computed to overcome the Maratos effect. Instead of filter method,   a nonmonotone line search is used to obtain the step length. Under some suitable conditions, not including the strict complementary condition,  the global convergence of the algorithm is obtained. Some numerical results are also reported in this paper.

An Inexact Smoothing Algorithm for Linear Second-Order Cone Complementarity Problems

ZHANG Jie, XU Cheng-Xian, RUI Shao-Ping

2011, 15(2):  95-102. 

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An inexact smoothing algorithm for second-order cone complementarity problems is proposed under the framework of smoothing methods. It is proved that the proposed algorithm has global convergence property. Numerical experiments demonstrate that the algorithm is effective for large-scale problems.

Higher-Order Characterizations for Set-Valued Optimization on Strictly Maximal Efficient Solutions

YANG Yang, XU Yi-Hong, XIONG Wei-Zhi

2011, 15(2):  103-109. 

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The strict efficiency of set-valued optimization is considered in real normed spaces. By applying the properties of higher-order derivatives, higher-order type necessary optimality condition is established for a set-valued optimization problem whose constraint condition is determined by a fixed set to attain its strictly maximal efficient solution. When objective function is concave, with the properties of strictly maximal efficient point, sufficient optimality condition is also derived.

Analysis for Stationary Queue Length of the M/T-SPH/1 Queue

ZHANG Hong-Bo, FENG Ping-Hua

2011, 15(2):  110-118. 

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The M/T-SPH/1 queueing model is studied, and the probability generating function for the stationary queue length of the model is given based on the quasi-birth-and-death process and operator-geometric method. Furthermore,  distribution of queue length is not a PH distribution, while the tail of the distribution has the characteristic of geometric decay under certain conditions.

Identification for A Non-Smooth Distributed Parameter System and Its Optimality Conditions

BAI Yi-La, LV Wei

2011, 15(2):  119-126. 

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The parameter identification problem of the temperature field of the transformer is a piecewise smooth identification problem. Taking the flow velocity as the identification parameter,  consider a parameter identification problem of temperature field of the transformer, prove the existence of the optimal parameter and the necessary optimality conditions, and present a theoretical basis for the numerical simulation of temperature field of transformers.