Table of Content

15 March 2015, Volume 19 Issue 1

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Original Articles

Characterizations of two classes of cone generalized pseudoinvexity

TANG Liping, YANG Xinmin

2015, 19(1):  1-8. 

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In this paper, a class of nonsmooth vector optimization problem with constraints is considered. The concepts of FJ-pseudoinvexity-I(II) in the sense of cone are introduced;  Gordan's theorem over general cone domains is established; and then, FJ-pseudoinvexity-I(II) are characterized by the relationships between FJ vector critical points and the (weak) efficient solutions of nonsmooth vector optimization.

Existence and stability of weakly Pareto-Nash equilibrium points for information sets generalized multiobjective games

JIA Wensheng, XIANG Shuwen

2015, 19(1):  9-17. 

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In this paper, the information sets generalized multiobjective game model is established by introducing information sets to the multiobjective game, and it is presented that the generalized multiobjective game, the generalized $n$-person game, the n-person noncooperative finite game as some special cases of  the information sets generalized multiobjective game. Then the existence theorem of weakly Pareto-Nash equilibrium points for the information sets generalized multiobjective game is proved by using Fan-Glicksberg fixed point theorem. Furthermore,  from the viewpoint of the essential solution and strongly essential solution, we prove that the generic stability and the existence of strongly essential component for information sets generalized multiobjective games.

A non-interior-point continuous algorithm with superlinear convergence for second-order cone programming

ZENG Youfang, TANG Chunming

2015, 19(1):  18-30. 

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In this paper, based on a new smoothing function of the well-known nonsmooth vector-valued min-function, a non-interior-point continuous algorithm for second-order cone programming is presented. The features of this method are as follows: firstly, the starting point can be chosen arbitrarily; secondly, at each iteration, only one system of linear equations is performed for searching an improving direction; finally, global, strong and superlinear convergence are obtained without assumption of strict complementarity. The numerical results demonstrate the effectiveness of the algorithm.

A multi-objective method for supplier pre-selection of platform-based products

MU Lifeng, CAO Yan

2015, 19(1):  31-44. 

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To a manufacture company, supplier selection is always a crucial decision problem which directly impacts the final profits of a company. In previous studies, suppliers are selected only from the view of particular components rather than the whole product. In addition, supplier selection is usually conducted in the stage of product production after the stage of product design in traditional process of product development. However, taking into consideration of supplier selection in the early stage of product development can effectively avoid the shortage of appropriate suppliers. In this paper, a new multi-objective approach to preliminary selection of supplier based on products platform is proposed and a multi-objective optimization model with the objectives of minimizing outsourcing cost, supply risk and leading time of product family from the perspective of whole product is established to help decision-makers improve the overall scheme of product design in early time of product development. In addition, the optimization model takes the impact of component-sharing into consideration, which is common in the product platform, on the result of preliminary selection of suppliers. NSGA-II algorithm is adopted to solve the established optimization model, and a case study is provided to illustrate the rationality and effectiveness of the proposed method and algorithm.

A hybrid method for a system of nonlinear equations with bound constraints

LIU Yuanwen, OU Yigui

2015, 19(1):  45-56. 

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Based on the nonmonotone technique and the Levenberg-Marquard method, this paper presents a new hybrid method for solving a system of nonlinear equations with bound constraints. Under some reasonable assumptions, the proposed algorithm is proven to be globally convergent. Preliminary numerical results indicate that this algorithm is effective.

Laplacian spectral characterizations of unicyclic graphs  H( p, t K_{ 1, m})

MEI Ruoxing, WANG Ligong, WANG Luhua, WANG Zhanqing

2015, 19(1):  57-64. 

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 Let $H(p,tK_{1,m})$ be a connected unicyclic graph with $p+mt$ vertices obtained from $C_{p}$ by attaching the center of star $K_{1,m}$ to each one of $t$ mutual adjacent vertices of the cycle $C_{p}$, respectively. In this paper, it is proved that the unicyclic graphs $H(p,p K_{1,4})$, $H(p,p K_{1,3})$, $H((p,(p-1) K_{1,3} )$ are determined by their Laplacian spectra, and when $ p $ is even number, the unicyclic graphs $H(p,2 K_{1,3})$, $H(p,(p-2)K_{1,3})$, $H(p,(p-3)K_{1,3})$ are also determined by their Laplacian spectra.

Multiperiod mergers and acquisitions (M{\&}A) decision in consideration of efficiency and scale under the incomplete information

SHI Hailiu, WANG Yingming, CHEN Shengqun

2015, 19(1):  65-76. 

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For problem of incomplete information and multiperiod information existing in the process of mergers and acquisitions (M{\&}A), an approach of enterprises M{\&}A decision is proposed in consideration of efficiency and scale. Incomplete and multiperiod evaluation information of two sides involving in M{\&}A is aggregated using evidence reasoning (ER), and whether the scale of a merger enterprise is too large is determined using DEA method, which is used to select feasible M{\&}A scheme, then the best targets is determined according to the value of competitive cross-efficiency of M{\&}A scheme. Finally, an illustrative example is given to show the feasibility and validity of the proposed method.

The strongly quasi \alpha-preinvex functions

WANG Haiying, FU Zufeng

2015, 19(1):  77-84. 

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In this paper, one important type of generalized convex functions termed as strongly quasi $\alpha$-preinvex functions is studied. The relationships among strongly quasi $\alpha$-preinvex functions and quasi $\alpha$-preinvex functions, and strictly quasi $\alpha$-preinvex functions and semistrictly quasi $\alpha$-preinvex functions have been discussed. Three important theorems of properties have been obtained for it on the middle point strongly quasi $\alpha$-preinvexity. We also present two important applications in terms of strongly quasi $\alpha$-preinvex functions in mathematical programming. These results complete the researches for the strongly quasi $\alpha$-preinvex functions to some extent.

The optimal management for defined benefit  pension of funds based on a Heston model

XIAO Jianwu

2015, 19(1):  85-91. 

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The portfolio decision and contribution plan are the paramount important problems of the defined benefit pension funds. So the paper creates a Heston stochastic volatility model for the defined pension funds management, and transfers the primal problem to the dual problem by applying optimal control theory and the Legendre transform. It provides an analytic solution to the primal optimal problem by studying the dual problem. At last, an optimal asset allocation strategy (between a risky asset and a riskless asset) and the least contribution policy are obtained.

The weakly pseudocontinuous of vector-valued functions and the application in multi-objective games

ZHANG Guang, WU Donghua, GAO Jing

2015, 19(1):  92-98. 

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In this paper, based on the study of pseudocontinuous real-valued functions, a definition of the weakly pseudocontinuous of vector-valued functions on cone is proposed. And a weakly pseudocontinuous relationship is established between the vector-valued functions on $R_+^k$ and the real-valued functions. As an application of the weakly pseudocontinuous of vector-valued functions on cone, the existence theorem of weakly Pareto-Nash equilibrium in multi-objective games with the pseudocontinuous payoffs is proved by the improved vector-valued Ky Fan points.

The sum of squares of the machine completion times minimization scheduling problem on two identical parallel machines

GU Cunchang, ZHANG Yuzhong

2015, 19(1):  99-107. 

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 In the process of two competitive companies zero-sum game, maximize the product of the two company proceeds, equivalent to minimize the sum of the squares of the machine completion times on two identical parallel machines. We consider an off-line modified delayed-start LPT algorithm: firstly, renumber the jobs in the non-decreasing processing time order; secondly, if the total processing time $P(2m)$ of the longest $2m$ jobs is less than $(2m+1)p_{2m+1}$, schedule the first $2m+1$ jobs optimally followed by the remaining jobs scheduled according to the LPT rule; otherwise, $P(2m)$ is not less than $ (2m+1)p_{2m+1}$, schedule the first $2m$ jobs optimally followed by the remaining jobs scheduled according to the LPT rule. We prove that the worst case ratio of this algorithm is not greater than $1+ (\frac{1}{2m+2} )^2$, and the bound is tight.

Balanced judicious partitions of graphs with $\Delta($G$)-\delta($G$)\leq$2

HU Xiaochen, HE Weili, HAO Rongxia

2015, 19(1):  108-116. 

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A bipartition $V_{1}$ and $V_{2}$ of $V(G)$ of a graph $G$ is balanced if $||V_{1}|-|V_{2}||\leq1$. Bollob\'{a}s and Scott conjectured that every graph with $m$ edges and minimum degree at least 2 admits a balanced bipartition $V_{1}$, $V_{2}$ such that {\rm max}$\{e(V_{1}), e(V_{2})\}\leq\frac{m}{3}$, where $e(V_{i})$ denoted the number of edges of $G$ with both ends in $V_{i}(i=1, 2)$. They showed that this conjecture held for regular graphs(i.e., when $\Delta(G)=\delta(G)$). Yan Juan and Xu Baogang showed that every ($k$, $k-1$)-biregular graph(i.e., $\Delta(G)-\delta(G)\leq1$) admitted a balanced bipartition into $V_{1}$, $V_{2}$ with about $\frac{m}{4}$ edges in each vertex class. In this paper we extend this result to graphs with $\Delta($G$)-\delta($G$)\leq2$.

Parallel approximate subgradient projection algorithm for convex feasibility problem

DANG Yazheng, XUE Zhonghui

2015, 19(1):  117-124. 

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In this paper, a relaxed parallel $\varepsilon$-subgradient projection algorithm which includes the over-relaxed case  and an accelerated parallel $\varepsilon$-subgradient projection algorithm  for solving convex feasibility problem (CFP) are presented. Compared with the previous subgradient projection algorithms, the algorithms presented in this paper use parallel process, i.e. in each iteration consider several approximation subgradient projections simultaneously.  Algorithms adopt over-relaxed iterative process and accelerated technique. Hence, they can reduce the amounts of data storage and   improve the convergence speed.  And we also discuss the convergence of the methods under some mild conditions. Finally, the results of numerical experiment indicate that the  algorithms are valid and have faster convergence speed than that of the algorithm in [18].

Edge colourings of embedded special graphs

SUN Lin, LUO Zhaoyang

2015, 19(1):  125-130. 

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Let $G$ be a graph embedded on a surface $\Sigma$ of Euler characteristic $\chi(\Sigma)\!\geq\!0$.\\ $\chi'(G)$ and $\Delta(G)$ denote the chromatic index and the maximum degree of $G$, respectively. The paper shows that $\Delta(G)=\chi'(G)$ if the graph $G$ with $\Delta(G)\geq 4$ satisfies the following properties: (1) any two triangles with distance at least two; (2) any $i$-cycle and $j$-cycle with distance at least one, $i,j\in\{3,4\}$; (3) $G$ contains no $5$-cycles.