Table of Content

15 March 2012, Volume 16 Issue 1

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Original Articles

Continuous-time Portfolio Selection with Loss Aversion in an Incomplete Market

Mi Hui, ZHANG Shu-Guang

2012, 16(1):  1-12. 

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 In this study we investigate a general continuous-time portfolio selection model with loss aversion in an incomplete market where the number of stocks is strictly less than the dimension of the underlying Brownian motion. The investor's preference facing market risks is defined by a S-shaped value function. By transforming the market into a complete one, we solve the optimal terminal wealth and the optimal wealth-portfolio pair of agents using  martingale method and  replicating technique. A special example with a two-piece power function and deterministic coefficients is presented to illustrate the general results. Last, the explicit expressions of the optimal solutions are given.

An Approximation Agorithm for the Stochastic Fault-Tolerant Fcility Placement Problem

SHAO Jia-Ting, Xu-Da-Chuan

2012, 16(1):  13-20. 

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In the deterministic fault-tolerant facility placement problem (FTFP),  we are given a set of locations  and a set of clients. We can open any number of different facilities with the same opening cost at each location. Each client j has an integer requirement r_j. Connecting  client j to different facilities at the same location is permitted. The objective is to open some facilities and connect each client j to r_j different facilities such that the total cost is minimized. In this paper, we consider the two-stage stochastic fault-tolerant facility placement problem (SFTFP)  with recourse in which the set of clients are unknown in advance. But there are finite scenarios materialized by a probability distribution. Each scenario specifies a subset of clients to be assigned. Moreover, each facility has two kinds of opening cost. In the first stage, we open a subset of facilities according to the stochastic information of the clients. In the second stage, we can open additional number of facilities when the actual information of the clients is revealed. We give a linear integral program and an LP-rounding based 5-approximation algorithm for the SFTFP.

An LVI-based Numerical Algorithm for Solving Quadratic Programming Problems

ZHANG Yu-Nong, LI Xue-Zhong, ZHANG Zhi-Jun, LI Jun

2012, 16(1):  21-30. 

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This paper presents and investigates a numerical algorithm (termed as 94LVI algorithm) for solving quadratic programming (QP) problems with linear equality and bound constraints. To do this, the constrained QP problems are firstly converted into linear variational inequalities (LVI), which are then converted into equivalent piecewise-linear projection equations (PLPE). After that, the resultant PLPE is solved by the presented 94LVI algorithm. The optimal numerical solutions to the QP problems are thus obtained. Furthermore, the theoretical proof of the global convergence of the 94LVI algorithm is presented. The numerical comparison results between the 94LVI algorithm and the active set algorithm are provided as well, which further demonstrates the efficacy and superiority of the presented algorithm for solving such QP problems.

Vertex vulnerability parameters of Kronecker products of complete multipartite graphs and complete graphs

TANG Dan, WANG He-Chao, DAN Er-Fang

2012, 16(1):  31-40. 

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Let G_1 and G_2 be two graphs. The Kronecker product G_1\times G_2 is defined as V(G_1\times G_2)=V(G_1)\times V(G_2) and E(G_1\times G_2)=\{(u_1,v_1)(u_2,v_2):u_1u_2\in E(G_1) and v_1v_2\in E(G_2)\}. In this paper we compute several vertex vulnerability parameters of Kronecker product of a complete p-partite graph K_{m_{1},m_{2},\ldots,m_{p}} and a complete graph K_n on n vertices, where m_{1}\leq m_{2}\leq \ldots \leq m_{p}, 2\leq p\leq n, and n\geq3. This result generalizes the previous result by Mamut and Vumar.

Positive Influencing Number of Partitioned Graphs

ZHAO Wei-Liang, ZHAO Yan-Cai

2012, 16(1):  41-48. 

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In this paper, a concept is introduced in order to model a class of problems in social networks. A social network consisting of positive and negative influential individuals is represented as a partitioned graph G=(V,E) with vertex set partition V=V⁺\cup V^-. A subset D\subseteq V^- is called a positive influencing set if every vertex  of  G has  in D\cup V⁺ at least half of its neighbors. The positive influencing number of G is the minimum cardinality of a positive influencing set. In the present paper, we show some sharp lower bounds for this parameter, and prove that this problem is NP-complete for partitioned bipartite graphs and partitioned chordal graphs, respectively.

On Properties of Quasi-semi-(E,F)-Convex Functions and Convex Programming

JIAN Jin-Bao, HU Qing-Juan, MA Peng-Fei, LI Jian-Ling

2012, 16(1):  49-55. 

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In 2004, Jian, Hu and Tang et al (Int. J. Pure Appl. Math., 2004, 14(4): 439-454) introduced the concept of quasi-semi-(E,F)-convex functions. In this paper, some important properties of quasi-semi-(E,F)-convex function and the associated quasi-semi-(E,F)-convex programming are further discussed. As a result,  some important results for this generalized convexity are established.

A New Penalty Function Based on Non-coercive Penalty Functions

Shang-You-Lin, LIU Mu-Hua, LI Pu

2012, 16(1):  56-66. 

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For the differentiable nonlinear programming problem, this paper proposes a new penalty function form of the approached  exact penalty function, presents
 with the gradual  approximation algorithm and evolutionary algorithm, and proves   that if the sequences of the approximation algorithm   exist accumulation point, it certainly is the optimal solution of original problem. In the weak assumptions,    we prove that the minimum sequences from the algorithm is bounded, and its accumulation points are the optimal    solution of the original problem and get that in the Mangasarian-Fromovitz qualification condition, through     limited iterations the minimum point is the feasible point.

Genetic Algorithm for Aircraft Landing Problem with Predetermined Time Window and Earliness/Tardiness Penalties

WANG Hong, LIN Dan, LI Min-Qiang

2012, 16(1):  67-76. 

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This paper considers Aircraft Landing Problem. Each aircraft lands within a predetermined time window and meets separation time requirements with other aircrafts. The objective is to minimize total weighted earliness and tardiness of all aircrafts. We suggest a genetic algorithm (GA) to solve this problem. Each chromosome contains two vectors: a sequence for landing the planes and an assignment of runways. A new decoding procedure is designed to determine the landing time for all aircraft. The computational experiments on 8 standard instances provided by the OR-Library show that the proposed algorithm outperforms the Scatter Search (SS) algorithm and the hybrid method combining Genetic Algorithms with Ant Colony Optimization (ACGA) presented in the literature. In order to evaluate the performance of the proposed decoding procedure, we compare the proposed algorithm with the GA that employs the decoding procedure applied in ACGA. The results show that the proposed decoding procedure can improve the quality of the solution for the considered problem. It is applicable developing the algorithm described in this paper for a similar scheduling problem with predetermined time window and earliness/tardiness penalties.

The Smooth Tree Option Pricing Model Based On the Minimum Cross Entropy

LI Ying-Hua, LI Xing-Si

2012, 16(1):  77-87. 

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To overcome the volatility of the binomial tree option price model's convergence, and to strengthen the predictive effect of the historical data information, we propose a novel tree model that is smooth and convergent. Based on the historical data information, the new model applies the minimum cross entropy formalism to derive the crucial parameters p,u and $d$ of the binomial tree option price model, and the backward induction is used to compute the option
price. Obviously, option price computed by the new model implies the historical data information. Because the minimum cross entropy formalism is a convex optimization problem, it has the unique optimal solution. Furthermore, the new model is also suitable for pricing the option in the incomplete market. Finally, compared with the CRR model, the new one can smoothly converge and have more accurate results in numerical examples. Moreover, the new model can
converge to the B-S formula for the call (put) European option (binary option).    

The Stability of the Solutions of Optimization Hub-and-Spoke Network Design Problem with Fixed Hub Arc Costs

WENG Ke-Rui

2012, 16(1):  88-96. 

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Hub-and-Spoke network design problem with fixed hub arc cost has a wide range of applications within the third party logistics, postal services and airline transportation. Current researches concentrated on hub location while this paper emphasizes on the fixed hub arc costs which reflects a fact that the transportation on hub arc must be provided with large-scale vehicles and therefore pay extra fixed costs. This paper constructs a mixed 0-1 integer programming model, and provided a heuristic algorithm based on Lagrangian relaxation。 We also extend the problem by add the route distance constraint which is very important on emergency logistics and express delivery. We solve the extended problem by modifying the original algorithm.

New Algorithm for the Continuing Dynamic Programming

ZHANG Peng

2012, 16(1):  97-105. 

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The paper proposes the discrete approximate iteration method to solve single-dimensional continuing dynamic programming model. At the same time, multidimensional continuing dynamic programming model is solved by the discrete approximate iteration method and bi-convergent method. The algorithm is as following: Firstly, let state value of one of state equations be unknown and the others be known. Secondly, use the discrete approximate iteration method to find the optimal value of the unknown state values and then continue iterating until all state equations have found optimal values. If the objective function is non-concave and non-convex, the convergence of the algorithm is proved. If the objective function is convex, the linear convergence of the algorithm is proved. At last, the effectiveness of the formation and the algorithm is proved by an example.

Factor Model Based Index Tracking and Empirical Analysis

CHEN Jie, Cui-Xue-Ting

2012, 16(1):  106-114. 

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Index tracking is a popular passive investment strategy widely used by index funds and institutional investors. In this paper, we present a cardinality constrained index tracking model based on factor model. The model minimizes the variance of the portfolio with constraints on factor betas and expected excess return. Taking into account the practical investment management, we also consider restrictions on the total number of stocks selected. The result of empirical analysis suggests that, by choosing different parameters, the optimal portfolios of our model can achieve low tracking error and excess return.

The Stability of the Solutions of Optimization Problem for Set-Valued Maps With Upper Semi-continuity Under Graphic Approximate

XIA Shun-You, XU De-Ping

2012, 16(1):  115-120. 

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In this paper, under the approximate condition of graphic topology, we first introduce the essential efficient solutions and the weakly essential efficient solutions of the optimization problem for upper semi-continuity maps with set-value. Second, by using the usco researching approach of generic stability, with the trembles of domain and map, the upper semi-continuity and compact properties of the weakly efficient solutions maps of this optimization problem are proved, under the approximate condition of graphic topology, and then it is generic lower semi-continuous, that is to say, in the sense of Baire Category, weakly efficient solutions maps of “most” this optimization problems are generic stability(i.e. essential) under the approximate condition of graphic topology. Last, we prove a necessary and sufficient condition of upper semi-continuity of the efficient solutions maps of this optimization problem.

A Single Machine Scheduling Problem with Subcontracting Options

ZHONG Wei-Ya, LIU Xiao-Lei, Huo-Zhi-Ming

2012, 16(1):  121-128. 

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In this paper, we study a single machine scheduling problem with subcontracting options as follows: there are n jobs which are available at time zero. Each job can be processed in house on a single machine or subcontracted to a subcontractor. If a job is subcontracted to the subcontractor, its delivery lead time is the same as the in-house processing time, but the processing cost is different from the in-house processing cost. The delivery lead time and processing cost of a subcontracted job is independent of the total workload of the subcontractor. The objective is to minimize the weighted sum of the maximal completion time (which is defined as the larger one of the maximal completion time of the in-house jobs and the maximal delivery lead time of the out-house jobs) and the total processing cost. This problem has been proved to be NP-hard. We first give a pseudo-polynomial time algorithm for this problem, then give an FPTAS.