In this paper,we study a class of Linear Fractional Programming on a nonempty bounded set,called the Problem(LFP),and design a branch and bound algorithm to find the global optimal solution of the problem(LFP).First,w...In this paper,we study a class of Linear Fractional Programming on a nonempty bounded set,called the Problem(LFP),and design a branch and bound algorithm to find the global optimal solution of the problem(LFP).First,we convert the problem(LFP)to the equivalent problem(EP2).Secondly,by applying the linear relaxation technique to the problem(EP2),the linear relaxation programming problem(LRP2Y)was obtained.Then,the overall framework of the algorithm is given,and the convergence and complexity of the algorithm are analyzed.Finally,experimental results are listed to illustrate the effectiveness of the algorithm.展开更多
In this paper, we propose an interactive method for solving the multilevel linear programming problems based on the intuitionistic fuzzy set theory. Firstly, the membership function and the non-membership function are...In this paper, we propose an interactive method for solving the multilevel linear programming problems based on the intuitionistic fuzzy set theory. Firstly, the membership function and the non-membership function are introduced to describe the uncertainty of the decision makers. Secondly, a satisfactory solution is derived by updating the minimum satisfactory degrees with considerations of the overall satisfactory balance among all levels. In addition, the steps of the proposed method are given in this paper. Finally, numerical examples illustrate the feasibility of this method.展开更多
An integer linear bilevel programming problem is firstly transformed into a binary linear bilevel programming problem, and then converted into a single-level binary implicit programming. An orthogonal genetic algorith...An integer linear bilevel programming problem is firstly transformed into a binary linear bilevel programming problem, and then converted into a single-level binary implicit programming. An orthogonal genetic algorithm is developed for solving the binary linear implicit programming problem based on the orthogonal design. The orthogonal design with the factor analysis, an experimental design method is applied to the genetic algorithm to make the algorithm more robust, statistical y sound and quickly convergent. A crossover operator formed by the orthogonal array and the factor analysis is presented. First, this crossover operator can generate a smal but representative sample of points as offspring. After al of the better genes of these offspring are selected, a best combination among these offspring is then generated. The simulation results show the effectiveness of the proposed algorithm.展开更多
Bilevel linear programming, which consists of the objective functions of the upper level and lower level, is a useful tool for modeling decentralized decision problems. Various methods are proposed for solving this pr...Bilevel linear programming, which consists of the objective functions of the upper level and lower level, is a useful tool for modeling decentralized decision problems. Various methods are proposed for solving this problem. Of all the algorithms, the ge- netic algorithm is an alternative to conventional approaches to find the solution of the bilevel linear programming. In this paper, we describe an adaptive genetic algorithm for solving the bilevel linear programming problem to overcome the difficulty of determining the probabilities of crossover and mutation. In addition, some techniques are adopted not only to deal with the difficulty that most of the chromosomes maybe infeasible in solving constrained optimization problem with genetic algorithm but also to improve the efficiency of the algorithm. The performance of this proposed algorithm is illustrated by the examples from references.展开更多
By using the theory of Euclidean Jordan algebras,based on a new class of smoothing functions,the QiSun-Zhou's smoothing Newton algorithm is extended to solve linear programming over symmetric cones(SCLP).The algor...By using the theory of Euclidean Jordan algebras,based on a new class of smoothing functions,the QiSun-Zhou's smoothing Newton algorithm is extended to solve linear programming over symmetric cones(SCLP).The algorithm is globally convergent under suitable assumptions.展开更多
Multiobjective Programming (MOP) has become famous among many researchers due to more practical and realistic applications. A lot of methods have been proposed especially during the past four decades. In this paper, w...Multiobjective Programming (MOP) has become famous among many researchers due to more practical and realistic applications. A lot of methods have been proposed especially during the past four decades. In this paper, we develop a new algorithm based on a new approach to solve MOP by starting from a utopian point, which is usually infeasible, and moving towards the feasible region via stepwise movements and a simple continuous interaction with decision maker. We consider the case where all objective functions and constraints are linear. The implementation of the pro-posed algorithm is demonstrated by two numerical examples.展开更多
A discrete differential evolution algorithm combined with the branch and bound method is developed to solve the integer linear bilevel programming problems, in which both upper level and lower level variables are forc...A discrete differential evolution algorithm combined with the branch and bound method is developed to solve the integer linear bilevel programming problems, in which both upper level and lower level variables are forced to be integer. An integer coding for upper level variables is adopted, and then a discrete differential evolution algorithm with an improved feasibility-based comparison is developed to directly explore the integer solution at the upper level. For a given upper level integer variable, the lower level integer programming problem is solved by the existing branch and bound algorithm to obtain the optimal integer solution at the lower level. In the same framework of the algorithm, two other constraint handling methods, i.e. the penalty function method and the feasibility-based comparison method are also tested. The experimental results demonstrate that the discrete differential evolution algorithm with different constraint handling methods is effective in finding the global optimal integer solutions, but the improved constraint handling method performs better than two compared constraint handling methods.展开更多
In order to solve linear interaction programming, the vertex-searching method is proposed in this paper. First, the existence of equilibrium is analyzed for the model of linear interaction programming. Then the conclu...In order to solve linear interaction programming, the vertex-searching method is proposed in this paper. First, the existence of equilibrium is analyzed for the model of linear interaction programming. Then the conclusion is obtained in which the equi- librium is in the boundary of the restriction region of linear interaction programming. Also, a searching equilibrium solution is deduced from the conclusion.展开更多
We establish polynomial complexity corrector algorithms for linear programming over bounds of the Mehrotra-type predictor- symmetric cones. We first slightly modify the maximum step size in the predictor step of the s...We establish polynomial complexity corrector algorithms for linear programming over bounds of the Mehrotra-type predictor- symmetric cones. We first slightly modify the maximum step size in the predictor step of the safeguard based Mehrotra-type algorithm for linear programming, that was proposed by Salahi et al. Then, using the machinery of Euclidean Jordan algebras, we extend the modified algorithm to symmetric cones. Based on the Nesterov-Todd direction, we obtain O(r log ε1) iteration complexity bound of this algorithm, where r is the rank of the Jordan algebras and ε is the required precision. We also present a new variant of Mehrotra-type algorithm using a new adaptive updating scheme of centering parameter and show that this algorithm enjoys the same order of complexity bound as the safeguard algorithm. We illustrate the numerical behaviour of the methods on some small examples.展开更多
Network virtualization is known as a promising technology to tackle the ossification of current Internet and will play an important role in the future network area. Virtual network embedding(VNE) is a key issue in net...Network virtualization is known as a promising technology to tackle the ossification of current Internet and will play an important role in the future network area. Virtual network embedding(VNE) is a key issue in network virtualization. VNE is NP-hard and former VNE algorithms are mostly heuristic in the literature.VNE exact algorithms have been developed in recent years. However, the constraints of exact VNE are only node capacity and link bandwidth.Based on these, this paper presents an exact VNE algorithm, ILP-LC, which is based on Integer Linear Programming(ILP), for embedding virtual network request with location constraints. This novel algorithm is aiming at mapping virtual network request(VNR) successfully as many as possible and consuming less substrate resources.The topology of each VNR is randomly generated by Waxman model. Simulation results show that the proposed ILP-LC algorithm outperforms the typical heuristic algorithms in terms of the VNR acceptance ratio, at least 15%.展开更多
In this paper, primal-dual interior-point algorithm with dynamic step size is implemented for linear programming (LP) problems. The algorithms are based on a few kernel functions, including both serf-regular functio...In this paper, primal-dual interior-point algorithm with dynamic step size is implemented for linear programming (LP) problems. The algorithms are based on a few kernel functions, including both serf-regular functions and non-serf-regular ones. The dynamic step size is compared with fixed step size for the algorithms in inner iteration of Newton step. Numerical tests show that the algorithms with dynaraic step size are more efficient than those with fixed step size.展开更多
In this paper, an Improved Affine-Scaling Interior Point Algorithm for Linear Programming has been proposed. Computational results of selected practical problems affirming the proposed algorithm have been provided. Th...In this paper, an Improved Affine-Scaling Interior Point Algorithm for Linear Programming has been proposed. Computational results of selected practical problems affirming the proposed algorithm have been provided. The proposed algorithm is accurate, faster and therefore reduces the number of iterations required to obtain an optimal solution of a given Linear Programming problem as compared to the already existing Affine-Scaling Interior Point Algorithm. The algorithm can be very useful for development of faster software packages for solving linear programming problems using the interior-point methods.展开更多
A primal-dual infeasible interior point algorithm for multiple objective linear programming(MOLP)problems was presented.In contrast to the current MOLP algorithm.moving through the interior of polytope but not confini...A primal-dual infeasible interior point algorithm for multiple objective linear programming(MOLP)problems was presented.In contrast to the current MOLP algorithm.moving through the interior of polytope but not confining the iterates within the feasible region in our proposed algorithm result in a solution approach that is quite different and less sensitive to problem size,so providing the potential to dramatically improve the practical computation effectiveness.展开更多
A new heuristic algorithm is proposed for solving general integer linear programming problems. In the algorithm, the objective function hyperplane is used as a cutting plane, and then by introducing a special set of a...A new heuristic algorithm is proposed for solving general integer linear programming problems. In the algorithm, the objective function hyperplane is used as a cutting plane, and then by introducing a special set of assistant sets, an efficient heuristic search for the solution to the integer linear program is carried out in the sets on the objective function hyperplane. A simple numerical example shows that the algorithm is efficient for some problems, and therefore, of practical interest.展开更多
In the last several years, there has been a marked improvement in the development of new algorithms for solving Linear Goal programming (LGP). This paper presents a survey of current methods for LGP.
Two existing methods for solving a class of fuzzy linear programming (FLP) problems involving symmetric trapezoidal fuzzy numbers without converting them to crisp linear programming problems are the fuzzy primal simpl...Two existing methods for solving a class of fuzzy linear programming (FLP) problems involving symmetric trapezoidal fuzzy numbers without converting them to crisp linear programming problems are the fuzzy primal simplex method proposed by Ganesan and Veeramani [1] and the fuzzy dual simplex method proposed by Ebrahimnejad and Nasseri [2]. The former method is not applicable when a primal basic feasible solution is not easily at hand and the later method needs to an initial dual basic feasible solution. In this paper, we develop a novel approach namely the primal-dual simplex algorithm to overcome mentioned shortcomings. A numerical example is given to illustrate the proposed approach.展开更多
This paper proposes a model-free online value iteration(VI)algorithm for solving stochastic linear quadratic control problems with ergodic cost functions,where the diffusion term in the dynamics equation is influenced...This paper proposes a model-free online value iteration(VI)algorithm for solving stochastic linear quadratic control problems with ergodic cost functions,where the diffusion term in the dynamics equation is influenced by both the state and control variables.First,we propose an offline VI algorithm based on the idea of stochastic approximation.However,this algorithm requires prior knowledge of the system parameters,which are not always readily available.To overcome this limitation,we then develop a(partially)model-free online learning algorithm based on VI.This algorithm only requires a single system trajectory and does not need the initial control to be stabilizing.By exploiting the growth rate of Itô’s integrals to handle the stochastic term generated by multiplicative noise,we provide a rigorous proof of the algorithm’s convergence.Finally,a simulation example is presented to validate the convergence of the proposed algorithms.展开更多
The paper presents a technique for solving the binary linear programming model in polynomial time. The general binary linear programming problem is transformed into a convex quadratic programming problem. The convex q...The paper presents a technique for solving the binary linear programming model in polynomial time. The general binary linear programming problem is transformed into a convex quadratic programming problem. The convex quadratic programming problem is then solved by interior point algorithms. This settles one of the open problems of whether P = NP or not. The worst case complexity of interior point algorithms for the convex quadratic problem is polynomial. It can also be shown that every liner integer problem can be converted into binary linear problem.展开更多
Balas and Mazzola linearization (BML) is widely used in devising cutting plane algorithms for quadratic 0-1 programs. In this article, we improve BML by first strengthening the primal formulation of BML and then consi...Balas and Mazzola linearization (BML) is widely used in devising cutting plane algorithms for quadratic 0-1 programs. In this article, we improve BML by first strengthening the primal formulation of BML and then considering the dual formulation. Additionally, a new cutting plane algorithm is proposed.展开更多
We apply the simplex algorithm which is a branch of linear programming to efficiently determine the allocation of resources required to operate a company in the software development field. The main aim of applying thi...We apply the simplex algorithm which is a branch of linear programming to efficiently determine the allocation of resources required to operate a company in the software development field. The main aim of applying this technique is to maximize the profit of a company under certain limitations. This can be done using the trial-and-error approach. However, this tedious process can be replaced by user-level tools such as Excel which are based on linear programming that will give more accurate results. Small software companies cannot afford to hire a high number of senior programmers to produce the required level of quality and to keep up with the demand for adding new features. On the other hand, lowering the quality of the product will reduce the number of customers and decrease profit. Another aspect is maximizing the utilization of hosting servers which are required for providing the services to customers since the cost of buying servers and maintaining them is extremely high. The simplex algorithm in linear programming will take the specified constraints into account to compute the optimal allocation of the available resources to maximize profit and limit the cost. This paper will present a model that uses the simplex algorithm with a set of constraints to determine how many projects of each type a company should take in one period of time.展开更多
基金Supported by the National Natural Science Foundation of China(Grant Nos.12571317 and 12071133).
摘要In this paper,we study a class of Linear Fractional Programming on a nonempty bounded set,called the Problem(LFP),and design a branch and bound algorithm to find the global optimal solution of the problem(LFP).First,we convert the problem(LFP)to the equivalent problem(EP2).Secondly,by applying the linear relaxation technique to the problem(EP2),the linear relaxation programming problem(LRP2Y)was obtained.Then,the overall framework of the algorithm is given,and the convergence and complexity of the algorithm are analyzed.Finally,experimental results are listed to illustrate the effectiveness of the algorithm.
基金Supported by the National Natural Science Foundation of China(71471140,71171150,71103135)
摘要In this paper, we propose an interactive method for solving the multilevel linear programming problems based on the intuitionistic fuzzy set theory. Firstly, the membership function and the non-membership function are introduced to describe the uncertainty of the decision makers. Secondly, a satisfactory solution is derived by updating the minimum satisfactory degrees with considerations of the overall satisfactory balance among all levels. In addition, the steps of the proposed method are given in this paper. Finally, numerical examples illustrate the feasibility of this method.
基金supported by the Fundamental Research Funds for the Central Universities(K50511700004)the Natural Science Basic Research Plan in Shaanxi Province of China(2013JM1022)
摘要An integer linear bilevel programming problem is firstly transformed into a binary linear bilevel programming problem, and then converted into a single-level binary implicit programming. An orthogonal genetic algorithm is developed for solving the binary linear implicit programming problem based on the orthogonal design. The orthogonal design with the factor analysis, an experimental design method is applied to the genetic algorithm to make the algorithm more robust, statistical y sound and quickly convergent. A crossover operator formed by the orthogonal array and the factor analysis is presented. First, this crossover operator can generate a smal but representative sample of points as offspring. After al of the better genes of these offspring are selected, a best combination among these offspring is then generated. The simulation results show the effectiveness of the proposed algorithm.
基金the National Natural Science Foundation of China(Nos.60574071 and70771080)
摘要Bilevel linear programming, which consists of the objective functions of the upper level and lower level, is a useful tool for modeling decentralized decision problems. Various methods are proposed for solving this problem. Of all the algorithms, the ge- netic algorithm is an alternative to conventional approaches to find the solution of the bilevel linear programming. In this paper, we describe an adaptive genetic algorithm for solving the bilevel linear programming problem to overcome the difficulty of determining the probabilities of crossover and mutation. In addition, some techniques are adopted not only to deal with the difficulty that most of the chromosomes maybe infeasible in solving constrained optimization problem with genetic algorithm but also to improve the efficiency of the algorithm. The performance of this proposed algorithm is illustrated by the examples from references.
基金Supported by Liu Hui Centre for Applied Mathematics,Nankai University and Tianjin University
摘要By using the theory of Euclidean Jordan algebras,based on a new class of smoothing functions,the QiSun-Zhou's smoothing Newton algorithm is extended to solve linear programming over symmetric cones(SCLP).The algorithm is globally convergent under suitable assumptions.
摘要Multiobjective Programming (MOP) has become famous among many researchers due to more practical and realistic applications. A lot of methods have been proposed especially during the past four decades. In this paper, we develop a new algorithm based on a new approach to solve MOP by starting from a utopian point, which is usually infeasible, and moving towards the feasible region via stepwise movements and a simple continuous interaction with decision maker. We consider the case where all objective functions and constraints are linear. The implementation of the pro-posed algorithm is demonstrated by two numerical examples.
基金supported by the Natural Science Basic Research Plan in Shaanxi Province of China(2013JM1022)the Fundamental Research Funds for the Central Universities(K50511700004)
摘要A discrete differential evolution algorithm combined with the branch and bound method is developed to solve the integer linear bilevel programming problems, in which both upper level and lower level variables are forced to be integer. An integer coding for upper level variables is adopted, and then a discrete differential evolution algorithm with an improved feasibility-based comparison is developed to directly explore the integer solution at the upper level. For a given upper level integer variable, the lower level integer programming problem is solved by the existing branch and bound algorithm to obtain the optimal integer solution at the lower level. In the same framework of the algorithm, two other constraint handling methods, i.e. the penalty function method and the feasibility-based comparison method are also tested. The experimental results demonstrate that the discrete differential evolution algorithm with different constraint handling methods is effective in finding the global optimal integer solutions, but the improved constraint handling method performs better than two compared constraint handling methods.
基金Supported by the National Soft Science Research Project of China (2006GXSZD085)
摘要In order to solve linear interaction programming, the vertex-searching method is proposed in this paper. First, the existence of equilibrium is analyzed for the model of linear interaction programming. Then the conclusion is obtained in which the equi- librium is in the boundary of the restriction region of linear interaction programming. Also, a searching equilibrium solution is deduced from the conclusion.
基金Supported by the National Natural Science Foundation of China(11471102,61301229)Supported by the Natural Science Foundation of Henan University of Science and Technology(2014QN039)
摘要We establish polynomial complexity corrector algorithms for linear programming over bounds of the Mehrotra-type predictor- symmetric cones. We first slightly modify the maximum step size in the predictor step of the safeguard based Mehrotra-type algorithm for linear programming, that was proposed by Salahi et al. Then, using the machinery of Euclidean Jordan algebras, we extend the modified algorithm to symmetric cones. Based on the Nesterov-Todd direction, we obtain O(r log ε1) iteration complexity bound of this algorithm, where r is the rank of the Jordan algebras and ε is the required precision. We also present a new variant of Mehrotra-type algorithm using a new adaptive updating scheme of centering parameter and show that this algorithm enjoys the same order of complexity bound as the safeguard algorithm. We illustrate the numerical behaviour of the methods on some small examples.
基金supported by the National Basic Research Program of China(973 Program)under Grant 2013CB329005
摘要Network virtualization is known as a promising technology to tackle the ossification of current Internet and will play an important role in the future network area. Virtual network embedding(VNE) is a key issue in network virtualization. VNE is NP-hard and former VNE algorithms are mostly heuristic in the literature.VNE exact algorithms have been developed in recent years. However, the constraints of exact VNE are only node capacity and link bandwidth.Based on these, this paper presents an exact VNE algorithm, ILP-LC, which is based on Integer Linear Programming(ILP), for embedding virtual network request with location constraints. This novel algorithm is aiming at mapping virtual network request(VNR) successfully as many as possible and consuming less substrate resources.The topology of each VNR is randomly generated by Waxman model. Simulation results show that the proposed ILP-LC algorithm outperforms the typical heuristic algorithms in terms of the VNR acceptance ratio, at least 15%.
基金Project supported by Dutch Organization for Scientific Research(Grant No .613 .000 .010)
摘要In this paper, primal-dual interior-point algorithm with dynamic step size is implemented for linear programming (LP) problems. The algorithms are based on a few kernel functions, including both serf-regular functions and non-serf-regular ones. The dynamic step size is compared with fixed step size for the algorithms in inner iteration of Newton step. Numerical tests show that the algorithms with dynaraic step size are more efficient than those with fixed step size.
摘要In this paper, an Improved Affine-Scaling Interior Point Algorithm for Linear Programming has been proposed. Computational results of selected practical problems affirming the proposed algorithm have been provided. The proposed algorithm is accurate, faster and therefore reduces the number of iterations required to obtain an optimal solution of a given Linear Programming problem as compared to the already existing Affine-Scaling Interior Point Algorithm. The algorithm can be very useful for development of faster software packages for solving linear programming problems using the interior-point methods.
基金Supported by the Doctoral Educational Foundation of China of the Ministry of Education(20020486035)
摘要A primal-dual infeasible interior point algorithm for multiple objective linear programming(MOLP)problems was presented.In contrast to the current MOLP algorithm.moving through the interior of polytope but not confining the iterates within the feasible region in our proposed algorithm result in a solution approach that is quite different and less sensitive to problem size,so providing the potential to dramatically improve the practical computation effectiveness.
摘要A new heuristic algorithm is proposed for solving general integer linear programming problems. In the algorithm, the objective function hyperplane is used as a cutting plane, and then by introducing a special set of assistant sets, an efficient heuristic search for the solution to the integer linear program is carried out in the sets on the objective function hyperplane. A simple numerical example shows that the algorithm is efficient for some problems, and therefore, of practical interest.
摘要In the last several years, there has been a marked improvement in the development of new algorithms for solving Linear Goal programming (LGP). This paper presents a survey of current methods for LGP.
摘要Two existing methods for solving a class of fuzzy linear programming (FLP) problems involving symmetric trapezoidal fuzzy numbers without converting them to crisp linear programming problems are the fuzzy primal simplex method proposed by Ganesan and Veeramani [1] and the fuzzy dual simplex method proposed by Ebrahimnejad and Nasseri [2]. The former method is not applicable when a primal basic feasible solution is not easily at hand and the later method needs to an initial dual basic feasible solution. In this paper, we develop a novel approach namely the primal-dual simplex algorithm to overcome mentioned shortcomings. A numerical example is given to illustrate the proposed approach.
基金supported by the National Natural Science Foundation of China(Grants No.62573266 and 62192753)the Natural Science Foundation of Shandong Province for Distinguished Young Scholars(Grant No.ZR2022JQ31).
摘要This paper proposes a model-free online value iteration(VI)algorithm for solving stochastic linear quadratic control problems with ergodic cost functions,where the diffusion term in the dynamics equation is influenced by both the state and control variables.First,we propose an offline VI algorithm based on the idea of stochastic approximation.However,this algorithm requires prior knowledge of the system parameters,which are not always readily available.To overcome this limitation,we then develop a(partially)model-free online learning algorithm based on VI.This algorithm only requires a single system trajectory and does not need the initial control to be stabilizing.By exploiting the growth rate of Itô’s integrals to handle the stochastic term generated by multiplicative noise,we provide a rigorous proof of the algorithm’s convergence.Finally,a simulation example is presented to validate the convergence of the proposed algorithms.
摘要The paper presents a technique for solving the binary linear programming model in polynomial time. The general binary linear programming problem is transformed into a convex quadratic programming problem. The convex quadratic programming problem is then solved by interior point algorithms. This settles one of the open problems of whether P = NP or not. The worst case complexity of interior point algorithms for the convex quadratic problem is polynomial. It can also be shown that every liner integer problem can be converted into binary linear problem.
摘要Balas and Mazzola linearization (BML) is widely used in devising cutting plane algorithms for quadratic 0-1 programs. In this article, we improve BML by first strengthening the primal formulation of BML and then considering the dual formulation. Additionally, a new cutting plane algorithm is proposed.
摘要We apply the simplex algorithm which is a branch of linear programming to efficiently determine the allocation of resources required to operate a company in the software development field. The main aim of applying this technique is to maximize the profit of a company under certain limitations. This can be done using the trial-and-error approach. However, this tedious process can be replaced by user-level tools such as Excel which are based on linear programming that will give more accurate results. Small software companies cannot afford to hire a high number of senior programmers to produce the required level of quality and to keep up with the demand for adding new features. On the other hand, lowering the quality of the product will reduce the number of customers and decrease profit. Another aspect is maximizing the utilization of hosting servers which are required for providing the services to customers since the cost of buying servers and maintaining them is extremely high. The simplex algorithm in linear programming will take the specified constraints into account to compute the optimal allocation of the available resources to maximize profit and limit the cost. This paper will present a model that uses the simplex algorithm with a set of constraints to determine how many projects of each type a company should take in one period of time.