The penalty function method, presented many years ago, is an important nu- merical method for the mathematical programming problems. In this article, we propose a dual-relax penalty function approach, which is signifi...The penalty function method, presented many years ago, is an important nu- merical method for the mathematical programming problems. In this article, we propose a dual-relax penalty function approach, which is significantly different from penalty func- tion approach existing for solving the bilevel programming, to solve the nonlinear bilevel programming with linear lower level problem. Our algorithm will redound to the error analysis for computing an approximate solution to the bilevel programming. The error estimate is obtained among the optimal objective function value of the dual-relax penalty problem and of the original bilevel programming problem. An example is illustrated to show the feasibility of the proposed approach.展开更多
In this article, the expected discounted penalty function Фδ,α (u) with constant interest δ and "discounted factor" exp(-αTδ) is considered. As a result, the integral equation of Фδ,α (u) is derived a...In this article, the expected discounted penalty function Фδ,α (u) with constant interest δ and "discounted factor" exp(-αTδ) is considered. As a result, the integral equation of Фδ,α (u) is derived and an exact solution for Фδ,α (0) is found. The relation between the joint density of the surplus immediately prior to ruin, and the deficit at ruin and the density of the surplus immediately prior to ruin is then obtained based on analytical methods.展开更多
The algorithm proposed by T. F. Colemen and A. R. Conn is improved in this paper, and the improved algorithm can solve nonlinear programming problem with quality constraints. It is shown that the improved algorithm po...The algorithm proposed by T. F. Colemen and A. R. Conn is improved in this paper, and the improved algorithm can solve nonlinear programming problem with quality constraints. It is shown that the improved algorithm possesses global convergence, and under some conditions, it possesses locally supperlinear convergence.展开更多
The exact minimax penalty function method is used to solve a noncon- vex differentiable optimization problem with both inequality and equality constraints. The conditions for exactness of the penalization for the exac...The exact minimax penalty function method is used to solve a noncon- vex differentiable optimization problem with both inequality and equality constraints. The conditions for exactness of the penalization for the exact minimax penalty function method are established by assuming that the functions constituting the considered con- strained optimization problem are invex with respect to the same function η (with the exception of those equality constraints for which the associated Lagrange multipliers are negative these functions should be assumed to be incave with respect to η). Thus, a threshold of the penalty parameter is given such that, for all penalty parameters exceeding this threshold, equivalence holds between the set of optimal solutions in the considered constrained optimization problem and the set of minimizer in its associated penalized problem with an exact minimax penalty function. It is shown that coercivity is not suf- ficient to prove the results.展开更多
In this paper, the general exact penalty functions in integer programming were studied. The conditions which ensure the exact penalty property for the general penalty function with one penalty parameter were given and...In this paper, the general exact penalty functions in integer programming were studied. The conditions which ensure the exact penalty property for the general penalty function with one penalty parameter were given and a general penalty function with two parameters was proposed.展开更多
The purpose of this paper is to consider the expected value of a discounted penalty due at ruin in the Erlang(2) risk process under constant interest force. An integro-differential equation satisfied by the expected...The purpose of this paper is to consider the expected value of a discounted penalty due at ruin in the Erlang(2) risk process under constant interest force. An integro-differential equation satisfied by the expected value and a second-order differential equation for the Laplace transform of the expected value are derived. In addition, the paper will present the recursive algorithm for the joint distribution of the surplus immediately before ruin and the deficit at ruin. Finally, by the differential equation, the defective renewal equation and the explicit expression for the expected value are given in the interest-free case.展开更多
In this paper, based on the inherent characteristic of the contention relation between flows in ad hoc networks, we introduce the notion of the link's interference set, extend the utility maximization problem represe...In this paper, based on the inherent characteristic of the contention relation between flows in ad hoc networks, we introduce the notion of the link's interference set, extend the utility maximization problem representing congestion control in wireline networks to ad hoc networks, apply the penalty function approach and the subgradient method to solve this problem, and propose the congestion control algorithm Penalty function-based Optical Congestion Control (POCC) which is implemented in NS2- simulator. Specifically, each link transmits periodically the information on its congestion state to its interference set; the set ; the sermon at each source adjusts the transmission rate based on the optimal tradeoffbetween the utility value and the congestion level which the interference set of the links that this session goes though suffers from. MATLAB-based simulation results showed that POCC can approach the globally optimal solution. The NS2-based simulation results showed that POCC outperforms default TCP and ATCP to achieve efficient and fair resource allocation in ad hoc networks.展开更多
In this paper, a new algorithm-approximate penalty function method is designed, which can be used to solve a bilevel optimization problem with linear constrained function. In this kind of bilevel optimization problem....In this paper, a new algorithm-approximate penalty function method is designed, which can be used to solve a bilevel optimization problem with linear constrained function. In this kind of bilevel optimization problem. the evaluation of the objective function is very difficult, so that only their approximate values can be obtained. This algorithm is obtained by combining penalty function method and approximation in bilevel programming. The presented algorithm is completely different from existing methods. That convergence for this algorithm is proved.展开更多
By the aid of the penalty function method, the equilibrium restriction conditions were introduced to the isoparametric hybrid finite element analysis, and the concrete application course of the penalty function method...By the aid of the penalty function method, the equilibrium restriction conditions were introduced to the isoparametric hybrid finite element analysis, and the concrete application course of the penalty function method in three-dimensional isoparametdc hybrid finite element was discussed. The separated penalty parameters method and the optimal hybrid element model with penalty balance were also presented. The penalty balance method can effectively refrain the parasitical stress on the premise of no additional degrees of freedom. The numeric experiment shows that the presented element not only is effective in improving greatly the numeric calculation precision of distorted grids but also has the universality.展开更多
A class of discontinuous penalty functions was proposed to solve constrained minimization problems with the integral approach to global optimization, m-mean value and v-variance optimality conditions of a constrained ...A class of discontinuous penalty functions was proposed to solve constrained minimization problems with the integral approach to global optimization, m-mean value and v-variance optimality conditions of a constrained and penalized minimization problem were investigated. A nonsequential algorithm was proposed. Numerical examples were given to illustrate the effectiveness of the algorithm.展开更多
In this paper, an approximate smoothing approach to the non-differentiable exact penalty function is proposed for the constrained optimization problem. A simple smoothed penalty algorithm is given, and its convergence...In this paper, an approximate smoothing approach to the non-differentiable exact penalty function is proposed for the constrained optimization problem. A simple smoothed penalty algorithm is given, and its convergence is discussed. A practical algorithm to compute approximate optimal solution is given as well as computational experiments to demonstrate its efficiency.展开更多
This paper introduces a new exact and smooth penalty function to tackle constrained min-max problems. By using this new penalty function and adding just one extra variable, a constrained rain-max problem is transforme...This paper introduces a new exact and smooth penalty function to tackle constrained min-max problems. By using this new penalty function and adding just one extra variable, a constrained rain-max problem is transformed into an unconstrained optimization one. It is proved that, under certain reasonable assumptions and when the penalty parameter is sufficiently large, the minimizer of this unconstrained optimization problem is equivalent to the minimizer of the original constrained one. Numerical results demonstrate that this penalty function method is an effective and promising approach for solving constrained finite min-max problems.展开更多
We present an approximation-exact penalty function method for solving the single stage stochastic programming problem with continuous random variable. The original problem is transformed into a determinate nonlinear p...We present an approximation-exact penalty function method for solving the single stage stochastic programming problem with continuous random variable. The original problem is transformed into a determinate nonlinear programming problem with a discrete random variable sequence, which is obtained by some discrete method. We construct an exact penalty function and obtain an unconstrained optimization. It avoids the difficulty in solution by the rapid growing of the number of constraints for discrete precision. Under lenient conditions, we prove the equivalence of the minimum solution of penalty function and the solution of the determinate programming, and prove that the solution sequences of the discrete problem converge to a solution to the original problem.展开更多
The penalty function method is one basic method for solving constrained nonlinear programming, in which simple smooth exact penalty functions draw much attention for their simpleness and smoothness. This article offer...The penalty function method is one basic method for solving constrained nonlinear programming, in which simple smooth exact penalty functions draw much attention for their simpleness and smoothness. This article offers a new kind of simple smooth approximative exact penalty function of general constrained nonlinear programmings and analyzes its properties.展开更多
In this paper, a new augmented Lagrangian penalty function for constrained optimization problems is studied. The dual properties of the augmented Lagrangian objective penalty function for constrained optimization prob...In this paper, a new augmented Lagrangian penalty function for constrained optimization problems is studied. The dual properties of the augmented Lagrangian objective penalty function for constrained optimization problems are proved. Under some conditions, the saddle point of the augmented Lagrangian objective penalty function satisfies the first-order Karush-Kuhn-Tucker (KKT) condition. Especially, when the KKT condition holds for convex programming its saddle point exists. Based on the augmented Lagrangian objective penalty function, an algorithm is developed for finding a global solution to an inequality constrained optimization problem and its global convergence is also proved under some conditions.展开更多
In this paper,we propose an iterative algorithm to find the optimal incentive mechanism for the principal-agent problem under moral hazard where the number of agent action profiles is infinite,and where there are an i...In this paper,we propose an iterative algorithm to find the optimal incentive mechanism for the principal-agent problem under moral hazard where the number of agent action profiles is infinite,and where there are an infinite number of results that can be observed by the principal.This principal-agent problem has an infinite number of incentive-compatibility constraints,and we transform it into an optimization problem with an infinite number of constraints called a semi-infinite programming problem.We then propose an exterior penalty function method to find the optimal solution to this semi-infinite programming and illustrate the convergence of this algorithm.By analyzing the optimal solution obtained by the proposed penalty function method,we can obtain the optimal incentive mechanism for the principal-agent problem with an infinite number of incentive-compatibility constraints under moral hazard.展开更多
In this papert a recursive quadratic programming algorithm is proposed andstudied.The line search functions used are Han's nondifferentiable penalty functionswith a second order penalty term. In order to avoid mar...In this papert a recursive quadratic programming algorithm is proposed andstudied.The line search functions used are Han's nondifferentiable penalty functionswith a second order penalty term. In order to avoid maratos effect,Fukushima's mixeddirection is used as the direction of line search.Finallyt we prove the global convergenceand the local second order convergence of the algorithm.展开更多
In this paper, we present an algorithm to solve the inequality constrained multi-objective programming (MP) by using a penalty function with objective parameters and constraint penalty parameter. First, the penalty fu...In this paper, we present an algorithm to solve the inequality constrained multi-objective programming (MP) by using a penalty function with objective parameters and constraint penalty parameter. First, the penalty function with objective parameters and constraint penalty parameter for MP and the corresponding unconstraint penalty optimization problem (UPOP) is defined. Under some conditions, a Pareto efficient solution (or a weakly-efficient solution) to UPOP is proved to be a Pareto efficient solution (or a weakly-efficient solution) to MP. The penalty function is proved to be exact under a stable condition. Then, we design an algorithm to solve MP and prove its convergence. Finally, numerical examples show that the algorithm may help decision makers to find a satisfactory solution to MP.展开更多
By using the penalty function method with objective parameters, the paper presents an interactive algorithm to solve the inequality constrained multi-objective programming (MP). The MP is transformed into a single obj...By using the penalty function method with objective parameters, the paper presents an interactive algorithm to solve the inequality constrained multi-objective programming (MP). The MP is transformed into a single objective optimal problem (SOOP) with inequality constrains;and it is proved that, under some conditions, an optimal solution to SOOP is a Pareto efficient solution to MP. Then, an interactive algorithm of MP is designed accordingly. Numerical examples show that the algorithm can find a satisfactory solution to MP with objective weight value adjusted by decision maker.展开更多
We propose a new unified path to approximately smoothing the nonsmooth exact penalty function in this paper. Based on the new smooth penalty function, we give a penalty algorithm to solve the constrained optimization ...We propose a new unified path to approximately smoothing the nonsmooth exact penalty function in this paper. Based on the new smooth penalty function, we give a penalty algorithm to solve the constrained optimization problem, and discuss the convergence of the algorithm under mild conditions.展开更多
基金supported by the National Science Foundation of China (70771080)Social Science Foundation of Ministry of Education (10YJC630233)
文摘The penalty function method, presented many years ago, is an important nu- merical method for the mathematical programming problems. In this article, we propose a dual-relax penalty function approach, which is significantly different from penalty func- tion approach existing for solving the bilevel programming, to solve the nonlinear bilevel programming with linear lower level problem. Our algorithm will redound to the error analysis for computing an approximate solution to the bilevel programming. The error estimate is obtained among the optimal objective function value of the dual-relax penalty problem and of the original bilevel programming problem. An example is illustrated to show the feasibility of the proposed approach.
文摘In this article, the expected discounted penalty function Фδ,α (u) with constant interest δ and "discounted factor" exp(-αTδ) is considered. As a result, the integral equation of Фδ,α (u) is derived and an exact solution for Фδ,α (0) is found. The relation between the joint density of the surplus immediately prior to ruin, and the deficit at ruin and the density of the surplus immediately prior to ruin is then obtained based on analytical methods.
基金the National+4 种基金 Natural Science Foundation of China
文摘The algorithm proposed by T. F. Colemen and A. R. Conn is improved in this paper, and the improved algorithm can solve nonlinear programming problem with quality constraints. It is shown that the improved algorithm possesses global convergence, and under some conditions, it possesses locally supperlinear convergence.
文摘The exact minimax penalty function method is used to solve a noncon- vex differentiable optimization problem with both inequality and equality constraints. The conditions for exactness of the penalization for the exact minimax penalty function method are established by assuming that the functions constituting the considered con- strained optimization problem are invex with respect to the same function η (with the exception of those equality constraints for which the associated Lagrange multipliers are negative these functions should be assumed to be incave with respect to η). Thus, a threshold of the penalty parameter is given such that, for all penalty parameters exceeding this threshold, equivalence holds between the set of optimal solutions in the considered constrained optimization problem and the set of minimizer in its associated penalized problem with an exact minimax penalty function. It is shown that coercivity is not suf- ficient to prove the results.
文摘In this paper, the general exact penalty functions in integer programming were studied. The conditions which ensure the exact penalty property for the general penalty function with one penalty parameter were given and a general penalty function with two parameters was proposed.
基金supported by the National Natural science Foundation of china(70271069)
文摘The purpose of this paper is to consider the expected value of a discounted penalty due at ruin in the Erlang(2) risk process under constant interest force. An integro-differential equation satisfied by the expected value and a second-order differential equation for the Laplace transform of the expected value are derived. In addition, the paper will present the recursive algorithm for the joint distribution of the surplus immediately before ruin and the deficit at ruin. Finally, by the differential equation, the defective renewal equation and the explicit expression for the expected value are given in the interest-free case.
文摘In this paper, based on the inherent characteristic of the contention relation between flows in ad hoc networks, we introduce the notion of the link's interference set, extend the utility maximization problem representing congestion control in wireline networks to ad hoc networks, apply the penalty function approach and the subgradient method to solve this problem, and propose the congestion control algorithm Penalty function-based Optical Congestion Control (POCC) which is implemented in NS2- simulator. Specifically, each link transmits periodically the information on its congestion state to its interference set; the set ; the sermon at each source adjusts the transmission rate based on the optimal tradeoffbetween the utility value and the congestion level which the interference set of the links that this session goes though suffers from. MATLAB-based simulation results showed that POCC can approach the globally optimal solution. The NS2-based simulation results showed that POCC outperforms default TCP and ATCP to achieve efficient and fair resource allocation in ad hoc networks.
文摘In this paper, a new algorithm-approximate penalty function method is designed, which can be used to solve a bilevel optimization problem with linear constrained function. In this kind of bilevel optimization problem. the evaluation of the objective function is very difficult, so that only their approximate values can be obtained. This algorithm is obtained by combining penalty function method and approximation in bilevel programming. The presented algorithm is completely different from existing methods. That convergence for this algorithm is proved.
文摘By the aid of the penalty function method, the equilibrium restriction conditions were introduced to the isoparametric hybrid finite element analysis, and the concrete application course of the penalty function method in three-dimensional isoparametdc hybrid finite element was discussed. The separated penalty parameters method and the optimal hybrid element model with penalty balance were also presented. The penalty balance method can effectively refrain the parasitical stress on the premise of no additional degrees of freedom. The numeric experiment shows that the presented element not only is effective in improving greatly the numeric calculation precision of distorted grids but also has the universality.
文摘A class of discontinuous penalty functions was proposed to solve constrained minimization problems with the integral approach to global optimization, m-mean value and v-variance optimality conditions of a constrained and penalized minimization problem were investigated. A nonsequential algorithm was proposed. Numerical examples were given to illustrate the effectiveness of the algorithm.
文摘In this paper, an approximate smoothing approach to the non-differentiable exact penalty function is proposed for the constrained optimization problem. A simple smoothed penalty algorithm is given, and its convergence is discussed. A practical algorithm to compute approximate optimal solution is given as well as computational experiments to demonstrate its efficiency.
基金supported by the Grant of the Academy of Mathematics and System Science of Chinese Academy of Sciences-The Hong Kong Polytechnic University Joint Research Institute (AMSS-PolyU)the Research Grands Council Grant of The Hong Kong Polytechnic University (No. 5365/09E)
文摘This paper introduces a new exact and smooth penalty function to tackle constrained min-max problems. By using this new penalty function and adding just one extra variable, a constrained rain-max problem is transformed into an unconstrained optimization one. It is proved that, under certain reasonable assumptions and when the penalty parameter is sufficiently large, the minimizer of this unconstrained optimization problem is equivalent to the minimizer of the original constrained one. Numerical results demonstrate that this penalty function method is an effective and promising approach for solving constrained finite min-max problems.
文摘We present an approximation-exact penalty function method for solving the single stage stochastic programming problem with continuous random variable. The original problem is transformed into a determinate nonlinear programming problem with a discrete random variable sequence, which is obtained by some discrete method. We construct an exact penalty function and obtain an unconstrained optimization. It avoids the difficulty in solution by the rapid growing of the number of constraints for discrete precision. Under lenient conditions, we prove the equivalence of the minimum solution of penalty function and the solution of the determinate programming, and prove that the solution sequences of the discrete problem converge to a solution to the original problem.
文摘The penalty function method is one basic method for solving constrained nonlinear programming, in which simple smooth exact penalty functions draw much attention for their simpleness and smoothness. This article offers a new kind of simple smooth approximative exact penalty function of general constrained nonlinear programmings and analyzes its properties.
文摘In this paper, a new augmented Lagrangian penalty function for constrained optimization problems is studied. The dual properties of the augmented Lagrangian objective penalty function for constrained optimization problems are proved. Under some conditions, the saddle point of the augmented Lagrangian objective penalty function satisfies the first-order Karush-Kuhn-Tucker (KKT) condition. Especially, when the KKT condition holds for convex programming its saddle point exists. Based on the augmented Lagrangian objective penalty function, an algorithm is developed for finding a global solution to an inequality constrained optimization problem and its global convergence is also proved under some conditions.
基金supported by National Natural Science Foundation of China(72031009 and 71871171)the National Social Science Foundation of China(20&ZD058).
文摘In this paper,we propose an iterative algorithm to find the optimal incentive mechanism for the principal-agent problem under moral hazard where the number of agent action profiles is infinite,and where there are an infinite number of results that can be observed by the principal.This principal-agent problem has an infinite number of incentive-compatibility constraints,and we transform it into an optimization problem with an infinite number of constraints called a semi-infinite programming problem.We then propose an exterior penalty function method to find the optimal solution to this semi-infinite programming and illustrate the convergence of this algorithm.By analyzing the optimal solution obtained by the proposed penalty function method,we can obtain the optimal incentive mechanism for the principal-agent problem with an infinite number of incentive-compatibility constraints under moral hazard.
文摘In this papert a recursive quadratic programming algorithm is proposed andstudied.The line search functions used are Han's nondifferentiable penalty functionswith a second order penalty term. In order to avoid maratos effect,Fukushima's mixeddirection is used as the direction of line search.Finallyt we prove the global convergenceand the local second order convergence of the algorithm.
文摘In this paper, we present an algorithm to solve the inequality constrained multi-objective programming (MP) by using a penalty function with objective parameters and constraint penalty parameter. First, the penalty function with objective parameters and constraint penalty parameter for MP and the corresponding unconstraint penalty optimization problem (UPOP) is defined. Under some conditions, a Pareto efficient solution (or a weakly-efficient solution) to UPOP is proved to be a Pareto efficient solution (or a weakly-efficient solution) to MP. The penalty function is proved to be exact under a stable condition. Then, we design an algorithm to solve MP and prove its convergence. Finally, numerical examples show that the algorithm may help decision makers to find a satisfactory solution to MP.
文摘By using the penalty function method with objective parameters, the paper presents an interactive algorithm to solve the inequality constrained multi-objective programming (MP). The MP is transformed into a single objective optimal problem (SOOP) with inequality constrains;and it is proved that, under some conditions, an optimal solution to SOOP is a Pareto efficient solution to MP. Then, an interactive algorithm of MP is designed accordingly. Numerical examples show that the algorithm can find a satisfactory solution to MP with objective weight value adjusted by decision maker.
文摘We propose a new unified path to approximately smoothing the nonsmooth exact penalty function in this paper. Based on the new smooth penalty function, we give a penalty algorithm to solve the constrained optimization problem, and discuss the convergence of the algorithm under mild conditions.