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
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 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, 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
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,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
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
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
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
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
The penalty function method of continuum shape optimization and its sensitivity analysis technique are presented. A relatively simple integrated shape optimization system is developed and used to optimize the design o...The penalty function method of continuum shape optimization and its sensitivity analysis technique are presented. A relatively simple integrated shape optimization system is developed and used to optimize the design of the inner frame shape of a three-axis test table. The result shows that the method converges well, and the system is stable and reliable.展开更多
In this paper, a logarithmic-exponential penalty function with two parameters for integer programming is discussed. We obtain the exact penalty properties and then establish the asymptotic strong nonlinear duality in ...In this paper, a logarithmic-exponential penalty function with two parameters for integer programming is discussed. We obtain the exact penalty properties and then establish the asymptotic strong nonlinear duality in the corresponding logarithmic-exponential dual formulation by using the obtained exact penalty properties. The discussion is based on the logarithmic-exponential nonlinear dual formulation proposed in [6].展开更多
This paper considers the expected discounted penalty function Φ(u) for the perturbed compound Poisson risk model with stochastic return on investments. After presenting an integro-differential equation that the exp...This paper considers the expected discounted penalty function Φ(u) for the perturbed compound Poisson risk model with stochastic return on investments. After presenting an integro-differential equation that the expected discounted penalty function satisfies, the paper derives the closed form solution by constructing an identical equation. The exact expression for Φ (0) is given using the Laplace transform technique when interest rate is constant. Applications of the results are given to the ruin probability and moments of the deficit at ruin.展开更多
In this paper, we consider a risk model in which each main claim may induce a delayed claim, called a by-claim. We assume that the time for the occurrence of a by-claim is random. We investigate the expected discounte...In this paper, we consider a risk model in which each main claim may induce a delayed claim, called a by-claim. We assume that the time for the occurrence of a by-claim is random. We investigate the expected discounted penalty function, and derive the defective renewal equation satisfied by it. We obtain some explicit results when the main claim and the by-claim are both exponentially distributed, respectively. We also present some numerical illustrations.展开更多
In this paper, we investigate the Gerber-Shiu discounted penalty function for the surplus process described by a piecewise deterministic Markov process (PDMP). We derive an integral equation for the Gerber-Shiu disc...In this paper, we investigate the Gerber-Shiu discounted penalty function for the surplus process described by a piecewise deterministic Markov process (PDMP). We derive an integral equation for the Gerber-Shiu discounted penalty function, and obtain the exact solution when the initial surplus is zero. Dickson formulae are also generalized to the present surplus process.展开更多
In this paper,a new objective penalty function approach is proposed for solving minimax programming problems with equality and inequality constraints.This new objective penalty function combines the objective penalty ...In this paper,a new objective penalty function approach is proposed for solving minimax programming problems with equality and inequality constraints.This new objective penalty function combines the objective penalty and constraint penalty.By the new objective penalty function,a constrained minimax problem is converted to minimizations of a sequence of continuously differentiable functions with a simple box constraint.One can thus apply any efficient gradient minimization methods to solve the minimizations with box constraint at each step of the sequence.Some relationships between the original constrained minimax problem and the corresponding minimization problems with box constraint are established.Based on these results,an algorithm for finding a global solution of the constrained minimax problems is proposed by integrating the particular structure of minimax problems and its global convergence is proved under some conditions.Furthermore,an algorithm is developed for finding a local solution of the constrained minimax problems,with its convergence proved under certain conditions.Preliminary results of numerical experiments with well-known test problems show that satisfactorilyapproximate solutions for some constrained minimax problems can be obtained.展开更多
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.
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, 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.
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.
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.
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.
文摘The penalty function method of continuum shape optimization and its sensitivity analysis technique are presented. A relatively simple integrated shape optimization system is developed and used to optimize the design of the inner frame shape of a three-axis test table. The result shows that the method converges well, and the system is stable and reliable.
基金Partially supported by the National Science Foundation of China (No.10271073)
文摘In this paper, a logarithmic-exponential penalty function with two parameters for integer programming is discussed. We obtain the exact penalty properties and then establish the asymptotic strong nonlinear duality in the corresponding logarithmic-exponential dual formulation by using the obtained exact penalty properties. The discussion is based on the logarithmic-exponential nonlinear dual formulation proposed in [6].
基金Supported by Key Project of National Social Science Fund (Grant No.06&ZD039)"Mathematics+X" Project of DUT
文摘This paper considers the expected discounted penalty function Φ(u) for the perturbed compound Poisson risk model with stochastic return on investments. After presenting an integro-differential equation that the expected discounted penalty function satisfies, the paper derives the closed form solution by constructing an identical equation. The exact expression for Φ (0) is given using the Laplace transform technique when interest rate is constant. Applications of the results are given to the ruin probability and moments of the deficit at ruin.
基金supported by 121 Young Doctorial Development Fund Project for Central University of Finance and Economics (No. QBJJJ201004)the 2011 research grant from the China Institute for Actuarial Science,Central University of Finance and Economics+2 种基金the Ministry of Education Project of Key Research Institute of Humanities and Social Sciences in Universities (No. 11JJD790004,No. 11JJD790053)The researchof Guojing Wang is supported by the Natural Science Foundation (No. KB2008155) of Jiangsu Province of Chinathe Research Fund for the Doctorial Program of Higher Education (No. 20093201110013)
文摘In this paper, we consider a risk model in which each main claim may induce a delayed claim, called a by-claim. We assume that the time for the occurrence of a by-claim is random. We investigate the expected discounted penalty function, and derive the defective renewal equation satisfied by it. We obtain some explicit results when the main claim and the by-claim are both exponentially distributed, respectively. We also present some numerical illustrations.
基金Supported by National Natural Science Foundation of China (Grant Nos. 10926161, 10901086, 10871102)National Basic Research Program of China (973 Program) 2007CB814905the Research Fund for the Doctorial Program of Higher Education
文摘In this paper, we investigate the Gerber-Shiu discounted penalty function for the surplus process described by a piecewise deterministic Markov process (PDMP). We derive an integral equation for the Gerber-Shiu discounted penalty function, and obtain the exact solution when the initial surplus is zero. Dickson formulae are also generalized to the present surplus process.
基金Supported by the National Natural Science Foundation of China (No.10771119)the Research Fund forthe Doctoral Program of Higher Education of China (No.20093705110002)
基金This research was supported by Natural Science Foundation of Chongqing(Nos.cstc2013jjB00001 and cstc2011jjA00010)by Chongqing Municipal Education Commission(No.KJ120616).
文摘In this paper,a new objective penalty function approach is proposed for solving minimax programming problems with equality and inequality constraints.This new objective penalty function combines the objective penalty and constraint penalty.By the new objective penalty function,a constrained minimax problem is converted to minimizations of a sequence of continuously differentiable functions with a simple box constraint.One can thus apply any efficient gradient minimization methods to solve the minimizations with box constraint at each step of the sequence.Some relationships between the original constrained minimax problem and the corresponding minimization problems with box constraint are established.Based on these results,an algorithm for finding a global solution of the constrained minimax problems is proposed by integrating the particular structure of minimax problems and its global convergence is proved under some conditions.Furthermore,an algorithm is developed for finding a local solution of the constrained minimax problems,with its convergence proved under certain conditions.Preliminary results of numerical experiments with well-known test problems show that satisfactorilyapproximate solutions for some constrained minimax problems can be obtained.