Due to the heterogeneity of rock masses and the variability of in situ stress,the traditional linear inversion method is insufficiently accurate to achieve high accuracy of the in situ stress field.To address this cha...Due to the heterogeneity of rock masses and the variability of in situ stress,the traditional linear inversion method is insufficiently accurate to achieve high accuracy of the in situ stress field.To address this challenge,nonlinear stress boundaries for a numerical model are determined through regression analysis of a series of nonlinear coefficient matrices,which are derived from the bubbling method.Considering the randomness and flexibility of the bubbling method,a parametric study is conducted to determine recommended ranges for these parameters,including the standard deviation(σb)of bubble radii,the non-uniform coefficient matrix number(λ)for nonlinear stress boundaries,and the number(m)and positions of in situ stress measurement points.A model case study provides a reference for the selection of these parameters.Additionally,when the nonlinear in situ stress inversion method is employed,stress distortion inevitably occurs near model boundaries,aligning with the Saint Venant's principle.Two strategies are proposed accordingly:employing a systematic reduction of nonlinear coefficients to achieve high inversion accuracy while minimizing significant stress distortion,and excluding regions with severe stress distortion near the model edges while utilizing the central part of the model for subsequent simulations.These two strategies have been successfully implemented in the nonlinear in situ stress inversion of the Xincheng Gold Mine and have achieved higher inversion accuracy than the linear method.Specifically,the linear and nonlinear inversion methods yield root mean square errors(RMSE)of 4.15 and 3.2,and inversion relative errors(δAve)of 22.08%and 17.55%,respectively.Therefore,the nonlinear inversion method outperforms the traditional multiple linear regression method,even in the presence of a systematic reduction in the nonlinear stress boundaries.展开更多
In this paper,we study the minimax linear fractional programming problem on a non-empty bounded set,called problem(MLFP),and we design a branch and bound algorithm to find a globally optimal solution of(MLFP).Firstly,...In this paper,we study the minimax linear fractional programming problem on a non-empty bounded set,called problem(MLFP),and we design a branch and bound algorithm to find a globally optimal solution of(MLFP).Firstly,we convert the problem(MLFP)to a problem(EP2)that is equivalent to it.Secondly,by applying the convex relaxation technique to problem(EP2),a convex quadratic relaxation problem(CQRP)is obtained.Then,the overall framework of the algorithm is given and its convergence is proved,the worst-case iteration number is also estimated.Finally,experimental data are listed to illustrate the effectiveness of the algorithm.展开更多
This paper presents a new dimension reduction strategy for medium and large-scale linear programming problems. The proposed method uses a subset of the original constraints and combines two algorithms: the weighted av...This paper presents a new dimension reduction strategy for medium and large-scale linear programming problems. The proposed method uses a subset of the original constraints and combines two algorithms: the weighted average and the cosine simplex algorithm. The first approach identifies binding constraints by using the weighted average of each constraint, whereas the second algorithm is based on the cosine similarity between the vector of the objective function and the constraints. These two approaches are complementary, and when used together, they locate the essential subset of initial constraints required for solving medium and large-scale linear programming problems. After reducing the dimension of the linear programming problem using the subset of the essential constraints, the solution method can be chosen from any suitable method for linear programming. The proposed approach was applied to a set of well-known benchmarks as well as more than 2000 random medium and large-scale linear programming problems. The results are promising, indicating that the new approach contributes to the reduction of both the size of the problems and the total number of iterations required. A tree-based classification model also confirmed the need for combining the two approaches. A detailed numerical example, the general numerical results, and the statistical analysis for the decision tree procedure are presented.展开更多
An improved genetic algorithm（IGA） based on a novel selection strategy to handle nonlinear programming problems is proposed.Each individual in selection process is represented as a three-dimensional feature vector w...An improved genetic algorithm（IGA） based on a novel selection strategy to handle nonlinear programming problems is proposed.Each individual in selection process is represented as a three-dimensional feature vector which is composed of objective function value,the degree of constraints violations and the number of constraints violations.It is easy to distinguish excellent individuals from general individuals by using an individuals＇ feature vector.Additionally,a local search（LS） process is incorporated into selection operation so as to find feasible solutions located in the neighboring areas of some infeasible solutions.The combination of IGA and LS should offer the advantage of both the quality of solutions and diversity of solutions.Experimental results over a set of benchmark problems demonstrate that IGA has better performance than other algorithms.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
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
In this paper, a class of augmented Lagrangiaus of Di Pillo and Grippo （DGALs） was considered, for solving equality-constrained problems via unconstrained minimization techniques. The relationship was further discus...In this paper, a class of augmented Lagrangiaus of Di Pillo and Grippo （DGALs） was considered, for solving equality-constrained problems via unconstrained minimization techniques. The relationship was further discussed between the uneonstrained minimizers of DGALs on the product space of problem variables and multipliers, and the solutions of the eonstrained problem and the corresponding values of the Lagrange multipliers. The resulting properties indicate more precisely that this class of DGALs is exact multiplier penalty functions. Therefore, a solution of the equslity-constralned problem and the corresponding values of the Lagrange multipliers can be found by performing a single unconstrained minimization of a DGAL on the product space of problem variables and multipliers.展开更多
In the present work, two new, （multi-）parametric programming （mp-P）-inspired algorithms for the solutionof mixed-integer nonlinear programming （MINLP） problems are developed, with their main focus being onproces...In the present work, two new, （multi-）parametric programming （mp-P）-inspired algorithms for the solutionof mixed-integer nonlinear programming （MINLP） problems are developed, with their main focus being onprocess synthesis problems. The algorithms are developed for the special case in which the nonlinearitiesarise because of logarithmic terms, with the first one being developed for the deterministic case, and thesecond for the parametric case （p-MINLP）. The key idea is to formulate and solve the square system of thefirst-order Karush-Kuhn-Tucker （KKT） conditions in an analytical way, by treating the binary variables and/or uncertain parameters as symbolic parameters. To this effect, symbolic manipulation and solution tech-niques are employed. In order to demonstrate the applicability and validity of the proposed algorithms, twoprocess synthesis case studies are examined. The corresponding solutions are then validated using state-of-the-art numerical MINLP solvers. For p-MINLP, the solution is given by an optimal solution as an explicitfunction of the uncertain parameters.展开更多
In this paper, on the basis of the logarithmic barrier function and KKT conditions, we propose a combined homotopy infeasible interior-point method （CHIIP） for convex nonlinear programming problems. For any convex n...In this paper, on the basis of the logarithmic barrier function and KKT conditions, we propose a combined homotopy infeasible interior-point method （CHIIP） for convex nonlinear programming problems. For any convex nonlinear programming, without strict convexity for the logarithmic barrier function, we get different solutions of the convex programming in different cases by CHIIP method.展开更多
A penalized interior point approach for constrained nonlinear programming is examined in this work. To overcome the difficulty of initialization for the interior point method, a problem equivalent to the primal proble...A penalized interior point approach for constrained nonlinear programming is examined in this work. To overcome the difficulty of initialization for the interior point method, a problem equivalent to the primal problem via incorporating an auxiliary variable is constructed. A combined approach of logarithm barrier and quadratic penalty function is proposed to solve the problem. Based on Newton＇s method, the global convergence of interior point and line search algorithm is proven. Only a finite number of iterations is required to reach an approximate optimal solution. Numerical tests are given to show the effectiveness of the method.展开更多
Nonlinear time series prediction is studied by using an improved least squares support vector machine （LSSVM） regression based on chaotic mutation evolutionary programming （CMEP） approach for parameter optimizatio...Nonlinear time series prediction is studied by using an improved least squares support vector machine （LSSVM） regression based on chaotic mutation evolutionary programming （CMEP） approach for parameter optimization. We analyze how the prediction error varies with different parameters （σ, γ） in LS-SVM. In order to select appropriate parameters for the prediction model, we employ CMEP algorithm. Finally, Nasdaq stock data are predicted by using this LS-SVM regression based on CMEP, and satisfactory results are obtained.展开更多
Filled function method is an approach to find the global minimum of nonlinear functions. Many Problems, such as computing,communication control, and management, in real applications naturally result in global optimiza...Filled function method is an approach to find the global minimum of nonlinear functions. Many Problems, such as computing,communication control, and management, in real applications naturally result in global optimization formulations in a form ofnonlinear global integer programming. This paper gives a modified filled function method to solve the nonlinear global integerprogramming problem. The properties of the proposed modified filled function are also discussed in this paper. The results ofpreliminary numerical experiments are also reported.展开更多
Converting the balance equation of the branch of a mine ventilation network into an equivalent nonlinearprogramming problem,this paper proves that the total sum of the energy loss in every branch will be a minimumwhen...Converting the balance equation of the branch of a mine ventilation network into an equivalent nonlinearprogramming problem,this paper proves that the total sum of the energy loss in every branch will be a minimumwhen the airflow distribution in the networks is in a balanced state.The energy means of solving the networkequations by nodal methods is also noted,and a theorem for the unique existence of the solution for a networkbalance equation is give.An example is used to explain these conclusions.展开更多
Planning for water quality management is important for facilitating sustainable socio-economic development;however, the planning is also complicated by a variety of uncertainties and nonlinearities. In this study, an ...Planning for water quality management is important for facilitating sustainable socio-economic development;however, the planning is also complicated by a variety of uncertainties and nonlinearities. In this study, an interval-parameter fuzzy robust nonlinear programming (IFRNP) model was developed for water quality management to deal with such difficulties. The developed model incorporated interval nonlinear programming (INP) and fuzzy robust programming (FRP) methods within a general optimization framework. The developed IFRNP model not only could explicitly deal with uncertainties represented as discrete interval numbers and fuzzy membership functions, but also was able to deal with nonlinearities in the objective function.展开更多
By redefining the multiplier associated with inequality constraint as a positive definite function of the originally-defined multiplier, say, u2_i, i=1, 2, ..., m, nonnegative constraints imposed on inequality constra...By redefining the multiplier associated with inequality constraint as a positive definite function of the originally-defined multiplier, say, u2_i, i=1, 2, ..., m, nonnegative constraints imposed on inequality constraints in Karush-Kuhn-Tucker necessary conditions are removed. For constructing the Lagrange neural network and Lagrange multiplier method, it is no longer necessary to convert inequality constraints into equality constraints by slack variables in order to reuse those results dedicated to equality constraints, and they can be similarly proved with minor modification. Utilizing this technique, a new type of Lagrange neural network and a new type of Lagrange multiplier method are devised, which both handle inequality constraints directly. Also, their stability and convergence are analyzed rigorously.展开更多
An exact augmented Lagrangian function for the nonlinear nonconvex programming problems with inequality constraints was discussed. Under suitable hypotheses, the relationship was established between the local unconstr...An exact augmented Lagrangian function for the nonlinear nonconvex programming problems with inequality constraints was discussed. Under suitable hypotheses, the relationship was established between the local unconstrained minimizers of the augmented Lagrangian function on the space of problem variables and the local minimizers of the original constrained problem. Furthermore, under some assumptions, the relationship was also established between the global solutions of the augmented Lagrangian function on some compact subset of the space of problem variables and the global solutions of the constrained problem. Therefore, f^om the theoretical point of view, a solution of the inequality constrained problem and the corresponding values of the Lagrange multipliers can be found by the well-known method of multipliers which resort to the unconstrained minimization of the augmented Lagrangian function presented.展开更多
In this paper a new approach for obtaining an approximation global optimum solution of zero-one nonlinear programming (0-1 NP) problem which we call it Parametric Linearization Approach (P.L.A) is proposed. By using t...In this paper a new approach for obtaining an approximation global optimum solution of zero-one nonlinear programming (0-1 NP) problem which we call it Parametric Linearization Approach (P.L.A) is proposed. By using this approach the problem is transformed to a sequence of linear programming problems. The approximately solution of the original 0-1 NP problem is obtained based on the optimum values of the objective functions of this sequence of linear programming problems defined by (P.L.A).展开更多
It’s well known that the solution of equations always uses complicated methods. In this paper the first integral method is used to find the actual solution of equations in a simple way, rather than the ex-complicated...It’s well known that the solution of equations always uses complicated methods. In this paper the first integral method is used to find the actual solution of equations in a simple way, rather than the ex-complicated ways. Therefore, the use of first integral method makes the solution more available and easy to investigate behavior waves through its solution. First integral method is used to find exact solutions to the general formula and the applications of the results to the linear and nonlinear equations.展开更多
A universal numerical approach for nonlinear mathematic programming problems is presented with an application of ratios of first-order differentials/differences of objective functions to constraint functions with resp...A universal numerical approach for nonlinear mathematic programming problems is presented with an application of ratios of first-order differentials/differences of objective functions to constraint functions with respect to design variables. This approach can be efficiently used to solve continuous and, in particular, discrete programmings with arbitrary design variables and constraints. As a search method, this approach requires only computations of the functions and their partial derivatives or differences with respect to design variables, rather than any solution of mathematic equations. The present approach has been applied on many numerical examples as well as on some classical operational problems such as one-dimensional and two-dimensional knap-sack problems, one-dimensional and two-dimensional resource-distribution problems, problems of working reliability of composite systems and loading problems of machine, and more efficient and reliable solutions are obtained than traditional methods. The present approach can be used without limitation of modeling scales of the problem. Optimum solutions can be guaranteed as long as the objective function, constraint functions and their First-order derivatives/differences exist in the feasible domain or feasible set. There are no failures of convergence and instability when this approach is adopted.展开更多
基金funded by the National Key R&D Program of China(Grant No.2022YFC2903904)the National Natural Science Foundation of China(Grant Nos.51904057 and U1906208).
文摘Due to the heterogeneity of rock masses and the variability of in situ stress,the traditional linear inversion method is insufficiently accurate to achieve high accuracy of the in situ stress field.To address this challenge,nonlinear stress boundaries for a numerical model are determined through regression analysis of a series of nonlinear coefficient matrices,which are derived from the bubbling method.Considering the randomness and flexibility of the bubbling method,a parametric study is conducted to determine recommended ranges for these parameters,including the standard deviation(σb)of bubble radii,the non-uniform coefficient matrix number(λ)for nonlinear stress boundaries,and the number(m)and positions of in situ stress measurement points.A model case study provides a reference for the selection of these parameters.Additionally,when the nonlinear in situ stress inversion method is employed,stress distortion inevitably occurs near model boundaries,aligning with the Saint Venant's principle.Two strategies are proposed accordingly:employing a systematic reduction of nonlinear coefficients to achieve high inversion accuracy while minimizing significant stress distortion,and excluding regions with severe stress distortion near the model edges while utilizing the central part of the model for subsequent simulations.These two strategies have been successfully implemented in the nonlinear in situ stress inversion of the Xincheng Gold Mine and have achieved higher inversion accuracy than the linear method.Specifically,the linear and nonlinear inversion methods yield root mean square errors(RMSE)of 4.15 and 3.2,and inversion relative errors(δAve)of 22.08%and 17.55%,respectively.Therefore,the nonlinear inversion method outperforms the traditional multiple linear regression method,even in the presence of a systematic reduction in the nonlinear stress boundaries.
基金Supported by the National Natural Science Foundation of China(Grant Nos.12071133 and 11871196).
文摘In this paper,we study the minimax linear fractional programming problem on a non-empty bounded set,called problem(MLFP),and we design a branch and bound algorithm to find a globally optimal solution of(MLFP).Firstly,we convert the problem(MLFP)to a problem(EP2)that is equivalent to it.Secondly,by applying the convex relaxation technique to problem(EP2),a convex quadratic relaxation problem(CQRP)is obtained.Then,the overall framework of the algorithm is given and its convergence is proved,the worst-case iteration number is also estimated.Finally,experimental data are listed to illustrate the effectiveness of the algorithm.
文摘This paper presents a new dimension reduction strategy for medium and large-scale linear programming problems. The proposed method uses a subset of the original constraints and combines two algorithms: the weighted average and the cosine simplex algorithm. The first approach identifies binding constraints by using the weighted average of each constraint, whereas the second algorithm is based on the cosine similarity between the vector of the objective function and the constraints. These two approaches are complementary, and when used together, they locate the essential subset of initial constraints required for solving medium and large-scale linear programming problems. After reducing the dimension of the linear programming problem using the subset of the essential constraints, the solution method can be chosen from any suitable method for linear programming. The proposed approach was applied to a set of well-known benchmarks as well as more than 2000 random medium and large-scale linear programming problems. The results are promising, indicating that the new approach contributes to the reduction of both the size of the problems and the total number of iterations required. A tree-based classification model also confirmed the need for combining the two approaches. A detailed numerical example, the general numerical results, and the statistical analysis for the decision tree procedure are presented.
基金supported by the National Natural Science Foundation of China (60632050)National Basic Research Program of Jiangsu Province University (08KJB520003)
文摘An improved genetic algorithm（IGA） based on a novel selection strategy to handle nonlinear programming problems is proposed.Each individual in selection process is represented as a three-dimensional feature vector which is composed of objective function value,the degree of constraints violations and the number of constraints violations.It is easy to distinguish excellent individuals from general individuals by using an individuals＇ feature vector.Additionally,a local search（LS） process is incorporated into selection operation so as to find feasible solutions located in the neighboring areas of some infeasible solutions.The combination of IGA and LS should offer the advantage of both the quality of solutions and diversity of solutions.Experimental results over a set of benchmark problems demonstrate that IGA has better performance than other algorithms.
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.
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In this paper, a class of augmented Lagrangiaus of Di Pillo and Grippo （DGALs） was considered, for solving equality-constrained problems via unconstrained minimization techniques. The relationship was further discussed between the uneonstrained minimizers of DGALs on the product space of problem variables and multipliers, and the solutions of the eonstrained problem and the corresponding values of the Lagrange multipliers. The resulting properties indicate more precisely that this class of DGALs is exact multiplier penalty functions. Therefore, a solution of the equslity-constralned problem and the corresponding values of the Lagrange multipliers can be found by performing a single unconstrained minimization of a DGAL on the product space of problem variables and multipliers.
基金financial support from EPSRC grants (EP/M027856/1 EP/M028240/1)
文摘In the present work, two new, （multi-）parametric programming （mp-P）-inspired algorithms for the solutionof mixed-integer nonlinear programming （MINLP） problems are developed, with their main focus being onprocess synthesis problems. The algorithms are developed for the special case in which the nonlinearitiesarise because of logarithmic terms, with the first one being developed for the deterministic case, and thesecond for the parametric case （p-MINLP）. The key idea is to formulate and solve the square system of thefirst-order Karush-Kuhn-Tucker （KKT） conditions in an analytical way, by treating the binary variables and/or uncertain parameters as symbolic parameters. To this effect, symbolic manipulation and solution tech-niques are employed. In order to demonstrate the applicability and validity of the proposed algorithms, twoprocess synthesis case studies are examined. The corresponding solutions are then validated using state-of-the-art numerical MINLP solvers. For p-MINLP, the solution is given by an optimal solution as an explicitfunction of the uncertain parameters.
文摘In this paper, on the basis of the logarithmic barrier function and KKT conditions, we propose a combined homotopy infeasible interior-point method （CHIIP） for convex nonlinear programming problems. For any convex nonlinear programming, without strict convexity for the logarithmic barrier function, we get different solutions of the convex programming in different cases by CHIIP method.
基金supported by the National Natural Science Foundation of China (Grant No.10771133)the Shanghai Leading Academic Discipline Project (Grant Nos.J50101, S30104)
文摘A penalized interior point approach for constrained nonlinear programming is examined in this work. To overcome the difficulty of initialization for the interior point method, a problem equivalent to the primal problem via incorporating an auxiliary variable is constructed. A combined approach of logarithm barrier and quadratic penalty function is proposed to solve the problem. Based on Newton＇s method, the global convergence of interior point and line search algorithm is proven. Only a finite number of iterations is required to reach an approximate optimal solution. Numerical tests are given to show the effectiveness of the method.
基金Supported by National High Technology Research and Development Program of China （863 Program） （2006AA04Z183）, National Nat- ural Science Foundation of China （60621001, 60534010, 60572070, 60774048, 60728307）, and the Program for Changjiang Scholars and Innovative Research Groups of China （60728307, 4031002）
基金The project supported by National Natural Science Foundation of China under Grant No. 90203008 and the Doctoral Foundation of the Ministry of Education of China
文摘Nonlinear time series prediction is studied by using an improved least squares support vector machine （LSSVM） regression based on chaotic mutation evolutionary programming （CMEP） approach for parameter optimization. We analyze how the prediction error varies with different parameters （σ, γ） in LS-SVM. In order to select appropriate parameters for the prediction model, we employ CMEP algorithm. Finally, Nasdaq stock data are predicted by using this LS-SVM regression based on CMEP, and satisfactory results are obtained.
文摘Filled function method is an approach to find the global minimum of nonlinear functions. Many Problems, such as computing,communication control, and management, in real applications naturally result in global optimization formulations in a form ofnonlinear global integer programming. This paper gives a modified filled function method to solve the nonlinear global integerprogramming problem. The properties of the proposed modified filled function are also discussed in this paper. The results ofpreliminary numerical experiments are also reported.
文摘Converting the balance equation of the branch of a mine ventilation network into an equivalent nonlinearprogramming problem,this paper proves that the total sum of the energy loss in every branch will be a minimumwhen the airflow distribution in the networks is in a balanced state.The energy means of solving the networkequations by nodal methods is also noted,and a theorem for the unique existence of the solution for a networkbalance equation is give.An example is used to explain these conclusions.
文摘Planning for water quality management is important for facilitating sustainable socio-economic development;however, the planning is also complicated by a variety of uncertainties and nonlinearities. In this study, an interval-parameter fuzzy robust nonlinear programming (IFRNP) model was developed for water quality management to deal with such difficulties. The developed model incorporated interval nonlinear programming (INP) and fuzzy robust programming (FRP) methods within a general optimization framework. The developed IFRNP model not only could explicitly deal with uncertainties represented as discrete interval numbers and fuzzy membership functions, but also was able to deal with nonlinearities in the objective function.
文摘By redefining the multiplier associated with inequality constraint as a positive definite function of the originally-defined multiplier, say, u2_i, i=1, 2, ..., m, nonnegative constraints imposed on inequality constraints in Karush-Kuhn-Tucker necessary conditions are removed. For constructing the Lagrange neural network and Lagrange multiplier method, it is no longer necessary to convert inequality constraints into equality constraints by slack variables in order to reuse those results dedicated to equality constraints, and they can be similarly proved with minor modification. Utilizing this technique, a new type of Lagrange neural network and a new type of Lagrange multiplier method are devised, which both handle inequality constraints directly. Also, their stability and convergence are analyzed rigorously.
文摘An exact augmented Lagrangian function for the nonlinear nonconvex programming problems with inequality constraints was discussed. Under suitable hypotheses, the relationship was established between the local unconstrained minimizers of the augmented Lagrangian function on the space of problem variables and the local minimizers of the original constrained problem. Furthermore, under some assumptions, the relationship was also established between the global solutions of the augmented Lagrangian function on some compact subset of the space of problem variables and the global solutions of the constrained problem. Therefore, f^om the theoretical point of view, a solution of the inequality constrained problem and the corresponding values of the Lagrange multipliers can be found by the well-known method of multipliers which resort to the unconstrained minimization of the augmented Lagrangian function presented.
文摘In this paper a new approach for obtaining an approximation global optimum solution of zero-one nonlinear programming (0-1 NP) problem which we call it Parametric Linearization Approach (P.L.A) is proposed. By using this approach the problem is transformed to a sequence of linear programming problems. The approximately solution of the original 0-1 NP problem is obtained based on the optimum values of the objective functions of this sequence of linear programming problems defined by (P.L.A).
文摘It’s well known that the solution of equations always uses complicated methods. In this paper the first integral method is used to find the actual solution of equations in a simple way, rather than the ex-complicated ways. Therefore, the use of first integral method makes the solution more available and easy to investigate behavior waves through its solution. First integral method is used to find exact solutions to the general formula and the applications of the results to the linear and nonlinear equations.
文摘A universal numerical approach for nonlinear mathematic programming problems is presented with an application of ratios of first-order differentials/differences of objective functions to constraint functions with respect to design variables. This approach can be efficiently used to solve continuous and, in particular, discrete programmings with arbitrary design variables and constraints. As a search method, this approach requires only computations of the functions and their partial derivatives or differences with respect to design variables, rather than any solution of mathematic equations. The present approach has been applied on many numerical examples as well as on some classical operational problems such as one-dimensional and two-dimensional knap-sack problems, one-dimensional and two-dimensional resource-distribution problems, problems of working reliability of composite systems and loading problems of machine, and more efficient and reliable solutions are obtained than traditional methods. The present approach can be used without limitation of modeling scales of the problem. Optimum solutions can be guaranteed as long as the objective function, constraint functions and their First-order derivatives/differences exist in the feasible domain or feasible set. There are no failures of convergence and instability when this approach is adopted.
