Global convergent algorithm for the bilevel linear fractional-linear programming based on modified convex simplex method

A global convergent algorithm is proposed to solve bilevel linear fractional-linear programming, which is a special class of bilevel programming. In our algorithm, replacing the lower level problem by its dual gap equaling to zero, the bilevel linear fractional-linear programming is transformed into a traditional sin- gle level programming problem, which can be transformed into a series of linear fractional programming problem. Thus, the modi- fied convex simplex method is used to solve the infinite linear fractional programming to obtain the global convergent solution of the original bilevel linear fractional-linear programming. Finally, an example demonstrates the feasibility of the proposed algorithm.

The Pivot Adaptive Method for Solving Linear Programming Problems

A new variant of the Adaptive Method (AM) of Gabasov is presented, to minimize the computation time. Unlike the original method and its some variants, we need not to compute the inverse of the basic matrix at each iteration, or to solve the linear systems with the basic matrix. In fact, to compute the new support feasible solution, the simplex pivoting rule is used by introducing a matrix that we will define. This variant is called “the Pivot Adaptive Method” (PAM);it allows presenting the resolution of a given problem under the shape of successive tables as we will see in example. The proofs that are not given by Gabasov will also be presented here, namely the proofs for the theorem of the optimality criterion and for the theorem of existence of an optimal support, and at the end, a brief comparison between our method and the Simplex Method will be given.

A NEW FRAMEWORK OF PRIMAL-DUAL INFEASIBLE INTERIOR-POINT METHOD FOR LINEAR PROGRAMMING

On the basis of the formulations of the logarithmic barrier function and the idea of following the path of minimizers for the logarithmic barrier family of problems the so called "centralpath" for linear programming, we propose a new framework of primal-dual infeasible interiorpoint method for linear programming problems. Without the strict convexity of the logarithmic barrier function, we get the following results: (a) if the homotopy parameterμcan not reach to zero,then the feasible set of these programming problems is empty; (b) if the strictly feasible set is nonempty and the solution set is bounded, then for any initial point x, we can obtain a solution of the problems by this method; (c) if the strictly feasible set is nonempty and the solution set is unbounded, then for any initial point x, we can obtain a (?)-solution; and(d) if the strictly feasible set is nonempty and the solution set is empty, then we can get the curve x(μ), which towards to the generalized solutions.

Diagnosis and Resolution of Infeasibility in the Constraint Method for Solving Multi Objective Linear Programming Problems

In this paper we discuss about infeasibility diagnosis and infeasibility resolution, when the constraint method is used for solving multi objective linear programming problems. We propose an algorithm for resolution of infeasibility, which is a combination of interactive, weighting and constraint methods.Numerical examples are provided to illustrate the techniques developed.

A New Interactive Method to Solve Multiobjective Linear Programming Problems

Multiobjective Programming (MOP) has become famous among many researchers due to more practical and realistic applications. A lot of methods have been proposed especially during the past four decades. In this paper, we develop a new algorithm based on a new approach to solve MOP by starting from a utopian point, which is usually infeasible, and moving towards the feasible region via stepwise movements and a simple continuous interaction with decision maker. We consider the case where all objective functions and constraints are linear. The implementation of the pro-posed algorithm is demonstrated by two numerical examples.

Solving Multi-Objective Linear Programming Problem by Statistical Averaging Method with the Help of Fuzzy Programming Method

A multi-objective linear programming problem is made from fuzzy linear programming problem. It is due the fact that it is used fuzzy programming method during the solution. The Multi objective linear programming problem can be converted into the single objective function by various methods as Chandra Sen’s method, weighted sum method, ranking function method, statistical averaging method. In this paper, Chandra Sen’s method and statistical averaging method both are used here for making single objective function from multi-objective function. Two multi-objective programming problems are solved to verify the result. One is numerical example and the other is real life example. Then the problems are solved by ordinary simplex method and fuzzy programming method. It can be seen that fuzzy programming method gives better optimal values than the ordinary simplex method.

Solving Fuzzy Multi-Objective Linear Programming Problem by Applying Statistical Method

In this paper, the statistical averaging method and the new statistical averaging methods have been used to solve the fuzzy multi-objective linear programming problems. These methods have been applied to form a single objective function from the fuzzy multi-objective linear programming problems. At first, a numerical example of solving fuzzy multi-objective linear programming problem has been provided to validate the maximum risk reduction by the proposed method. The proposed method has been applied to assess the risk of damage due to natural calamities like flood, cyclone, sidor, and storms at the coastal areas in Bangladesh. The proposed method of solving the fuzzy multi-objective linear programming problems by the statistical method has been compared with the Chandra Sen’s method. The numerical results show that the proposed method maximizes the risk reduction capacity better than Chandra Sen’s method.

A Dynamic Active-Set Method for Linear Programming

An efficient active-set approach is presented for both nonnegative and general linear programming by adding varying numbers of constraints at each iteration. Computational experiments demonstrate that the proposed approach is significantly faster than previous active-set and standard linear programming algorithms.

Levenberg-Marquardt Method for Mathematical Programs with Linearly Complementarity Constraints

In this paper, a new method for solving a mathematical programming problem with linearly complementarity constraints (MPLCC) is introduced, which applies the Levenberg-Marquardt (L-M) method to solve the B-stationary condition of original problem. Under the MPEC-LICQ, the proposed method is proved convergent to B-stationary point of MPLCC.

Linear Programming建模研讨

研究用图解法、simplex method和匈牙利法建立Linear Programming的数学模型并求得了最优解.结果表明:对仅有两个变量的Linear Programming,既可通过图解法求得最优解;也可用单纯形表简便地求得最优解;而对任务和人数不等的assignment problem,则用匈牙利法求最优解.

Discrete differential evolution algorithm for integer linear bilevel programming problems

A discrete differential evolution algorithm combined with the branch and bound method is developed to solve the integer linear bilevel programming problems, in which both upper level and lower level variables are forced to be integer. An integer coding for upper level variables is adopted, and then a discrete differential evolution algorithm with an improved feasibility-based comparison is developed to directly explore the integer solution at the upper level. For a given upper level integer variable, the lower level integer programming problem is solved by the existing branch and bound algorithm to obtain the optimal integer solution at the lower level. In the same framework of the algorithm, two other constraint handling methods, i.e. the penalty function method and the feasibility-based comparison method are also tested. The experimental results demonstrate that the discrete differential evolution algorithm with different constraint handling methods is effective in finding the global optimal integer solutions, but the improved constraint handling method performs better than two compared constraint handling methods.

A Glorious Literature on Linear Goal Programming Algorithms

In the last several years, there has been a marked improvement in the development of new algorithms for solving Linear Goal programming (LGP). This paper presents a survey of current methods for LGP.

Linear Programming for Optimum PID Controller Tuning

This work presents a new methodology based on Linear Programming (LP) to tune Proportional-Integral-Derivative (PID) control parameters. From a specification of a desired output time domain of the plant, a linear optimization system is proposed to adjust the PID controller leading the output signal to stable operation condition with minimum oscillations. The constraint set used in the optimization process is defined by using numerical integration approach. The generated optimization problem is convex and easily solved using an interior point algorithm. Results obtained using familiar plants from literature have shown that the proposed linear programming problem is very effective for tuning PID controllers.

Primal-Dual Interior-Point Algorithms with Dynamic Step-Size Based on Kernel Functions for Linear Programming

In this paper, primal-dual interior-point algorithm with dynamic step size is implemented for linear programming （LP） problems. The algorithms are based on a few kernel functions, including both serf-regular functions and non-serf-regular ones. The dynamic step size is compared with fixed step size for the algorithms in inner iteration of Newton step. Numerical tests show that the algorithms with dynaraic step size are more efficient than those with fixed step size.

A Modified Centered Climbing Algorithm for Linear Programming

In this paper, we propose a modified centered climbing algorithm (MCCA) for linear programs, which improves the centered climbing algorithm (CCA) developed for linear programs recently. MCCA implements a specific climbing scheme where a violated constraint is probed by means of the centered vector used by CCA. Computational comparison is made with CCA and the simplex method. Numerical tests show that, on average CPU time, MCCA runs faster than both CCA and the simplex method in terms of tested problems. In addition, a simple initialization technique is introduced.

A Way to Find All the Optimal Solutions in Linear Programming

With the expression theorem of convex polyhedron, this paper gives the general expression for the solutions in standard linear programming problems. And the calculation procedures in determining the optimal solutions are also given.

Constraint Optimal Selection Techniques (COSTs) for Linear Programming

We describe a new active-set, cutting-plane Constraint Optimal Selection Technique (COST) for solving general linear programming problems. We describe strategies to bound the initial problem and simultaneously add multiple constraints. We give an interpretation of the new COST’s selection rule, which considers both the depth of constraints as well as their angles from the objective function. We provide computational comparisons of the COST with existing linear programming algorithms, including other COSTs in the literature, for some large-scale problems. Finally, we discuss conclusions and future research.

Posterior Constraint Selection for Nonnegative Linear Programming

Posterior constraint optimal selection techniques (COSTs) are developed for nonnegative linear programming problems (NNLPs), and a geometric interpretation is provided. The posterior approach is used in both a dynamic and non-dynamic active-set framework. The computational performance of these methods is compared with the CPLEX standard linear programming algorithms, with two most-violated constraint approaches, and with previously developed COST algorithms for large-scale problems.

An Improved Affine-Scaling Interior Point Algorithm for Linear Programming

In this paper, an Improved Affine-Scaling Interior Point Algorithm for Linear Programming has been proposed. Computational results of selected practical problems affirming the proposed algorithm have been provided. The proposed algorithm is accurate, faster and therefore reduces the number of iterations required to obtain an optimal solution of a given Linear Programming problem as compared to the already existing Affine-Scaling Interior Point Algorithm. The algorithm can be very useful for development of faster software packages for solving linear programming problems using the interior-point methods.

Partial Pricing Rule Simplex Method with Deficient Basis

A new partial pricing column rule is proposed to the basis-deficiency-allowing simplex method developed by Pan.Computational results obtained with a set of small problems and a set of standard NETLIB problems show its promise of success.