Showing papers on "Linear-fractional programming published in 2008"

A simplex based algorithm to solve separated continuous linear programs

Abstract: We consider the separated continuous linear programming problem with linear data. We characterize the form of its optimal solution, and present an algorithm which solves it in a finite number of steps, using an analog of the simplex method, in the space of bounded measurable functions.

Genetic algorithm based on simplex method for solving linear-quadratic bilevel programming problem

Abstract: The bilevel programming problems are useful tools for solving the hierarchy decision problems In this paper, a genetic algorithm based on the simplex method is constructed to solve the linear-quadratic bilevel programming problem (LQBP) By use of Kuhn-Tucker conditions of the lower level programming, the LQBP is transformed into a single level programming which can be simplified to a linear programming by the chromosome according to the rule Thus, in our proposed genetic algorithm, only the linear programming is solved by the simplex method to obtain the feasibility and fitness value of the chromosome Finally, the feasibility of the proposed approach is demonstrated by the example

Geometric Duality in Multiple Objective Linear Programming

Abstract: We develop in this article a geometric approach to duality in multiple objective linear programming. This approach is based on a very old idea, the duality of polytopes, which can be traced back to the old Greeks. We show that there is an inclusion-reversing one-to-one map between the minimal faces of the image of the primal objective and the maximal faces of the image of the dual objective map.

Piecewise linear dynamic programming for constrained POMDPs

Abstract: We describe an exact dynamic programming update for constrained partially observable Markov decision processes (CPOMDPs). State-of-the-art exact solution of unconstrained POMDPs relies on implicit enumeration of the vectors in the piecewise linear value function, and pruning operations to obtain a minimal representation of the updated value function. In dynamic programming for CPOMDPs, each vector takes two valuations, one with respect to the objective function and another with respect to the constraint function. The dynamic programming update consists of finding, for each belief state, the vector that has the best objective function valuation while still satisfying the constraint function. Whereas the pruning operation in an unconstrained POMDP requires solution of a linear program, the pruning operation for CPOMDPs requires solution of a mixed integer linear program.

A method of solving fully fuzzified linear fractional programming problems

Abstract: In this paper, we propose a method of solving the fully fuzzified linear fractional programming problems, where all the parameters and variables are triangular fuzzy numbers. We transform the problem of maximizing a function with triangular fuzzy value into a deterministic multiple objective linear fractional programming problem with quadratic constraints. We apply the extension principle of Zadeh to add fuzzy numbers, an approximate version of the same principle to multiply and divide fuzzy numbers and the Kerre’s method to evaluate a fuzzy constraint. The results obtained by Buckley and Feuring in 2000 applied to fractional programming and disjunctive constraints are taken into consideration here. The method needs to add extra zero-one variables for treating disjunctive constraints. In order to illustrate our method we consider a numerical example.

On the Shortest Linear Straight-Line Program for Computing Linear Forms

Abstract: We study the complexity of the Shortest Linear Program (SLP) problem, which is to minimize the number of linear operations necessary to compute a set of linear forms. SLP is shown to be NP-hard. Furthermore, a special case of the corresponding decision problem is shown to be Max SNP-Complete. Algorithms producing cancellation-free straight-line programs, those in which there is never any cancellation of variables in GF(2), have been proposed for circuit minimization for various cryptographic applications. We show that such algorithms have approximation ratios of at least 3/2 and therefore cannot be expected to yield optimal solutions to non-trivial inputs.

Redundancy Allocation for Series-Parallel Systems Using Integer Linear Programming

Abstract: We consider the problem of maximizing the reliability of a series-parallel system given cost and weight constraints on the system. The number of components in each subsystem, and the choice of components are the decision variables. In this paper, we propose an integer linear programming approach that gives an approximate feasible solution, close to the optimal solution, together with an upper bound on the optimal reliability. We show that integer linear programming is a useful approach for solving this reliability problem. The mathematical programming model is relatively simple. Its implementation is immediate by using a mathematical programming language, and integer linear programming software. And the computational experiments show that the performance of this approach is excellent based on a comparison with previous results.

An Exact Method for a Discrete Multiobjective Linear Fractional Optimization

Abstract: Integer linear fractional programming problem with multiple objective (MOILFP) is an important field of research and has not received as much attention as did multiple objective linear fractional programming. In this work, we develop a branch and cut algorithm based on continuous fractional optimization, for generating the whole integer efficient solutions of the MOILFP problem. The basic idea of the computation phase of the algorithm is to optimize one of the fractional objective functions, then generate an integer feasible solution. Using the reduced gradients of the objective functions, an efficient cut is built and a part of the feasible domain not containing efficient solutions is truncated by adding this cut. A sample problem is solved using this algorithm, and the main practical advantages of the algorithm are indicated.

A DC Programming Approach for Mixed-Integer Linear Programs

Abstract: In this paper, we propose a new efficient algorithm for globally solving a class of Mixed Integer Program (MIP). If the objective function is linear with both continuous variables and integer variables, then the problem is called a Mixed Integer Linear Program (MILP). Researches on MILP are important in both theoretical and practical aspects. Our approach for solving a general MILP is based on DC Programming and DC Algorithms. Using a suitable penalty parameter, we can reformulate MILP as a DC programming problem. By virtue of the state of the art in DC Programming research, a very efficient local nonconvex optimization method called DC Algorithm (DCA) was used. Furthermore, a robust global optimization algorithm (GOA-DCA): A hybrid method which combines DCA with a suitable Branch-and-Bound (B&B) method for globally solving general MILP problem is investigated. Moreover, this new solution method for MILP is also applicable to the Integer Linear Program (ILP). An illustrative example and some computational results, which show the robustness, the efficiency and the globality of our algorithm, are reported.

Fuzzy Multiple Objective Linear Programming

Abstract: In this chapter, first a literature review on the fuzzy multi-objective linear programming (FMOLP) and then its mathematical modeling with an application is given. FMOLP is one of the multi-objective modeling techniques most frequently used in the literature. The possible values of the parameters in FMOLP are imprecisely or ambiguously known to the experts. Therefore, it would be more appropriate for these parameters to be represented as fuzzy numerical data that can be represented by fuzzy numbers.

Modified big-M method to recognize the infeasibility of linear programming models

Abstract: This paper provides an effective modification to the big-M method which leads to reducing the iterations of this method, when it is used to recognize the infeasibility of linear systems.

Solving Stereo Matching Problems Using Interior Point Methods

Abstract: This paper describes an approach to reformulating the stereo matching problem as a large scale Linear Program. The approach proceeds by approximating the match cost function associated with each pixel with a piecewise linear convex function. Regularization terms related to the first and second derivative of the disparity field are also captured with piecewise linear penalty terms. The resulting large scale linear program can be tackled using interior point methods and the associated Newton Steps involve matrices that reflect the structure of the underlying pixel grid. The proposed scheme effectively exploits the structure of these matrices to solve these linear systems efficiently.

Open-loop and closed-loop optimization of linear control systems

Abstract: A canonical optimal control problem for linear systems with timevarying coefficients is considered in the class of discrete controls. On the basis of linear programming methods, two primal and two dual methods of constructing optimal open-loop controls are proposed. A method of synthesis of optimal feedback control is described. Results are illustrated by a fourth-order problem; estimates of efficiency of proposed methods are given.

Primal-dual stability in continuous linear optimization

Abstract: Any linear (ordinary or semi-infinite) optimization problem, and also its dual problem, can be classified as either inconsistent or bounded or unbounded, giving rise to nine duality states, three of them being precluded by the weak duality theorem. The remaining six duality states are possible in linear semi-infinite programming whereas two of them are precluded in linear programming as a consequence of the existence theorem and the non-homogeneous Farkas Lemma. This paper characterizes the linear programs and the continuous linear semi-infinite programs whose duality state is preserved by sufficiently small perturbations of all the data. Moreover, it shows that almost all linear programs satisfy this stability property.

The maximum return-on-investment plant location problem with market share

Abstract: This paper examines the plant location problem under the objective of maximizing return-on-investment. However, in place of the standard assumption that all demands must be satisfied, we impose a minimum acceptable level on market share. The model presented takes the form of a linear fractional mixed integer program. Based on properties of the model, a local search procedure is developed to solve the problem heuristically. Variable neighbourhood search and tabu search heuristics are also developed and tested. Thus, a useful extension of the simple plant location problem is examined, and heuristics are developed for the first time to solve realistic instances of this problem.

A random fuzzy programming models based on possibilistic programming

Abstract: This paper considers linear programming problems where each coefficient of the objective function is expressed by a random fuzzy variable. New decision making models are proposed based on stochastic and possibilistic programming in order to maximize both of possibility and probability with respect to the objective function value. It is shown that each of the proposed models is transformed into a deterministic equivalent one. Solution algorithms using convex programming techniques and/or the bisection method are provided for obtaining an optimal solution of each model.

Frequency domain controller design by linear programming guaranteeing quadratic stability

Abstract: In a recent work, a frequency method based on linear programming was proposed to design fixed-order linearly parameterized controllers for stable linear multi-model SISO systems. The method is based on the shaping of the open-loop transfer functions in the Nyquist diagram under a set of linear constraints guaranteeing a lower bound on the crossover frequency and a linear stability margin. In this paper, this method is extended to guarantee quadratic stability. For this purpose, new linear constraints based on the phase difference of the characteristic polynomials of the closed-loop systems are added in the Nyquist diagram. A simulation example illustrates the effectiveness of the proposed approach.

Linear programming problem with interval coefficients and an interpretation for its constraints

Abstract: In this paper, we introduce a Satisfaction Function (SF) to compare interval values on the basis ofTseng and Klein’s idea. The SF estimates the degree to which arithmetic comparisons between two intervalvalues are satisfied. Then, we define two other functions called Lower and Upper SF based on the SF. Weapply these functions in order to present a new interpretation of inequality constraints with intervalcoefficients in an interval linear programming problem. This problem is as an extension of the classical linear programming problem to an inexact environment. On the basis of definitions of the SF, the lower and upper SF and their properties, we reduce the inequality constraints with interval coefficients in their satisfactory crisp equivalent forms and define a satisfactory solution to the problem. Finally, a numerical example is given and its results are compared with other approaches

Solving Pseudo-Boolean Problems with SCIP

Abstract: Pseudo-Boolean problems generalize SAT problems by allowing linear constraints and a linear objective function. Different solvers, mainly having their roots in the SAT domain, have been proposed and compared,for instance, in Pseudo-Boolean evaluations. One can also formulate Pseudo-Boolean models as integer programming models. That is,Pseudo-Boolean problems lie on the border between the SAT domain and the integer programming field. In this paper, we approach Pseudo-Boolean problems from the integer programming side. We introduce the framework SCIP that implements constraint integer programming techniques. It integrates methods from constraint programming, integer programming, and SAT-solving: the solution of linear programming relaxations, propagation of linear as well as nonlinear constraints, and conflict analysis. We argue that this approach is suitable for Pseudo-Boolean instances containing general linear constraints, while it is less efficient for pure SAT problems. We present extensive computational experiments on the test set used for the Pseudo-Boolean evaluation 2007. We show that our approach is very efficient for optimization instances and competitive for feasibility problems. For the nonlinear parts, we also investigate the influence of linear programming relaxations and propagation methods on the performance. It turns out that both techniques are helpful for obtaining an efficient solution method.

Transmission Network Bi-level Programming Model Considering Economy and Reliability and Hybrid Algorithm

Abstract: Transmission network determinate bi-level linear programming model considering economy and reliability is established in the paper, reliability problem is added to economy programming problem as constraints in this model, it changes traditional transmission programming building style, and optimizes planning scheme in economy under high reliability constraints. The minimum investment cost of transmission lines is used as the upper programming objective, and its constraints are the number of right-of-ways restricts; the follower programming objective is minimization of load curtailment in load buses, and its constraints are traditional operation restricts, which strictly satisfy N?1 security criterion and let optimal planning scheme meet N?1 secure operation requirements. Hybrid algorithm which integrates improved niche genetic algorithm (INGA) with prime-dual interior point method (PDIPM) is proposed to solve the above model. Niche genetic algorithm is adopted to deal with integer variables of upper programming and search global optimality, prime-dual interior point method is adopted to solve the follower programming quickly, the algorithm speed and convergence are improved. The results of 18-bus system and 46-bus system prove that the proposed model and algorithm are valid.

Mathematical Programming Formulations for the Bottleneck Hyperplane Clustering Problem

Abstract: We discuss a mixed-integer nonlinear programming formulation for the problem of covering a set of points with a given number of slabs of minimum width, known as the bottleneck variant of the hyperplane clustering problem. We derive several linear approximations, which we solve using a standard mixed-integer linear programming solver. A computational comparison of the performance of the different linearizations is provided.