Extension of primal-dual interior point method based on a kernel function for linear fractional problem

In this chapter, we consider several special problems which appear in linear fractional programming. Thus, in Section 8.1, we consider a problem of linear fractional programming in which the variables of the objective function appear in absolute-value, and the constraints lack the condition for non-negativity. We show that, under certain conditions, it is possible to apply the Simplex algorithm developed for the standard linear fractional problem. In Section 8.2, we consider the separable linear fractional programming problem, and we reduce it to the solution of a sequence of quadratic programming problems. In Section 8.3, we study the linear fractional programming problem with disjunctive constraints for which we present a solution method and we construct the Dual problem.

Linear fractional programming has been an important planning tool for the past four decades. The main contribution of this study is to show, under some assumptions, for a linear programming problem, that there are two different dual problems (one linear programming and one linear fractional functional programming) that are equivalent. In other words, we formulate a linear programming problem that is equivalent to the general linear fractional functional programming problem. These equivalent models have some interesting properties which help us to prove the related duality theorems in an easy manner. A traditional data envelopment analysis (DEA) model is taken, as an instance, to illustrate the applicability of the proposed approach.

In this paper, we are interested in solving a linear fractional program by two different approaches. The first one is based on interior point methods which makes it possible to solve an equivalent linear program to the linear fractional program. The second one allows us to solve a variational inequalities problem equivalent to the linear fractional program by an efficient projection method. Numerical tests were carried out by the two approaches and a comparative study was carried out. The numerical tests show clearly that interior point methods are more efficient than of projection one.

Piecewise linear programming via interior points

The revised harmonious fuzzy technique (RHFT) is a method used to solve fuzzy optimization problems. It was capitalized as an extension of the classical linear programming technique to handle constraints and objectives that are fuzzy. The harmonious fuzzy technique HFT aims to find a solution that satisfies the uncertain restraints and optimizes the uncertain objectives while taking into account the uncertainty or fuzziness of the problem parameters. This work demonstrates how the RHFT can be utilized to dexterously solve “fully fuzzy multi-goal linear fractional programming (FFMOLFP) problems”. Initially, the FFMOLFP problem can be converted to “single goal linear fractional programming (SOLFP) problems” consuming the modified brittle linear technique. Second, the RHFT is applied to converted brittle problems into linear programming problem, which follow, “the single-goal problem” is made on so on applied the revised harmonious fuzzy for apiece level. at the end, the obtained LPP will be solved by applied the simplex algorithm. To illustrate the application of this method, two examples will be provided. Also, the numerical results are simulated by comparing between proposed method and efficient ranking function methods for fully fuzzy linear fractional programming problems FFLFPP

Revisiting Karnik–Mendel Algorithms in the framework of Linear Fractional Programming

Interior point methods in DEA to determine non-zero multiplier weights

The study deals with the multi-choice mathematical programming problem, where the right hand side of the constraints is multi-choice in nature. However, the problem of multi-choice linear programming cannot be solved directly by standard linear or nonlinear programming techniques. The aim of this paper is to transform such problems to a standard mathematical linear programming problem. For each constraint, exactly one parameter value is selected out of a multiple number of parameter values. This process of selection can be established in different ways. In this paper, we present a new simple technique enabling us to handle such problem as a mixed integer linear programming problem and consequently solve them by using standard linear programming software. Our main aim depends on inserting a specific number of binary variables and using them to construct a linear combination which gives just one parameter among the multiple choice values for each choice of the values of the binary variables. A numerical example is presented to illustrate our analysis.

This paper establishes a framework for solving some optimization problems with linear constraints using simplex-type methods. The problems include those found in linear programs, linear fractional programs, and generalized linear fractional programs. In this study, these problems refer to a standard form of minimizing a single parameter subject to parameterized linear equations. Based on the analysis of parameterized basis-based solutions, a unified simplex-type approach is proposed. The adaptability of the parameterized model and that of the solution procedure are discussed. In particular, the proposed algorithm can prevent cycling when compared with the conventional simplex method used for solving linear programs.

In this paper, we study a new approach for solving linear fractional programming problem (LFP) by converting it into a single Linear Programming (LP) Problem, which can be solved by using any type of linear fractional programming technique. In the objective function of an LFP, if β is negative, the available methods are failed to solve, while our proposed method is capable of solving such problems. In the present paper, we propose a new method and develop FORTRAN programs to solve the problem. The optimal LFP solution procedure is illustrated with numerical examples and also by a computer program. We also compare our method with other available methods for solving LFP problems. Our proposed method of linear fractional programming (LFP) problem is very simple and easy to understand and apply.

(ProQuest: ... denotes formulae omitted.)1. IntroductionThe development of sophisticated software to solve linear optimization problems by interior point methods has started since the early works on this subject. There are three main research lines aimed at improving the efficiency of such methods for solving large-scale problems: reduction of the total number of iterations, techniques to obtain a fast iteration and specific methods for particular classes of problems.This work addresses the second one. Iterative methods are used to solve the linear systems of equations which are the most expensive step at each iteration of interior point methods. Since such systems are very ill-conditioned near a solution, the design of specially tailored preconditioners is an important implementation issue. On the other hand, since the early linear systems do not present the same features, it is advisable to adopt hybrid preconditioners that begin as a generic preconditioner and adapt during the course of the iteration, becoming ever more specialized as convergence takes place (Bocanegra et al., 2007).During the initial iterations a controlled Cholesky factorization is adopted (Campos & Birkett, 1998). Its major advantage is the control parameter that allows the preconditioner to vary all way from a diagonal preconditioner to the full Cholesky factorization, if desired. At the onset of convergence, a splitting preconditioner is used (Oliveira & Sorensen, 2005). Its major advantage is its excellent behavior near a solution of the linear program. However, this desirable feature has a price: the preconditioner could be very expensive to compute. A careful implementation must be performed in order to achieve competitive numerical results regarding both: speed and robustness. An effective implementation of the splitting preconditioner depends crucially upon finding a suitable set of linearly independent columns to form a nonsingular matrix, to be factored, from the constraint matrix.There are several techniques for finding such a set of columns such as the delayed update form for the LU factorization, the symbolic dependent columns, the symbolic independent columns, the combination of symbolic dependent and independent columns and strongly connected components. Some are well known and already applied in other contexts (Coleman & Pothen, 1987; Duff & Reid, 1986; El-Bakry et al., 1994). Others were developed to compute the splitting preconditioner (Oliveira, 1997; Oliveira & Sorensen, 2005). Among the techniques used is the study of the nonzero structure of the constraint matrix to speed up the numerical factorization, such as using key columns, symbolically dependent and independent columns, finding strongly connected components (Oliveira & Sorensen, 2005). Other implementation issues, include ways for changing preconditioners, are also discussed in (Ghidini et al., 2012; Velazco et al., 2010).The choice of the controlled factorization is justified due to the possibility of computing an inexpensive preconditioner in the initial interior point iterations and, as the linear systems become more ill conditioned, the controlled preconditioner can be improved with just the change of a parameter value. Numerical experiments illustrating the effectiveness of such strategies in order to solve large scale linear programming problems are presented in Bocanegra et al. (2007) and Velazco et al. (2010).This work is organized as follows: Section 2 presents the predictor-corrector interior point method, defines its search directions and explains how to solve the resulting linear systems of equations. The controlled Cholesky factorization and splitting preconditioners are discussed in this section. Sections 3 and 4 study several techniques in order to achieve an efficient implementation of the splitting preconditioner. In Section 5 the numerical experiments are shown and discussed. Conclusions follow in Section 6. …

Karmarkar's interior point method as a computation method for solving linear programming (LP) has attracted interest in the operation research community, due to its efficiency, reliability, and accuracy. This paper presents an extended quadratic interior point (EQIP) method, based on improvement of initial condition for solving both linear and quadratic programming problems, to solve power system optimization problem (PSOP), such as economic dispatch (ED) and VAr planning (VP) problems. The EQIP method is able to accommodate the nonlinearity in objectives and constraints. The scheme is demonstrated on several IEEE standard systems and is capable of achieving fast convergence and improvement in computational speed over an existing efficient Simplex, such as the MINOS code. The number of iterations during the computation is relatively insensitive to numbers of controls and constraints. Moreover, the proposed EQIP method guarantees a global optimum within the interior feasible region. >

In this paper, we studied fuzzy linear fractional programming (FLFP) problems with trapezoidal fuzzy numbers where the objective functions are fuzzy numbers and the constraints are real numbers. In this study, in order to obtain the fuzzy optimal solution with unrestricted variables and parameters, a new efficient method for FLFP problem has been proposed. These proposed methods are based on crisp linear fractional programming and newly transformation technique is also used. A computational procedure has been presented to obtain an optimal solution. To show the efficiency of our proposed method a real life example has been illustrated.

In this paper, a solution procedure is proposed to solve neutrosophic linear fractional programming problem (NLFP) where cost of the objective function, the resources and the technological coefficients are triangular neutrosophic numbers. Here, the NLFP problem is transformed into an equivalent crisp multi-objective linear fractional programming (MOLFP) problem. By using proposed approach, the transformed MOLFP problem is reduced to a single objective linear programming (LP) problem which can be solved easily by suitable linear programming technique. The proposed procedure illustrated through a numerical example.

Multi‐period planning problems in the oil and refinery industry are typically large, sparse, staircase/band diagonal structured and nonlinear optimization problems. Successive linear programming (SLP) type methods have been widely used for solving these planning problems. But, it has long been recognized that the simplex method used in solving linear programs requires a large number of iterations for staircase/band diagonal structured problems. In this paper, we report results of an application of a recently developed interior point method that promises to be many times faster than the simplex method for multi‐period planning problems. However, to facilitate the use of interior point method in the current SLP algorithms a hybrid method combining the interior point method and the simplex method is developed. Therefore, the results determined with this hybrid method are qualitatively equivalent to that obtained with the simplex method alone. The CPU times corresponding to the hybrid method are compared with the CPU times of simplex and dual affine methods. The new hybrid method generates a basic feasible solution of the linear programming problem and is approximately 7 times faster than the simplex method on the tested planning problems. Moreover, the interior point and hybrid methods become faster as the problem size increases.