A new look at fractional programming

The parametric approach to fractional programming problems is examined and a new format is proposed for it. The latter reflects the fact that the approach as a whole capitalizes on a first-order necessary and sufficient optimality condition pertaining to differentiable pseudolinear functions.     

Continuous-Time Generalized Fractional Programming Problems. Part I: Basic Theory

This study, that will be presented as two parts, develops a computational approach to a class of continuous-time generalized fractional programming problems. The parametric method for finite-dimensional generalized fractional programming is extended to problems posed in function spaces. The developed method is a hybrid of the parametric method and discretization approach. In this paper (Part I), some properties of continuous-time optimization problems in parametric form pertaining to continuous-time generalized fractional programming problems are derived. These properties make it possible to develop a computational procedure for continuous-time generalized fractional programming problems. However, it is notoriously difficult to find the exact solutions of continuous-time optimization problems. In the accompanying paper (Part II), a further computational procedure with approximation will be proposed. This procedure will yield bounds on errors introduced by the numerical approximation. In addition, both the size of discretization and the precision of an approximation approach depend on predefined parameters.     

Using the parametric approach to solve the continuous-time linear fractional max–min problems

A numerical algorithm based on parametric approach is proposed in this paper to solve a class of continuous-time linear fractional max-min programming problems. We shall transform this original problem into a continuous-time non-fractional programming problem, which unfortunately happens to be a continuous-time nonlinear programming problem. In order to tackle this nonlinear problem, we propose the auxiliary problem that will be formulated as a parametric continuous-time linear programming problem. We also introduce a dual problem of this parametric continuous-time linear programming problem in which the weak duality theorem also holds true. We introduce the discrete approximation method to solve the primal and dual pair of parametric continuous-time linear programming problems by using the recurrence method. Finally, we provide two numerical examples to demonstrate the usefulness of this algorithm.     

An iterative approach to solve multiobjective linear fractional programming problems

This paper suggests an iterative parametric approach for solving multiobjective linear fractional programming (MOLFP) problems which only uses linear programming to obtain efficient solutions and always converges to an efficient solution. A numerical example shows that this approach performs better than some existing algorithms. Randomly generated MOLFP problems are also solved to demonstrate the performance of new introduced algorithm.     

Method of Centers for Generalized Fractional Programming

We propose an algorithm to solve generalized fractional programming problems. The proposed algorithm combines the parametric approach and the Huard method of centers. A minimum of the problem is obtained by solving a sequence of unconstrained optimization problems.     

Parametric simplex algorithms for solving a special class of nonconvex minimization problems

It is shown that parametric linear programming algorithms work efficiently for a class of nonconvex quadratic programming problems called generalized linear multiplicative programming problems, whose objective function is the sum of a linear function and a product of two linear functions. Also, it is shown that the global minimum of the sum of the two linear fractional functions over a polytope can be obtained by a similar algorithm. Our numerical experiments reveal that these problems can be solved in much the same computational time as that of solving associated linear programs. Furthermore, we will show that the same approach can be extended to a more general class of nonconvex quadratic programming problems.     

Parametric computation of a fuzzy set solution to a class of fuzzy linear fractional optimization problems

The class of fuzzy linear fractional optimization problems with fuzzy coefficients in the objective function is considered in this paper. We propose a parametric method for computing the membership values of the extreme points in the fuzzy set solution to such problems. We replace the exhaustive computation of the membership values—found in the literature for solving the same class of problems—by a parametric analysis of the efficiency of the feasible basic solutions to the bi-objective linear fractional programming problem through the optimality test in a related linear programming problem, thus simplifying the computation. An illustrative example from the field of production planning is included in the paper to complete the theoretical presentation of the solving approach, but also to emphasize how many real life problems may be modelled mathematically using fuzzy linear fractional optimization.     

Solving Fractional Multicriteria Optimization Problems with Sum of Squares Convex Polynomial Data

This paper focuses on the study of finding efficient solutions in fractional multicriteria optimization problems with sum of squares convex polynomial data. We first relax the fractional multicriteria optimization problems to fractional scalar ones. Then, using the parametric approach, we transform the fractional scalar problems into non-fractional problems. Consequently, we prove that, under a suitable regularity condition, the optimal solution of each non-fractional scalar problem can be found by solving its associated single semidefinite programming problem. Finally, we show that finding efficient solutions in the fractional multicriteria optimization problems is tractable by employing the epsilon constraint method. In particular, if the denominators of each component of the objective functions are same, then we observe that efficient solutions in such a problem can be effectively found by using the hybrid method. Some numerical examples are given to illustrate our results.     

Generalized linear multiplicative and fractional programming

This paper addresses itself to the algorithm for minimizing the sum of a convex function and a product of two linear functions over a polytope. It is shown that this nonconvex minimization problem can be solved by solving a sequence of convex programming problems. The basic idea of this algorithm is to embed the original problem into a problem in higher dimension and apply a parametric programming (path following) approach. Also it is shown that the same idea can be applied to a generalized linear fractional programming problem whose objective function is the sum of a convex function and a linear fractional function.     

Using the Dinkelbach-type algorithm to solve the continuous-time linear fractional programming problems

A Dinkelbach-type algorithm is proposed in this paper to solve a class of continuous-time linear fractional programming problems. We shall transform this original problem into a continuous-time non-fractional programming problem, which unfortunately happens to be a continuous-time nonlinear programming problem. In order to tackle this nonlinear problem, we propose the auxiliary problem that will be formulated as parametric continuous-time linear programming problem. We also introduce a dual problem of this parametric continuous-time linear programming problem in which the weak duality theorem also holds true. We introduce the discrete approximation method to solve the primal and dual pair of parametric continuous-time linear programming problems by using the recurrence method. Finally, we provide two numerical examples to demonstrate the usefulness of this practical algorithm.     

Complex Minimax Fractional Programming of Analytic Functions

We prove that a minmax fractional programming problem is equivalent to a minimax nonfractional parametric problem for a given parameter in complex space. Using a parametric approach, we establish the Kuhn-Tucker type necessary optimality conditions and prove the existence theorem of optimality for complex minimax fractional programming in the framework of generalized convexity. Subsequently, we apply the optimality conditions to formulate a one-parameter dual problem and prove weak duality, strong duality, and strict converse duality theorems involving generalized convex complex functions. This research was partly supported by NSC, Taiwan.     

Duality for nonlinear fractional programs

Summary This paper develops duality results for nonlinear fractional programming problems. This is accomplished by using some known results connecting the solutions of a nonlinear fractional program with those of a suitably defined parametric convex program.     

Optimality conditions and duality models for generalized fractional programming problems containing locally subdifferentiable and ρ:-convex functions

Both parametric and nonparametric necessary and sufficient optimality conditions are established for a class of nonsmooth generalized fractional programming problems containing ρ-convex functions. Subsequently, these optimality criteria are utilized as a basis for constructing two parametric and four parameter-free duality models and proving appropriate duality theorems. Several classes of generalized fractional programming problems, including those with arbitrary norms, square roots of positive semidefinite quadratic forms, support functions, continuous max functions, and discrete max functions, which can be viewed as special cases of the main problem are briefly discussed. The optimality and duality results developed here also contain, as special cases, similar results for nonsmooth problems with fractional, discrete max, and conventional objective functions which are particular cases of the main problem considered in this paper     

OPTIMALITY CONDITIONS AND DUALITY MODELS FOR A CLASS OF NONSMOOTH CONTINUOUS-TIME GENERALIZED FRACTIONAL PROGRAMMING PROBLEMS

Abstract

Both parametric and parameter-free stationary-point-type and saddle-point-type necessary and sufficient optimality conditions are established for a class of nonsmooth continuous-time generalized fractional programming problems with Volterra-type integral inequality and nonnegativity constraints. These optimality criteria are then utilized for constructing ten parametric and parameter-free Wolfe-type and Lagrangian-type dual problems and for proving weak, strong, and strict converse duality theorems. Furthermore, it is briefly pointed out how similar optimality and duality results can be obtained for two important special cases of the main problem containing arbitrary norms and square roots of positive semidefinite quadratic forms. All the results developed here are also applicable to continuous-time programming problems with fractional, discrete max, and conventional objective functions, which are special cases of the main problem studied in this paper.     

Optimizing a General Optimal Replacement Model by Fractional Programming Techniques

In this paper we adapt the well-known parametric approachfrom fractional programming to solvea class of fractional programs with a noncompact feasible region.Such fractional problems belong to an important class ofsingle component preventive maintenance models.Moreover, for a special but important subclass we showthat the subproblems occurring in this parametric approachare easy solvable.To solve the problem directly we also propose for a relatedsubclass a specialized version of the bisection method.Finally, we present some computational results for these twomethods applied to an inspection model and a minimal repair modelhaving both a unimodal failure rate.     

An Efficient Algorithm for Quadratic Sum-of-Ratios Fractional Programs Problem

The quadratic sum-of-ratios fractional program problem has a broad range of applications in practical problems. This article will present an e?cient branch-and-bound algorithm for globally solving the quadratic sum-of-ratios fractional program problem. In this algorithm, lower bounds are computed by solving a series of parametric relaxation linear programming problems, which are established by utilizing new parametric linearizing technique. To enhance the computational speed of the proposed algorithm, a rectangle reducing tactic is used to reject a part of the investigated rectangle or the whole rectangle where there does not contain any global optimal solution of the quadratic sum-of-ratios fractional program problem. Compared with the known approaches, the proposed algorithm does not need to introduce new variables and constraints. Therefore, the proposed algorithm is more suitable for application in engineering.     

Generalized fractional programming duality: A parametric approach

Using a parametric approach, duality is presented for a minimax fractional programming problem that involves several ratios in the objective function.The first author is thankful to Natural Science and Engineering Research Council of Canada for financial support through Grant A-5319, and the authors are thankful to the anonymous referees for useful suggestions.     

Duality Without a Constraint Qualification for Minimax Fractional Programming

Using a parametric approach, we establish the necessary and sufficient conditions for generalized fractional programming without the need of a constraint qualification. Subsequently, these optimality criteria are utilized as a basis for constructing a parametric dual model and two other parameter-free dual models. Several duality theorems are established.     

Solving special nonlinear fractional programming problems via parametric linear programming

A new procedure is developed to solve a generalized linear fractional programming problem. We find the optimal solutions in two steps. First we solve a parametric linear programming problem. Using the results of this step we then define a simple optimization problem in the second step. This yields an optimal value which together with the results of the parametric analysis provides the optimal solutions of the considered fractional programming problem.     

具有(F,α,ρ,d)—凸的分式规划问题的最优性条件和对偶性

给出了一类非线性分式规划问题的参数形式和非参数形式的最优性条件,在此基础上,构造出了一个参数对偶模型和一个非参数对偶模型,并分别证明了其相应的对偶定理,这些结果是建立在次线性函数和广义凸函数的基础上的.