The Sum of a Linear and a Linear Fractional Function: Pseudoconvexity on the Nonnegative Orthant and Solution Methods

The aim of the paper is to present sequential methods for a pseudoconvex optimization problem whose objective function is the sum of a linear and a linear fractional function and the feasible region is a polyhedron, not necessarily compact. Since the sum of a linear and a linear fractional function is not in general pseudoconvex, we first derive conditions characterizing its pseudoconvexity on the nonnegative orthant. We prove that the sum of a linear and a linear fractional function is pseudoconvex if and only if it assumes particular canonical forms. Then, theoretical properties regarding the existence of a minimum point and its location are established, together with necessary and sufficient conditions for the infimum to be finite. The obtained results allow us to suggest simplex- like sequential methods for solving optimization problems having as objective function the proposed canonical forms

Generating the efficient frontier of a class of bicriteria generalized fractional programming

In this paper, a particular class of bicriteria maximization problems over a compact polyhedron is considered. The first component of the objective function is the ratio of powers of affine functions and the second one is linear. Several theoretical properties are provided, such as the pseudoconcavity of the first criterium of the objective function, the connectedness and compactness of both the efficient frontier and the set of efficient points. The obtained results allow us to propose a new simplex-like solution method for generating the whole efficient frontier; to better clarify the use of the suggested algorithm, several examples are described and the results of a computational test are presented

Simplex-like sequential methods for a class of generalized fractional programs

A sequential method for a class of generalized fractional programming problems is proposed. The considered objective function is the ratio of powers of affine functions and the feasible region is a polyhedron, not necessarily bounded. Theoretical properties of the optimization problem are first established and the maximal domains of pseudoconcavity are characterized. When the objective function is pseudoconcave in the feasible region, the proposed algorithm takes advantage of the nice optimization properties of pseudoconcave functions; the particular structure of the objective function allows to provide a simplex-like algorithm even when the objective function is not pseudoconcave. Computational results validate the nice performance of the proposed algorithm

On the bicriteria maximization problem

The bicriteria maximization problem has been studied mainly by several authors mainly with the aim of establishing the connectedness of the set E of all efficient points. In this paper we will introduce a parametric real-valued function which allows us either to derive a parametric representation of E in a general form or to find sequential methods for generating E

A sequential method for a class of pseudoconcave fractional problems

Fractional programming, Pseudoconcavity, Post-optimality analysis, 90C32, 26B25,

Some Classes of Pseudoconvex Fractional Functions via the Charnes and Cooper Transformation

Using a very recent approach based on the Charnes-Cooper trasformation we characterize the pseudoconvexity of the sum between a quadratic fractional function and a linear one. Furthemore we prove that the ratio between a quadratic fractional function and the cube of an affine one is pseudoconvex if and only if the product between a quadratic fractional function and an affine one is pseudoconvex and we provide a sort of canonical form for this latter class of functions. Benefiting by the new results we\ are able to characterize the pseudoconvexity of the ratio between a quadratic fractional function and the cube of an affine one

Optimality and constraint qualifications in vector optimization

We propose a unifying approach in deriving constraint qualifications and theorem of the alternative. We first introduce a separation theorem between a subspace and the non-positive orthant, and then we use it to derive a new constraint qualification for a smooth vector optimization problem with inequality constraints. The proposed condition is weaker than the existing conditions stated in the recent literature. According with the strict relationship between generalized convexity and constraint qualifications, we introduce a new class of generalized convex vector functions. This allows us to obtain some new constraint qualifications in a more general form than the ones related to componentwise generalized convexity. Finally, the introduced separation theorem allows us to derive some of the known theorems of the alternative which are used in the literature to get constraint qualifications