The optimal level solutions method

Abstract

In this paper the method of optimal level solutions, introduced by Cambini and Martein for fractional programming problems [5,·81, is developed in a general framework. In such a framework all the algorithms based on the optimal level solutions approach stated so far in literature can be easily embedded and their properties proved at once. INTRODUCTIONIn the late 80's and in the 90's a lot of papers were devoted to the development of algorithms to solve a mathematical programming problem by means of the examination of a . suitably defined subset of the feasible region: more precisely, the set of optimal points of a corresponding parametric program is explored instead of the whole feasible region. Cambini and Martein in their pioneering papers [5,8,10], referring in particular to the linear fractional program, called such subset the optimal level solutions set.The known algorithms based on this idea are of simplex like type and, at least in the special cases investigated till now, computationally comparable with the simplex method for linear programs (for computational experiments see [4,16,17,23]). This paper deals with the generalization of the concept of optimal level solutions to a wide class of mathematical programming problems: some general properties of these solutions and of the corresponding optimal value function are stated in sections 2 and 3.The stated properties are essential in order to develop algorithms for thE? solution of the considered programs. The general structure of such algorithms will be discussed in section 4, while in section 5 it will

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

“…Quoting [2], the algorithm "generates a sequence of finite optimal level solutions x 1 , x 2 ..., x k , which corresponds to increasing levels of the original objective function"; then either the maximum point does not exist or it coincides with the last point of the sequence". This approach has been first conceived for linear fractional problems and then applied to suggest several sequential methods for generalized fractional programming (see for example Ellero [9] and Cambini Martein [2] and references therein, Cambini Sodini [5,6] and Carosi Martein [7]). Referring to problem P, it is worth noticing that the denominator function d T x + d 0 is lower bounded on X, so that the linear problem…”

“…The proposed algorithm is based on the so called "optimal level solutions" method which has been first conceived for linear fractional problems by Cambini and Martein [1,2] and then applied to solve generalized fractional problems. In his survey, Ellero [9] proposes a unified framework encompassing several contributions of the literature; all of them are based on the "optimal level solutions" method and they successfully solve different classes of generalized fractional problems. More recent analyses, following this approach, have been performed for solving rank-two problems (see for example Cambini Sodini [6]) and rank-three ones (see for example Cambini Sodini [5] and Carosi Martein [7]).…”

Untitled

“…The parameter ξ is said to be a feasible level if the set X ξ is nonempty. An optimal solution of problem P ξ is called an optimal level solution [5,10,16]. For any given ξ ∈ , the optimal solution for problem P ξ can be computed by means of any solution algorithm for strictly convex quadratic problems.…”

“…For the sake of completeness, let us now briefly recall the optimal level solution approach (see, for example, [10]). It is trivial that the optimal solution of problem P is also an optimal level solution and that, in particular, it is the optimal level solution with the smallest value; the idea of this approach is then to scan all the feasible levels, studying the corresponding optimal level solutions, until the minimizer of the problem is reached or a feasible halfline carrying f (x) down to −∞ is found.…”

“…The aim of this paper is to show that problem P can be solved with a parametric approach based on the so called "optimal level solutions method", which has been widely used to solve several classes of optimization problems (see, for example, [5,16], see also [10] for a survey on these methods).…”