A Unifying Approach to Solve a Class of Parametrically-Convexifiable Problems

Abstract

The aim of this paper is to show how a wide class of generalized quadratic programs can be solved, in a unifying framework, by means of the so-called optimal level solutions method. In other words, the problems are solved by analyz- ing, explicitly or implicitly, the optimal solutions of particular quadratic strictly convex parametric subproblems. In particular, it is pointed out that some of these problems share the same set of optimal level solutions. A solution algorithm is pro- posed and fully described. The results achieved are then deepened in the particular case of box constrained problems

On the minimization of a class of generalized linear functions on a flow polytope

“…Obviously, an optimal solution of problem P is also an optimal level solution and, 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. Starting from an incumbent optimal level solution, this can be done by means of a sensitivity analysis on the parameter ξ, which allows us to move in the various steps through several optimal level solutions until the optimal solution is found (see [4]).…”

“…The objective function is then evaluated over the segment of optimal level solutions x (θ), θ ∈ [0, F R ], in order to improve the incumbent optimal solution. Notice that θ := arg min θ∈[0,FR] z(θ), where z(θ) = f (x (θ)), can be implemented numerically, and eventually improved for specific functions f (x) (see [3,4,15]).…”

A unifying approach to solve a class of rank-three programs involving linear and quadratic functions

“…Such an approach has been applied to study various classes of problems in [4,7,22,[25][26][27]. Later, the optimal level solution method has been applied to study more general classes of problems, providing also computational results obtained by extensive numerical experiences [8][9][10][11][13][14][15][16][17][18][19].…”

“…With this aim, some further assumptions on problem P are usually assumed in order to let the Karush-Kuhn-Tucker conditions become both necessary and sufficient. Specifically speaking, functions g ξ (x) can be assumed to be pseudoconvex or, following the lines proposed in [10], can be transformed to equivalent quadratic convex functions γ ξ (x) such that arg min x∈Xξ g ξ (x) = arg min x∈Xξ γ ξ (x) (in this last case problem P has been said to be "parametrically-convexifiable").…”

“…These examples point out the wideness of the class of problems P which, at the best of our knowledge, have never been studied in the literature by means of a unifying framework. Notice also that even in the particular case φ(ξ) = ξ the problems considered in this paper differ from the rank-two ones studied in [8][9][10][11]15]. The solution method proposed to solve this class of problems is based on the so called "optimal level solutions" method (see [6,9,[13][14][15][16][17]26]).…”