The Symmetric Formulation of the Simplex Method for Quadratic Programming

Abstract : For the solution of convex quadratic programming problem, a number of efficient methods have been developed. The most well-known methods are the Simplex method for quadratic programming, discovered by Dantzig and, together with the closely related dual method, further developed by van de Panne and Whinston, and methods developed by Beale, Houthakker and Wolfe. The authors have shown that the methods by Beale and Houthakker can be considered as variants of the Simplex method for quadratic programming or are closely related to it. Compared with the Simplex tableaux used in linear programming, quadratic programming tableaux have a larger size. A tableau for a linear programming problem with n variables and m constraints had (m + l) (n + l) nontrivial elements, while a Simplex tableau for a quadratic programming problem with the same number of variables and constraints has (m + n + l) elements. In the Simplex method for quadratic programming, a considerable number of tableaux will be in standard form, which means that the tableau can be divided in symmetric and skew-symmetric parts, so that the number of elements to be computed and stored is reduced by nearly one half. However, nonstandard tableaux do not have these symmetry properties, so that all elements of these tableaux must be computed. This paper gives a reformulation of the Simplex method in which all tableaux are in standard form, so that use can be made of the symmetry properties in every tableau. The actual number of nontrivial elements in a quadratic Simplex tableau is therefore decreased by a factor of 2. This symmetric formulation has other advantages as well. (Author)