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)

Abstract : The paper reformulates Houthakker's capacity method for quadratic programming in the framework of the Simplex and dual methods for quadratic programming, thereby greatly reducing the conceptual and computational complexities of the method. It is shown that the method is applicable for all convex quadratic programming problems, including the case of a semi-definite matrix of the quadratic form and that of constraints in equality form. The method reduces in the linear programming case to a parametric version of the dual method. (Author)

The paper discusses the allocation of output among consumption and two types of capital with different gestation periods. Along an optimal path, we show that the imputed prices of capital goods, from the time they start production, do not exceed the prices of output, which are not less than the marginal instantaneous utility of consumption. A simple numerical example helps to illustrate some further implications of the model. RECENT PAPERS on optimal growth consider models of allocation of resources between consumption and investment. It is invariably assumed that investment results in an instantaneous increase in the stock of capital. Such assumptions obscure differences in the gestation periods among various capital goods. In [1] we discuss how a growth problem with time lags can be formulated and interpreted and explain the derivation of the necessary conditions for optimization. In this note we study the effects of differences in gestation periods on optimal investment plans for a growth problem including depreciation and population growth. Consider an economy where two capital goods and labor are used in the production of a single commodity. The per capita output at time t is given by the production function: f (kl, k2), where k, is the per capita stock of capital of type one, and k2 is the per capita stock of capital of type two. From now on, all variables will be per capita and we drop the designation. We assume the following: