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In an earlier paper, R.W. Cottle and the author proposed a special principal pivoting algorithm for a class of large, structured linear complementarity problems. The method was applied with encouraging results to data relating to the free boundary problem for infinite journal bearings. The purpose of the present paper is to compare the empirical performance of the principal pivoting method with that of two other pertinent methods. One of the latter employs LU factorizations and is adaptive in the sense that each iteration exploits the factorization associated with its predecessor. The other is a modification of the point successive overrelaxation technique. Recommendations based on the reported computational experience are made. (Author).

Let T denote the d-dimensional integer lattice. Let X = (X(t); t belongs to T) be a process taking values on a countable space E with probability law P. The pair (X, P) is called a random field. For each finite subset J of T, let Q sub J denote the conditional distribution of (X(t); t belongs to J) given the values of X(t), t belongs to J. The collection Q = (Q sub J; J subset T) is called the conditional distribution of (X, P). A characterization is given for the collection of all random fields with a given conditional distribution Q. (Author).

It was shown elsewhere by Dantzig that if there exits one complementary tree in the undirected network G(G epsilon eta(n)) then there exists at least two. The proof there is by means of an algorithm which finds a different complementary tree from a given one. It is shown in this paper that using an extended form of Dantzig's algorithm can lead to a stronger result: if there exists one complementary tree in G(G epsilon eta(n)) then there exists at least four. Also some examples are provided to establish an upper bound on the smallest number of complementary trees in a network G(G epsilon eta(n)) which has at least one complementary tree.

The authors empirically compared ten pivot selection rules for representing the inverse of a sparse basis in triangularized product form. On examples drawn from actual applications, one of the rules yield inverses that were only slightly less sparse than the original basis. The rule was used in the M5 mathematical programming system and has resulted in substantial reduction in running time.

The paper considers the problem of portfolio selection for a rish averting investor wishing to allocate his resources among several investment opportunities in order to maximize the expected utility of final wealth. When the investment returns are joint normally distributed and there is a riskless asset it is known that the investment proportions in the risky assets are independent of the utility function. These proportions are generally unique and may be obtained from the solution of a fractional program or a linear complementary problem. Given the investor's utility function the optimal investment proportions in all assets may be found by solving a stochastic program having one random variable and one decision variable. The two stage procedure provides an efficient computational scheme to solve large scale portfolio problems. Data on the major pooled Canadian equity pension funds was used to provide an empirical test of the suggested solution approach using several common utility function classes. (Author).

It has long been known that the problem of determining the equilibrium composition of a solution of chemically reacting species could be formulated as a constrained minimum problem. Previous methods for solving the chemical equilibrium problem in this form have had much success. However, all such methods run into trouble whenever degeneracy or near-degeneracy occurs during the computational procedure. The paper shows that the constrained minimum formulation of the chemical equilibrium problem is equivalent to a generalized linear program which can in turn be replaced by a quadratic program. In these alternative forms, degeneracy is more easily accomodated than in previous methods. (Author).