An algorithm for solving min‐cost or max‐flow multicommodity flow problems is described. It is a specialization of the simplex method, which takes advantage of the special structure of the multicommodity problem. The only nongraph or nonadditive operations in a cycle involve the inverse of a working basis, whose dimension is the number of currently saturated arcs. Efficient relations for updating this inverse are derived.

:When applied to a problem which has more than one local optimal solution, most nonlinear programming algorithms will terminate with the first local solution found.Several methods have been suggested for extending the search to find the global optimum of such a nonlinear program.In this report we present the results of some numerical experiments designed to compare the performance of various strategies for finding the global solution.

INI'RODUCTIONT i e constraints of large linear programs can often be partitioned into independent subsets, except for relatively few coupling rows and coupling columns. The individual subsets may, for example, arise from constraints on the activity levels of subdivisions of a large corporation. Alternatively, such blocks may arise from activities in different time periods. The coupling rows may arise from limitations on shared resources or from combining the outputs of subdivisions to meet overall demands. The coupling columns arise from activities which involve different time periods (e.g., storage), or which involve different subdivisions (e.g., transportation or assembly). The case with only coupling rows or coupling columns, but not both has received much attention [8], [9]. A smaller amount of work has been done on the problem with both coupling rows and columns. Ritter has proposed a dual method [2], [7]. ExceFt for the preliminary work of Webber and White [lo] and Heesterman [4], there is no primal algorii hm which exploits the structure of this problem. There is a need for such an algorithm, since such problems occur often in practice. A primal method is desirable since in large problems slow convergence may force termination of the algorithm prior to optimality.' I he algorithm proposed here is an extension of the generalized upper bounding method for problems without coupling columns proposed in [ 5 ] and [ 6 ] . It produces the same sequence of extreme point solutions as the primal simplex method, and hence has the desirable convergence properties of that slgorithm. However, the operations within each simplex iteration are organized to take maximal advar tage of problem structure. Because of this structure it is possible to perform the computations while maintaining a compact representation of the basis inverse. In particular it is sufficient to maintain and update at each cycle a working basis inverse, inverses from each block, and possibly another matrix, V in (4). The dimension of the working basis need never be more than the number of coupling rows in the problem plus the number of coupling columns in the current basis. Hence its dimension may change from cycle to cycle. At most one of the block inverses need be updated at any interation. Given these quantities, all information needed to carry out a simplex iteration can easily be obtained.Further, efficient relations are given for updating the working basis and block inverses and V for the next cycle.'1, significant amount of computational work has been done on a special class of production and inventory problems. Problems as large as 362 rows by 3225 columns were solved. Computation times 411

As an approach to solving nonlinear programs, we study a class of functions known to be exact penalty functions for a proper choice of the parameters. The goal is to iteratively determine the correct parameter values. A basic algorithm has been developed. We have proved that this algorithm converges for concave programs, and in the limited computational tests performed to date it has always converged for nonconcave programs also.Suggestions for continuing the work are given.

A modification of Hartman, J. K., L. S. Lasdon. 1970. A generalized upper bounding method for doubly coupled linear programs. Naval Res. Logist. Quart. 17 (4, December) to rearrange matrices into a form useful for linear programming.