A generalized upper bounding method for doubly coupled linear programs

Abstract

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

Dynamic factorization in large-scale optimization

“…McBride (1972) and Graves and McBride (1976) formalize and generalize the factorization approach. They view factorization as a unifying framework for tableau-based simplex specializations and illustrate this by developing a variation of the GUB algorithm ofDantzig and Van Slyke and a GUB algorithm for the doubly-coupled linear programs of Hartman and Lasdon ( 1970) . They present a new algorithm for the set partitioning LP and an equalityconstrained form ofthe pure network with side constraints model.…”

Structured markovian decision problems

“…Hence, the efficient, specialized algorithm can be applied. This specialization is related to similar work on structured linear programst as studied in [6]. The algorithm, in essence, exploits the special nature of Markovian decision problems to modify and simplify an applicable, structured policy iteration algorithm.…”

“…Note that in the proof of Lemma 2.2, we give a simple means of permuting the columns of Bs to achieve the diagonal structure. This points out a simplification possible in the problem at hand which is not achievable in the general linear programming problem of similar structure as discussed in [6].…”

“…The efficiency of the above policy iteration algorithm, except for the simplification noted above, should be comparable to that of the hart man-Lasdon [6] decomposition procedure for L.P.'s. Further, improvements over the L.Y.…”

Metodo primal-dual de pontos interiores aplicado ao problema de multifluxo

“…Técnicas deste tipo, como por exemplo "compact inverse methods" ou 1 'Generalized Upper Bounding" (GUB) estão descritas em Lasdon [41]. As idéias básicas da partição primai se encontram em Bennett [8], Kaul (34], Saigal [66] e Hartman e Lasdon [22,23]. Ẽ s t a s idéias foram mais tarde desenvolvidas e adaptadas para problemas de multifluxo por Maier [45], Kennington [36], Helgason e Kennington [26], Ali, Helgason, Kennington e Lall [4] e McCallum [48].…”