Serial decomposition

The vector f can be selected to be either nonlinear or affme. Although VI formulations of equilibrium problems over Cartesian products of sets have been derived for example, Dafermos, , b; Pang, ; Na- see gumey, b; Nagumey and Aronson, and se- rial decomposition algorithms implemented Pang, Downloaded from hpc. In section 1, we outline the general multicommodity market equilibrium model. In section 2, we discuss the adaptation of the serial and parallel decomposition algo- rithms used in the numerical work.

In section 3, we de- scribe the computing environment, and provide compu- tational experience on single and multiple CPUs for randomly generated linear and nonlinear multicom- modity market equilibrium problems. We conclude with a discussion. Results and Discussion 1. Single commodity models along these lines are solved in Nagurney b, a and Nagurney and Aronson We assume that J different commodi- ties, typically denoted by k, are produced at supply markets and are consumed at n demand markets.

The typical supply and demand market will be denoted, re- spectively, by i and j. The state of the system will be described by a number of vectors as follows: 39 Downloaded from hpc. Finally, also consider the general situation where we the transaction cost of any commodity between any pair of supply and demand markets may depend upon ship- ments generated by any commodity between any pair of supply and demand markets.

Thus, we assume that the transaction cost c is related to the multicommodity ship- ment pattern Q through a known continuous transaction cost function In the standard spatial price equilibrium model cf. Ta- kayama and Judge, , the transaction cost repre- sents only the transportation cost associated with ship- ping commodity k from supply market i to demand market j.

In the subsequent section we adapt the serial and cially effec6ve when the resulting subproblems have special s8ucWm parallel diagonal decomposition algorithms outlined in the Introduction to be used for the solution of this that can be further exploited compu- model.

For purposes of computational comparison, we cationally. In the case of the multi- will assume, henceforth, that the strong monotonicity as- commodity market equilibrium model, sumption 2 holds.

In the case that the transaction cost also includes the transportation cost, which reflects congestion, then we also expect the condition to hold. Although the condition appears to be the most restrictive in terms of the demand price func- tions, we note that positive definiteness has been also as- sumed in the work of Takayama and Judge Hence, the assumption 2 is in the same spirit. Finally, we note that by assuming monotonicity, we are guaranteed the existence of a unique solution.

Hence, the relative efficiencies of the decomposition al- gorithms are based on the performance times required to compute the identical equilibrium within a conver- gence tolerance.

These, in turn, can be solved via the supply and de- mand market equilibration algorithms introduced re- cently by Dafermos and Nagumey , see also Na- gurney, b, c , which have the remarkable Downloaded from hpc.

For a theoretical analysis of such equilibration algorithms, see Eydeland and Na- gumey Specifically, the adaptations of the diagonal decom- position schemes, referred to henceforth as DS and DP to reflect diagonal serial and diagonal parallel decompo- sition, respectively, take on the following forms. At iteration t, construct new supply price, demand price, and transaction cost functions for commodity 1, which are linear, and solve the VI subproblem, which is a quadratic programming problem.

Then construct new supply price and demand price and transaction cost functions for commodity 2, using the latest computed values for the previously solved commodity sub- problems, and so on, until equilibrium conditions 11 hold within a prespecified tolerance.

The VI subproblem at iteration t, for commodity k, is hence Downloaded from hpc. Update the functions according to 21 , 22 , and 23 for all the commodities. Continue in this fashion until equilibrium conditions 11 hold within a prespecified tolerance. The resulting VI subproblem is identical to 19 with the functions 21 , 22 , and Hence, the en- suing problem is also equivalent to the solution of a qua- dratic programming problem of the form For con- vergence results, see Dafermos and Pang The problems solved represent the largest 44 Downloaded from hpc.

The problems we consid- ered were large-scale with as many as 32, variables and over , data elements. All of the examples were stored in main memory. In this section we are interested in the relative performance of DS and DP in a nonparallel environment.

We con- ducted these computational tests for the sake of com- pleteness and insight. Here we emphasize that we use CPU time as a measurement, since these experiments are conducted in a multiuser environment, hence the elapsed times would be inordinately large and meaningless.

In terms of applications, this condi- tion means that the supply price of commodity k at supply market 1 depends primarily upon the supply of commodity k at supply market i; the demand price of commodity k at demand market j depends primarily upon the demand for commodity k at demand market j; and the transaction cost for commodity k between supply market i and demand market j depends pri- marily on the quantity of the commodity shipped be- tween the pair of markets.

We would expect such a con- dition to hold in many applications. Observe that in the linear case, the DS and DP methods and their counterparts for the projection method choice 2 induce identical algorithms.

We generated random examples ranging from 10 supply markets and 10 demand markets 20 markets to 50 supply markets and 50 demand markets markets total , in increments of 10 supply and 10 de- mand markets 20 markets , with the number of com- modities being set at 3, 6, 9, and Linear, asymmetric, single commodity problems of similar market number dimensions had been solved by Nagumey b using serial decomposition by demand markets and by supply markets and the projection method.

Pang pre- sented numerical results with a complementarity algo- rithm for substantially simpler single-price spatial price equilibrium problems, in which the transaction cost con- sists of only a fixed transportation cost term M, and these terms must satisfy a restrictive triangle inequality see also Theise and Jones, The supply price, demand price, and transaction cost functions were generated randomly as follows, so that a strict diagonal dominance condition held in order to guarantee uniqueness of the solution.

The diagonal term, 1, in each supply price function 24 , was gener- ated uniformly in the range [, 30,]. The fixed supply price term, tk, was generated in the range [, 1,].

The diagonal term, - mjf, in each demand price function 25 , was generated uniformly in the range [ - 50, - 5,].

The fixed demand price term, A was generated in the range [50, ,]. The diagonal term, -Jlk in each transaction cost function 26 , was generated uniformly in the range [20, 6,].

The Dafermos and Nagumey demand market equilibration algorithm, which had performed well in computational tests see also Nagumey, b, c; and Eydeland and Nagumey, , was used to solve the quadratic programming problems 20 in this exper- iment, as well as all subsequent experiments.

The termi- nation criterion utilized in these and all subsequent ex- amples was that the equilibrium conditions 11 held within a tolerance set equal to This is reasonable be- cause of the large spread of the functions. In Table la the number of cross-terms for each function 24 , 25 , and 26 in each example is set equal to 5, whereas the number of cross-terms for the functions in the examples in Table 1 b is set at Downloaded from hpc.

We considered multicommodity market equilibrium problems with nonlinear supply price, demand price, and transaction cost functions of the form These functions also generated to ensure that the were strong monotonicity condition held. The linear terms in the functions 27 , 28 , and 29 , were generated in the same manner as their counterparts in 24 , 25 , and The termination criterion utilized was identical to the one used in the linear tests.

In Table 2a the number of cross-terms for each 48 Downloaded from hpc. DS is a Gauss-Seidel type of algorithm in that it updates the functions as soon as the new commodity information is available. The su- perior performance of a Gauss-Seidel-type decomposi- tion algorithm, as compared to the projection method, had been shown earlier by Nagurney b for the single commodity problem.

Both DS and DP were ro- bust and converged for all of the examples. In the next section we utilize two speedup measures to explore the efficiency of the DP algorithm when implemented on the parallel processors of the IBM E. Hence, it is not amenable to parallelization.

DP, on the other hand, can be implemented in parallel, with each or several commodity subproblem s assigned to a processor. In particular, we utilize the IBM E in a production environment with a high Strategic User priority to eliminate conten- tion , in which up to 3 CPUs are allocated to our experi- ments.

We note that here we utilize elapsed times, rather than CPU times, as was done in section 3. Indeed, it is well known that the CPU time will increase in any parallel processing environment when multiple processors, rather than a single processor, are used.

The construction of new functions for each com- modity cf. In this series of experiments we consider equilibrium problems in which the number of commodities is greater than, or equal to, the number of processors. In the case where the number of commodities exceeds the number of pro- cessors, the commodity subproblems are assigned asynchronously to a processor, once it becomes available; otherwise, each processor is allocated a single commodity. We utilized the Parallel Fortran compiler, VS 2.

The speedup measures were defined as follows: where Ta is the elapsed time to solve the problem using the serial algorithm DS and TN is the elapsed time to solve the problem using the parallel algorithm DP on N processors in the same software and hardware environ- ment. We also defined Fig. In fact, these data were taken when we were the only logged-on Strategic User. Indeed, our measurements of CPU time on this system were within 0. We selected four examples that we had solved pre- viously ; specifcally, three from Table la, where example 1 is linear with markets and 3 commodities, ex- ample 2 is linear with markets and 6 commodities, and example 3 is linear with markets and 12 com- modities ; and one from Table 2a, where example 4 is nonlinear with markets and 12 commodities.

We varied the number of processors from one to three and computed the two speedups for the four examples. The Fig. We also note that since examples 1, 2, and 3 in Table 3 correspond to the examples in row 5, columns 1, 2, and 4, respectively, in Table la, and example 4 corresponds to the example in row 5, column 4 in Table 2a, one may want to derive meaning from the ratio of the reported times.

E is the efficiency. However, one on a must realize that the elapsed times reported in Table 3 and the CPU times reported in Tables la and 2a are not directly comparable; hence such ratios do not represent speedups obtained in terms of parallelization. Indeed, we would expect the elapsed times for the examples in the experiments reported in Tables la, 1 b, 2a, and 2b would be large, as noted above, because of the multiuser interactive environment.

Of course, one can expect im- provements in compilers for parallel processing in the future. These multicommodity problems are the largest of this level of generality solved to date. The computational results conducted on the IBM E demonstrated that both algorithms are robust and effi- cient. Moreover, the speedups obtained are promising. This work represents the first time that parallel VI decomposition algorithms have been implemented in parallel and their relative efficiencies vis a vis serial de- composition algorithms evaluated.

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View code. Example More experiments are detailed in our paper, in source code, we provide a toy example to show how to use our serial-EMD. The whole serial-EMD consists of three phase: Concatenate Phase: In this phase, the original multi-dimensional signals will be concatenated from head to tail for each signal with a proper transition.

Deconcatenate Phase: In this phase, the concatenated imfs will be deconcatenated to generate imfs for each signal.

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