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**A parallel algorithm for constrained concave quadratic global minimization.**
*(English)*
Zbl 0665.90071

The global minimization of large-scale concave quadratic problems over a bounded polyhedral set using a parallel branch and bound approach is considered. The objective function consists of both a concave part (nonlinear variables) and a strictly linear part, which are coupled by the linear constraints. These large-scale problems are characterized by having the number of linear variables much greater than the number of nonlinear variables. A linear and underestimating function to the concave part of the objective is easily constructed and minimized over the feasible domain to get both upper and lower bounds on the global minimum function value. At each minor iteration of the algorithm, the feasible domain is divided into subregions and linear underestimating problems over each subregion are solved in parallel. Branch and bound techniques can then be used to eliminate parts of the feasible domain from consideration and improve the upper and lower bounds. It is shown that the algorithm guarantees that a solution is obtained to within any specified tolerance in a finite number of steps. Computational results are presented for problems with 25 and 50 nonlinear variables and up to 400 linear variables. These results were obtained on a four processor CRAY2 using both sequential and parallel implementations of the algorithm. The average parallel solution time was approximately 15 seconds for problems with 400 linear variables and a relative tolerance of 0.001. For a relative tolerance of 0.1, the average computation time appears to increase only linearly with the number of linear variables.

### Keywords:

global minimization; large-scale concave quadratic problems; bounded polyhedral set; parallel branch and bound; Computational results
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\textit{A. T. Phillips} and \textit{J. B. Rosen}, Math. Program. 42, No. 2 (B), 421--448 (1988; Zbl 0665.90071)

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### References:

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[9] | A.T. Phillips and J.B. Rosen, ”A parallel algorithm for constrained concave quadratic global minimization,” Technical Report 87-48, Computer Science Department, University of Minnesota, Minneapolis, MN (1987a). · Zbl 0665.90071 |

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