# A primal-dual Newton-type algorithm for geometric programs with equality constraints

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## Abstract

An augmented Lagrangian algorithm is used to find local solutions of geometric programming problems with equality constraints (GPE). The algorithm is Newton's method for unconstrained minimization. The complexity of the algorithm is defined by the number of multiplications and divisions. By analyzing the algorithm we obtain results about the influence of each parameter in the GPE problem on the complexity of an iteration. An attempt is made to estimate the number of iterations needed for convergence. In practice, certain hypotheses are tested, such as the influence of the penalty coefficient update formula, the distance of the starting point from the optimum, and the stopping criterion. For these tests, a random problem generator was constructed, many problems were run, and the results were analyzed by statistical methods.

## Key Words

Geometric programming equality constraints penalty functions Newton's method augmented Lagrangian method## Preview

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## References

- 1.Tapia, R. A.,
*Diagonalized Multiplier Methods and Quasi-Newton Methods for Constrained Optimization*, Journal of Optimization Theory and Applications, Vol. 22, pp. 135–194, 1977.Google Scholar - 2.Blau, G. E., andWilde, D. J.,
*A Lagrangian Algorithm for Equality Constrained Generalized Polynomial Optimization*, AIChE Journal, Vol. 17, pp. 235–240, 1971.Google Scholar - 3.Rijckaert, M. J.,
*Engineering Applications of Geometric Programming, Part 2*, Optimization and Design, Edited by M. Avriel, M. J. Rijckaert, and D. J. Wilde, Prentice-Hall, Englewood Cliffs, New Jersey, 1971.Google Scholar - 4.Rijckaert, M. J., andMartens, X. M.,
*A Comparison of Generalized Geometric Programming Algorithms*, Journal of Optimization Theory and Applications, Vol. 26, pp. 205–242, 1978.Google Scholar - 5.Abrams, R. A.,
*Consistency, Superconsistency, and Dual Degeneracy in Posynomial Geometric Programming*, Operations Research, Vol. 24, pp. 325–335, 1976.Google Scholar - 6.Beightler, C. S., andPhilips, D. T.,
*Applied Geometric Programming*, John Wiley and Sons, New York, New York, 1976.Google Scholar - 7.Bertsekas, D. P.,
*Multiplier Methods: A Survey*, Automatica, Vol. 12, pp. 133–145, 1976.Google Scholar - 8.Tapia, R. A.,
*A Stable Approach to Newton's Method for General Mathematical Programming Problems in R*^{n}, Journal of Optimization Theory and Applications, Vol. 14, pp. 453–476, 1974.Google Scholar - 9.Buys, J. D.,
*Dual Algorithms for Constrained Optimization*, University of Leiden, PhD Thesis, 1972.Google Scholar - 10.Hestenes, M. R.,
*Multiplier and Gradient Methods*, Journal of Optimization Theory and Applications, Vol. 4, pp. 303–320, 1969.Google Scholar - 11.
- 12.Powell, M. J. D.,
*A Method for Nonlinear Constraints in Minimization Problems*, Optimization, Edited by R. Fletcher, Academic Press, London, England, 1969.Google Scholar - 13.Powell, M. J. D.,
*Algorithm for Nonlinear Constraints That Use Lagrangian Functions*, Mathematical Programming, Vol. 14, pp. 224–248, 1978.Google Scholar - 14.Kort, B. W.,
*Combined Primal-Dual and Penalty Function Algorithms for Nonlinear Programming*, Stanford University, PhD Thesis, 1975.Google Scholar - 15.Rockafellar, R. T.,
*Augmented Lagrange Multiplier Functions and Duality in Nonconvex Programming*, SIAM Journal of Control, Vol. 12, pp. 268–284, 1974.Google Scholar - 16.Glad, T., andPolak, E.,
*A Multiplier Method with Automatic Limitation on Penalty Growth*, Mathematical Programming, Vol. 17, pp. 140–155, 1979.Google Scholar - 17.Bunch, J. R., Kaufman, L., andParlett, B. N.,
*Decomposition of a Symmetric Matrix*, Numerische Mathematik, Vol. 27, pp. 95–109, 1976.Google Scholar - 18.Ortega, J. M., andRheinboldt, W. C.,
*Iterative Solution of Nonlinear Equations in Several Variables*, Academic Press, New York, New York, 1970.Google Scholar - 19.More, J. J., andSorensen, D. C.,
*Newton's Method*, Argonne National Laboratory, Argonne, Illinois, Report No. ANL-82-8, 1982.Google Scholar - 20.Crowder, H. P., Dembo, R. S., andMulvey, J. M.,
*Reporting Computational Experiments in Mathematical Programming*, Mathematical Programming, Vol. 15, pp. 316–329, 1978.Google Scholar - 21.Ratner, M., Lasdon, L. S., andJain, A.,
*Solving Geometric Programs Using GRG: Results and Comparisons*, Advances in Geometric Programming, Edited by M. Avriel, Plenum Press, New York, New York, 1980.Google Scholar - 22.Colville, A. R.,
*A Comparative Study of Nonlinear Programming Codes*, IBM, New York Scientific Center, Report No. 320–2949, 1968.Google Scholar - 23.Lidor, G.,
*Test Problem Generator for Nonlinear Programming Algorithms*, City College, New York, New York, Department of Computer Sciences, 1979.Google Scholar - 24.Nie, N. E., Hull, C. H., Jenkins, J. G., Steinbrenner, K., andBent, O. H.,
*SPSS—Statistical Package for the Social Sciences*, McGraw-Hill, New York, New York, 1975.Google Scholar