Abstract
This paper describes several parallel algorithms for solving nonlinear programming problems. Two approaches where parallelism can successfully be introduced have been explored: a quadratic approximation method based on penalty function and a dual method. These methods are improved by using two algorithms originally proposed for solving unconstrained problems: the parallel variable metric algorithm and the parallel Jacobson-Oksman algorithm. Even though general problems are dealt with, particular emphasis is placed on the potential of these parallel methods for separable programming problems. The numerical effectiveness of the algorithms is demonstrated on a set of test problems using a Cray-1S vector computer and serial computers (with respect to sequential versions of the same methods).
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Berger, Ph., Dayde, M., andFraboul, C.,Experience in Parallelizing Numerical Algorithms for MIMD Architectures: Use of Asynchronous Methods, La Recherche Aérospatiale, No. 5, pp. 325–340, 1985.
Dixon, L. C. W.,The Place of Parallel Computation in Numerical Optimization, EEC/CNR Summer School, Bergamo University, Bergamo, Italy, 1981.
Lootsma, F. A.,State-of-the-Art in Parallel Unconstrained Optimization, Parallel Computing 85, Edited by M. Feilmeir, G. Joubert, and U. Schendel, Elsevier Science Publishers, Amsterdam, Holland, 1986.
Lootsma, F. A.,Parallel Algorithms for Unconstrained Optimization and Constrained Nonlinear Optimization, Quaderni del Dipartimento di Matematica, Statistica, Informatica e Applicazioni, Istituto Universitario di Bergamo, Bergamo, Italy, 1986.
Mitanker, W. L.,A Survey of Parallelism in Numerical Analysis, SIAM Review, Vol. 13, No. 4, pp. 524–546, 1971.
Schnabel, R. B.,Parallel Computing in Optimization, Computational Mathematical Programming, Edited by K. Schittkowski, Springer-Verlag, Berlin, Germany, pp. 357–382, 1985.
Berger, Ph., Dayde, M., andDunyach, J. C.,Optimum Design Using a Parallel Algorithm for Nonlinear Programming Problems, Parallel Computing 85, Edited by M. Feilmeir, G. Joubert, and U. Schendel, Elsevier Science Publishers, Amsterdam, Holland, 1986.
Berger, Ph., Dayde, M., andDunyach, J. C.,Evluation and Comparison of Parallel Algorithms in Structural Optimization, Innovative Numerical Methods in Engineering, Edited by R. P. Shaw, J. Periaux, A. Chaudouet, J. Wu, C. Marino, and C. A. Brebbia, Springer-Verlag, Berlin, Germany, pp. 567–572, 1986.
Straeter, T. A.,A Parallel Variable Metric Optimization Algorithm, NASA, Technical Report No. D-7329, 1973.
Travassos, R., andKaufman, H.,Parallel Algorithms for Solving Nonlinear Two-Point Boundary-Value Problems Which Arise in Optimal Control, Journal of Optimization Theory and Applications, Vol. 30, No. 1, pp. 53–71, 1980.
Fletcher, R., andPowell, M. J. D.,A Rapidly Convergent Descent Method for Minimization, Computer Journal, Vol. 6, No. 2, pp. 163–168, 1963.
Toint, Ph. L.,Sparsity Exploiting Quasi-Newton Methods for Unconstrained Optimization, Nonlinear Optimization, Theory and Algorithms, Edited by L. C. W. Dixon, E. Spedicato, and G. P. Szegö, Birkhäuser, Boston, Massachusetts, 1980.
Dayde, M.,Parallélisation d'Algorithmes d'Optimisation pour des Problèmes de Conception Optimale, Thèse de Doctorat de l'INPT, Toulouse, France, 1986.
Straeter, T. A., andMarkos, A. J.,A Parallel Jacobson-Oksman Optimization Algorithm, NASA, Technical Report No. D-8020, 1975.
Charalambous, C.,Unconstrained Optimization Based on Homogeneous Models, Mathematical Programming, Vol. 5, No. 2, pp. 189–198, 1973.
Jacobson, D. H., andOksman, W.,An Algorithm That Minimizes Homogeneous Functions of N Variables in N + 2 Iterations and Rapidly Minimizes General Functions, Harvard University, Technical Report No. 618, 1970.
Zoutendijk, G.,Some Algorithms Based on the Principle of Feasible Directions, Nonlinear Programming, Edited by J. B. Rosen, O. L. Mangasarian, and K. Ritter, Academic Press, London, England, 1970.
Rosen, J. B., andKreuser, J.,A Gradient Projection Algorithm for Nonlinear Programming, Numerical Methods for Nonlinear Optimization, Edited by F. A. Lootsma, Academic Press, London, England, 1972.
Wolfe, P.,Methods of Nonlinear Programming: The Reduced Gradient Method, Recent Advances in Mathematical Programming, Edited by G. W. Graves and P. Wolfe, McGraw-Hill, New York, New York, pp. 67–86, 1963.
Abadie, J., andCarpenter, J.,Generalization of the Wolfe Reduced Gradient Method to the Case of Nonlinear Constraints, Optimization, Edited by R. Fletcher, Academic Press, New York, New York, pp. 37–47, 1969.
Fiacco, A. V., andMcCormick, G. P.,Nonlinear Programming, Wiley, New York, New York, 1968.
Lootsma, F. A.,A Survey of Methods for Solving Constrained Minimization Problems via Unconstrained Minimization, Numerical Methods for Nonlinear Optimization, Edited by F. A. Lootsma, Academic Press, London, England, pp. 313–348, 1972.
Bertsekas, D. P.,Penalty and Multiplier Methods, Nonlinear Optimization, Theory and Algorithms, Edited by L. C. W. Dixon, E. Spedicato, and G. P. Szegö, Birkhhäuser, Boston, Massachusetts, 1980.
Bartholomew-Biggs, M. C.,Recursive Quadratic Programming Based on Penalty Functions for Constrained Optimization, Nonlinear Optimization, Theory and Algorithms, Edited by L. C. W. Dixon, E. Spedicato, and G. P. Szegö, Birkhäuser, Boston, Massachusetts, 1980.
Fletcher, R.,An Ideal Penalty Function for Constrained Optimization, Nonlinear Programming 2, Edited by O. L. Mangasarian, R. R. Meyer, and S. M. Robinson, Academic Press, London, England, 1975.
Powell, M. J. D.,Variable Metric Methods for Constrained Optimization, Nonlinear Optimization, Theory and Algorithms, Edited by L. C. W. Dixon, E. Spedicato, and G. P. Szegö, Birkhäuser, Boston, Massachusetts, 1980.
Stoer, J.,Principles of Sequential Quadratic Programming Methods for Solving Nonlinear Programs, Computational Mathematical Programming, Edited by K. Schittkowski, Springer-Verlag, Berlin, Germany, pp. 165–208, 1985.
Fleury, C.,Reconciliation of Mathematical Programming and Optimality Criteria Approaches to Structural Optimization, Report No. SA-86, University of Liège, Aerospace Laboratory, Liège, Belgium, 1980.
Fleury, C.,Dual Methods of Convex Programming, International Journal for Numerical Methods in Engineering, Vol. 14, No. 12, pp. 1761–1783, 1979.
Betts, J. T.,A Gradient Projection-Multiplier Method for Nonlinear Programming, Journal of Optimization Theory and Applications, Vol. 24, No. 1, pp. 523–548, 1978.
Fox, H.,Nonlinear Optimization with Linear Constraints Using a Projection Method, NASA, Technical Report No. 2086, 1982.
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Communicated by L. C. W. Dixon
These studies were sponsored in part by the CERT. The author would particularly like to thank Ph. Berger (LSI-ENSEEIHT), the researchers of the DERI (CERT) and of the Groupe Structures, Aerospatiale, for their assistance.
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Dayde, M. Parallel algorithms for nonlinear programming problems. J Optim Theory Appl 61, 23–46 (1989). https://doi.org/10.1007/BF00940841
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DOI: https://doi.org/10.1007/BF00940841