The diagonal quadratic approximation (DQA) algorithm is extended for the case of risk-averse utility and other nonlinear functions associated with stochastic programs. The method breaks the stochastic program into a sequence of smaller quadratic programming subproblems that can be executed in parallel. Each subproblem is solved approximately by means of a convex version of a primal-dual interior-point code (LOQO). Convergence of the distributed DQA method is discussed.
All communication takes place among neighboring processors rather than via a master routine leading to an efficient distributed implementation. Results with a realworld airline planning model possessing a convex objective, 155,320 linear constraints and 303,600 variables, show the DQA algorithm’s efficiency. The interior point direct solver (convex-LOQO) is shown to solve moderate-size stochastic programs in a small number of iterations (under 50).

MSC codes

  1. 90C15
  2. 90C30
  3. 90C25
  4. 68Q22


  1. stochastic programming
  2. decomposition
  3. parallel computation
  4. interior point methods.

Get full access to this article

View all available purchase options and get full access to this article.


A. Beguelin, J. Dongarra, R. Geist, R. Manchek, V. Sunderam, A Users' Guide to PVM Parallel Virtual Machine, Tech. Report, ORNL/TM-11826, Engineering Physics and Mathematics Div., Mathematical Sciences Section, Oak Ridge National Laboratory, Oak Ridge, TN, 1991
Adam J. Berger, John M. Mulvey, Andrzej Ruszczyński, W. W. Hager, D. W. Hearn, P. M. Pardalaos, Restarting strategies for the DQA algorithmLarge scale optimization (Gainesville, FL, 1993), Kluwer Acad. Publ., Dordrecht, 1994, 1–25
Dimitri P. Bertsekas, Constrained optimization and Lagrange multiplier methods, Computer Science and Applied Mathematics, Academic Press Inc. [Harcourt Brace Jovanovich Publishers], New York, 1982xiii+395
John R. Birge, Decomposition and partitioning methods for multistage stochastic linear programs, Oper. Res., 33 (1985), 989–1007
J. R. Birge, D. Holmes, Efficient solution of two-stage stochastic linear programs using interior point methods, Comput. Optim. Appl., 1 (1992), 245–276
T. J. Carpenter, Masters Thesis, Practical Interior Point Methods for Quadratic Programming, Ph.D. dissertation, Dept. of Civil Engineering and Operations Research, Princeton University, Princeton, NJ, 1992
Tamra J. Carpenter, Irvin J. Lustig, John M. Mulvey, David F. Shanno, Separable quadratic programming via a primal-dual interior point method and its use in a sequential procedure, ORSA J. Comput., 5 (1993), 182–191
L. F. Escudero, P. V. Kamesam, On solving stochastic production planning problems via scenario modelling, Tech. Report, IBM T. J. Watson Research Center, Yorktown Heights, NY, 1993
A. S. El-Bakry, R. A. Tapir, T. Tsuchiya, Y. Zhang, On the formulation and theory of the primal-dual Newton interior-point method for nonlinear programming, Tech. report, TR92-40, Rice University, Houston, TX, 1993
Linos F. Frantzeskakis, Warren B. Powell, A successive linear approximation procedure for stochastic, dynamic vehicle allocation problems, Transportation Sci., 24 (1990), 40–57
J. Gondzio, A. Ruszczyński, Sensitivity method for basis inverse representation in multistage stochastic linear programming problems, J. Optim. Theory Appl., 74 (1992), 221–242
Elizabeth R. Jessup, Dafeng Yang, Stavros A. Zenios, Parallel factorization of structured matrices arising in stochastic programming, SIAM J. Optim., 4 (1994), 833–846
R. Keeney, H. Raiffa, Decisions with multiple objectives: preferences and value tradeoffs, John Wiley & Sons, New York-London-Sydney, 1976xxviii+569 pp. ISBN 0-471-46510-0
M. Kusy, W. Ziemba, A bank asset liability model, Oper. Res., 34 (1986), 356–376
Irvin J. Lustig, Roy E. Marsten, David F. Shanno, On implementing Mehrotra's predictor-corrector interior-point method for linear programming, SIAM J. Optim., 2 (1992), 435–449
Irvin J. Lustig, John M. Mulvey, Tamra J. Carpenter, Formulating two-stage stochastic programs for interior point methods, Oper. Res., 39 (1991), 757–770
Harry M. Markowitz, Portfolio selection: Efficient diversification of investments, Cowles Foundation for Research in Economics at Yale University, Monograph 16, John Wiley & Sons Inc., New York, 1959x+344
Renato D. C. Monteiro, Ilan Adler, An extension of Karmarkar type algorithm to a class of convex separable programming problems with global linear rate of convergence, Math. Oper. Res., 15 (1990), 408–422
John M. Mulvey, Andrzej Ruszczyński, A diagonal quadratic approximation method for large-scale linear programs, Oper. Res. Lett., 12 (1992), 205–215
J. M. Mulvey, A. Ruszczynski, A new scenario decomposition methods or large-scale stochastic optimization, 1991, Tech. re-port SOR 91-19, Dept. of Civil Engineering and Operations Research, Princeton University, Princeton, NJ, Oper. Res., to appear
J. M. Mulvey, R. J. Vanderbei, S. A. Zenio, Robust optimization of large scale systems, 1991, Report 91-06-04, Decision Sciences Department, The Wharton School, University of Pennsylvania, Philadelphia, Oper. Res., to appear
J. M. Mulvey, H. Vladimirou, Stochastic network programming for financial planning problems, Management Sci., 38 (1992), 1642–1664
John M. Mulvey, Hercules Vladimirou, Solving multistage stochastic networks: an application of scenario aggregation, Networks, 21 (1991), 619–643
S. Nielsen, S. A. Zenios, A massively parallel algorithm for nonlinear stochastic network problems, Oper. Res., 41 (1993), 319–337
Warren B. Powell, B. L. Golden, A. A. Assad, A comparative review of alternative algorithms for the dynamic vehicle allocation problemVehicle routing: methods and studies, Stud. Management Sci. Systems, Vol. 16, North-Holland, Amsterdam, 1988, 249–291
R. T. Rockafellar, Roger J.-B. Wets, Scenarios and policy aggregation in optimization under uncertainty, Math. Oper. Res., 16 (1991), 119–147
Andrzej Ruszczyński, A regularized decomposition method for minimizing a sum of polyhedral functions, Math. Programming, 35 (1986), 309–333
Andrzej Ruszczyński, Parallel decomposition of multistage stochastic programming problems, Math. Programming, 58 (1993), 201–228
A. Ruszczyński, Augmented Lagrangian decomposition for sparse convex optimization, 1992, working paper WP-92-75, IIASA, Laxenburg, Mathematics of Oper. Res., to appear
S. Sen, R. Doverspike, S. Cosares, Network planning with random demand, Report, University of Arizona, Tucson, 1992, December
V. S. Sunderam, PVM: A framework for parallel distributed computing, Concurrency: Practice and Experience, 2 (1990), 315–339
G. Stephanopoulos, W. Westerberg, The use of Hestenes' method of multipliers to resolve dual gaps in engineering system optimization, J. Optimization Theory Appl., 15 (1975), 285–309
R. J. Vanderbei, LOQO User's Manual, Tech. Report, SOR 92-5, Dept. of Civil Engineering and Operations Research, Princeton University, Princeton, NJ, 1992
Robert J. Vanderbei, Tamra J. Carpenter, Symmetric indefinite systems for interior point methods, Math. Programming, 58 (1993), 1–32
R. J.-B. Wets, Yu. Ermoliev, R. J.-B. Wets, Large scale linear programming techniquesNumerical techniques for stochastic optimization, Springer Ser. Comput. Math., Vol. 10, Springer, Berlin, 1988, 65–93

Information & Authors


Published In

cover image SIAM Journal on Optimization
SIAM Journal on Optimization
Pages: 735 - 753
ISSN (online): 1095-7189


Submitted: 7 September 1993
Accepted: 22 April 1994
Published online: 13 July 2006

MSC codes

  1. 90C15
  2. 90C30
  3. 90C25
  4. 68Q22


  1. stochastic programming
  2. decomposition
  3. parallel computation
  4. interior point methods.



Metrics & Citations



If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.

Cited By







Copy the content Link

Share with email

Email a colleague

Share on social media

The SIAM Publications Library now uses SIAM Single Sign-On for individuals. If you do not have existing SIAM credentials, create your SIAM account https://my.siam.org.