An SQP Algorithm for Recourse-based Stochastic Nonlinear Programming

Xinshun Ma, Cunzhe Liu


The stochastic nonlinear programming problem with completed recourse and nonlinear constraints is studied in this paper. We present a sequential quadratic programming method for solving the problem based on the certainty extended nonlinear model. This algorithm is obtained by combing the active set method and filter method. The convergence of the method is established under some standard assumptions. Moreover, a practical design is presented and numerical results are provided.


stochastic programming; nonlinear constraints; SQP

Full Text:



R.M. Van Slyke and R.J.-B. Wets, “L-shaped linear programs with applications to optimal control and stochastic programming”, SIAM Journal of Applied Mathematics, 1969, 17 (4): 638–663.

P. Kall and S.W. Wallace, “Stochastic Programming”, John Wiley & Sons, New York, 1994.

J.R. Birge and F. Louveaux, “Introduction to stochastic programming”, Berlin:Springer, 1997.

R.T. Rockafellar and R.J-B. Wets, “A dual solution procedure for quadratic stochastic programs with simple recourse”, Berlin:Springer, 1983, 252-265.

R.T. Rockafellar and R.J-B. Wets, “A Lagrangian finite-generation technique for solving linear-quadratic problems in stochastic programming”, Math. Prog. Study, 1986, 28:63-93.

R.T. Rockafellar and R.J-B. Wets, “Linear-quadratic problems with stochastic penalties: the finite generation algorithm”, Berlin:Springer, 1987, 545-560.

R.T. Rockafellar, “Linear-quadratic programming and optimal control”, SIAM J. Contr. Optim, 1987, 25:781-814.

R.T. Rockafellar and R.J-B. Wets, “Generalized linear-quadratic problems of deterministic and stochastic optimal control in discrete time”, SIAM J. Contr. Optim, 1990, 28:810-822.

X. Chen, L. Qi and S. Womersley, “Newton’s method for quadratic stochastic programs with recourse”, Journal of Computational and Applied Mathematics, 1995, 60:29-46.

L. Qi and S. Womersley, “An SQP algorithm for extended linear-quadratic problems in stochastic programming”, Annals of Operations Research, 1995, 56:251-285.

Z. Wei, L. Qi and X. Chen, “An SQP-Type method and its application in stochastic programs”, Journal of Optimization theory and applications, 2003, 205-228.

U.V. Shanbhag, “Decomposition and sampling methods for stochastic equilibrium problems”, phD thesis, Department of Management Science and Engineering (Operations Research), Stanford University, 2006.

P. Li Yong and H. Huang Guo, “Interval-parameter Two-stage stochastic nonlinear programming for water resources management under uncertainty”, Water Resour Manage, 2008, 22:681-698.

Heinz W. Engl, “Existence of measurable optima in stochastic nonlinear programming and control”, Appl. Math. Optim, 1979, 5:271-281.

R.T. Rockafellar and R.J.-B. Wets, “Stochastic convex programming: Kuhn-Tucker conditions”, Journal of Mathematical Economics, 1975,2: 349-370.

A.A. Kulkarni and U.V. Shanbhag, “Recourse-based stochastic nonlinear programming properties and Benders-SQP algorithms”, Computational Optimization and Applications , 2012,51: 77-123.

D.Q. Mayne and E. Polak, “A superlinearly convergent algorithm for constrained optimization problem”, Math. Prog. Study, 1982, 16:45-61.

M.J.D. Powell and Y. Yuan, “A recursive quadratic programming algorithm that use differentiable exact penalty function”, Math. Prog, 1986, 35:265-278.

N. Maratos, “Exact penalty function algorithms for finite dimensional and control optimization problems”, phD thesis, Imperial College Sci. Tech. University of London, 1978.

R. Fletcher and S. Leyffer, “Nonlinear programming without a penalty function”, Mathematical Programming, 2002, 91:239-269.

G.C. Broyden, “The convergence of a class of double rand minimization algorithms : 2.the new algorithm”, J. Inst. Math. Appl., 1970, 6:222-231.

R. Fletcher, “A new approach to variable metric algorithms”, Computer J., 1970, 13:317-322.

X. Chen and S. Womersley, “Random test problems and parallel methods for quadratic programs and quadratic stochastic programs”, Optim, Methods Softw. 2000,13:275-306.

W. Hock and K. Schittkowski, “Test examples for nonlinear programming codes”, Springer, New York, 1981.

L. Qi, C.Ling, X. Tong and G. Zhou, “A smoothing projected Newton-type algorithm for semi-infinite programming”, Comput Optim Appl, 2009, 42:1-30.

Diwekar Urmila and David Amy, “Bonue algorithm for large scale stochastic nonlinear programming problems”, Spring Berlin, 2015.

Lululemon Black Friday cheap nfl jerseys Lululemon factory Outlet ny Black Friday discount tiffany outlet wholesale soccer jerseys online oakley black friday cheap nhl jerseys china cheap nfl jerseys north face black friday sale cheap nfl jerseys online Jordans Black Friday Sale 2015 Cheap Moncler Cyber Monday moncler outlet cheap soccer jerseys moncler outlet black friday cheap authentic nfl jerseys north face cyber monday Louboutin Black Friday canada wholesale cheap nfl jerseys lululemon cyber monday 2015 cheap nfl jerseys from china 2015 Cheap Moncler Black Friday Sale Moncler Cyber Monday 2015 cheap jerseys Lululemon Cyber Monday Sale jordans cyber monday deals 2015 cheap nike nfl jerseys Black Friday deals Lululemon 2015 jordan black friday 2015 Moncler Jackets Black Friday Sale 2015 Louboutin Pas Cher Black Friday 2015 Canada Lululemon north face black friday cheap wholesale soccer jerseys