The augmented Lagrangian method can be used for solving recourse problems and obtaining their normal solution in solving twostage stochastic linear programming problems. The augmented Lagrangian objective function of a stochastic linear problem is not twice differentiable which precludes the use of a Newton method. In this paper, we apply the smoothing techniques and a fast NewtonArmijo algorithm for solving an unconstrained smooth reformulation of this problem. Computational results and comparisons are given to show the effectiveness and speed of the algorithm.
In stochastic programming, some data are random variables with specific possibility distribution [
In this paper, we consider the following twostage stochastic linear program (slp) with recourse which involves the calculation of an expectation over a discrete set of scenarios:
We assume that the problem (
In general, the recourse function
We now describe our notation. Let
As mentioned the objective function of (
Using dual of the problem (
For functions
For each
The gradient of function
in which
To prove (a), refer to [
Also, (b) and (c) can be easily proved considering that function
Using the approximated recourse function
Consider the problem (
According to Theorem
In the augmented Lagrangian method, the unconstrained maximization problem is solved which gives the project of a point on the solution set of the problem (
Assume that
Considering that the objective function of the problem (
Consider the following maximization problem
Also, in special conditions, the solution for the problem (
Assume that the solution set
According to the theorems mentioned above, augmented Lagrangian method presents the following iteration process for solving the problem (
We note that the problem (
Chen and Mangasarian [
We begin with a simple lemma that bounds the square difference between the plus function
For
Consider the problems (
For any
Considering the advantage of the twice differentiability of the objective function of the problem (
In each iteration of the process (
In the algorithm, the Hessian matrix may be singular, thus we use a modified Newton. The direction in each iteration for solving (
Choose a
number.
Choose a
The proposed algorithm was applied to solve some recourse problems. Table
Comparative between smooth augmented Lagrangian Newton method (SALN) and CPLEX solver.
N. P  Recourse problem 
Solver 



Time (second) 

P1 

SALN 




CPLEX 





P2 

SALN 




CPLEX 





P3 

SALN 




CPLEX 





P4 

SALN 




CPLEX 





P5 

SALN 




CPLEX 





P6 

SALN 




CPLEX 



 
P7 

SALN 




CPLEX 



 
P8 

SALN 




CPLEX 



 
P9 

SALN 




CPLEX 



 
P10 

SALN 




CPLEX 



 
P11 

SALN 




CPLEX 



 
P12 

SALN 




CPLEX 



 
P13 

SALN 




CPLEX 



 
P14 

SALN 




CPLEX 





P15 

SALN 




CPLEX 





P16 

SALN 




CPLEX 





P17 

SALN 




CPLEX 





P18 

SALN 




CPLEX 





P19 

SALN 




CPLEX 





P20 

SALN 




CPLEX 



 
P21 

SALN 




CPLEX 



 
P22 

SALN 




CPLEX 



 
P23 

SALN 




CPLEX 



 
P24 

SALN 




CPLEX 



 
P25 

SALN 




CPLEX 




The test generator generates recourse problems. These problems are generated using the MATLAB code show in Algorithm
The algorithm considered for solving several recourse problems was run on a computer with 2.5 dualcore CPU and 4 GB memory in MATLAB 7.8 programming environment. Also, in the generated problems, recourse matrix
In Table
In this paper, a smooth reformulation process, based on augmented Lagrangian algorithm, was proposed for obtaining the normal solution of recourse problem of a stochastic linear programming. This smooth iterative process allows us to use a quadratically convergent Newton algorithm, which accelerates obtaining the normal solution.
Table
The authors would like to thank the reviewers for their helpful comments.