Smoothing Techniques and Augmented Lagrangian Method for Recourse Problem of Two-Stage Stochastic Linear Programming

Algorithm 1

Newton method with the Armijo rule.

Choose a 𝑧₀∈𝑅^𝑚2, 𝛼>1,𝜄>1, 𝜖>0 be error tolerance and 𝛿 is a small positive

number.

𝑖=0;

While  𝑧_𝑖−𝑧_𝑖−1∞≥𝜖

Choose a 𝑝₀∈𝑅^𝑛2 and set 𝑘=0.

While  ∇_𝑝𝑆(𝑝_𝑘,𝛽,𝑧_𝑖,𝜉,𝜔,𝛼)_∞≥𝜖

Choose  𝜆_𝑘=  max{𝑠,𝑠𝜎,𝑠𝜎²,…} such that

𝑆(𝑝_𝑘,𝛽,𝑧_𝑖,𝜉,𝜔,𝛼)−𝑆(𝑝_𝑘+𝜆_𝑘𝑑_𝑘,𝛽,𝑧_𝑖,𝜉,𝜔,𝛼)≥−𝜆_𝑖𝜇∇(𝑆(𝑝_𝑘,𝛽,𝑧_𝑖,𝜉,𝜔,𝛼))^𝑇𝑑_𝑘,

 where,

𝑑_𝑘=−(∇²𝑆(𝑝_𝑘,𝛽,𝑧_𝑖,𝜉,𝜔,𝛼,)−𝛿𝐼_𝑛2)⁻¹∇𝑆(𝑝_𝑘,𝛽,𝑧_𝑖,𝜉,𝜔,𝛼),𝑠>0  be a constant,

𝜎∈(0,1)  and 𝜇∈(0,1).

Put 𝑝_𝑘+1=𝑝_𝑘+𝜆_𝑘𝑑_𝑘, 𝑘=𝑘+1  and 𝛼=𝜄𝛼.

end

Set 𝑝_𝑖+1=𝑝_𝑘+1, 𝑧_𝑖+1=𝜑(𝑧_𝑖+𝑊^𝑇(𝜔)𝑝_𝑖+1+𝛽𝜉,𝛼)  and 𝑖=𝑖+1.

end