Nonmonotone Adaptive Barzilai-Borwein Gradient Algorithm for Compressed Sensing

Algorithm 1

NABBL1 for (3).

Initialization: Give initial point 𝑥₀∈𝑅^𝑛, set parameters 𝜇>0, 𝛼>0 and 𝜌∈(0,1),

𝛿∈(0,1), ℎ∈(0,1] and positive integer 𝑚. Set 𝑘=0.

Step  1. If 𝑑_𝑘2=0, then stop. Otherwise, continue.

Step  2. Compute 𝑑_𝑘 via (18).

Step  3. Compute 𝛼_𝑘 via following nonmonotone line search

   𝐹𝑥_𝑘+𝛼_𝑘𝑑_𝑘≤max_{(𝑘)0≤𝑗≤𝑚}𝐹𝑥_𝑘−𝑗+𝛿𝛼_𝑘∇𝑓𝑥_𝑘^𝑇𝑑_𝑘+𝜇𝑥_𝑘+ℎ𝑑_𝑘1−𝜇𝑥_𝑘1ℎ,

where the smallest nonnegative integer 𝑗_𝑘 such as the stepsize 𝛼_𝑘=𝛼𝜌^𝑗𝑘 satisfing above,

𝑚(0)=0 and 0≤𝑚(𝑘)≤min{𝑚(𝑘−1)+1,𝑚}.

Step  4. Let 𝑥_𝑘+1=𝑥_𝑘+𝛼_𝑘𝑑_𝑘.

Step  5. Let 𝑘=𝑘+1. Go to Step  1.