Accelerated Double Direction Method for Solving Unconstrained Optimization Problems

Algorithm 3

Accelerated Double Direction method (ADD method).

Require:  0<𝜌<1, 0<𝜏<1, 𝑥₀

 (1) Set 𝑘=0, compute 𝑓(𝑥₀), 𝑔₀ and take 𝛾₀=1.

 (2) If 𝑔_𝑘<𝜖, then go to Step  9, else continue by the next Step.

 (3) Find the stepsize 𝛼_𝑘 applying by Algorithm 1.

 (4) Compute 𝑑_𝑘 by Algorithm 2.

 (5) Compute 𝑥_𝑘+1 using (8), 𝑓(𝑥_𝑘+1) and 𝑔_𝑘+1.

 (6) Determine the scalar 𝛾_𝑘+1 using (13).

 (7) If 𝛾_𝑘+1<0, then take 𝛾_𝑘+1=1.

 (8) Set 𝑘∶=𝑘+1, go to Step  2.

 (9) Return 𝑥_𝑘+1 and 𝑓(𝑥_𝑘+1).