Graph-Based Symbolic Technique and Its Application in the Frequency Response Bound Analysis of Analog Integrated Circuits

Algorithm 5

Projected gradient method.

(1) if  𝑃𝑥_𝑐−∇𝑓𝑥_𝑐−𝑥_𝑐=0, that is, 𝑥_𝑐 is stationary then

(2)   Stop

(3) Compute 𝑑_𝑐=𝑃𝑥_𝑐−𝛼_𝑐∇𝑓𝑥_𝑐−𝑥_𝑐

(4) Set 𝜆←1

(5)  Set 𝑥₊=𝑥₊+𝜆𝑥_𝑐

(6) if  𝑓(𝑥₊)≤𝑓(𝑥_𝑐)+𝛾𝑥₊−𝑥_𝑐,∇𝑓(𝑥_𝑐)  then

(7)   𝜆_𝑐=𝜆, 𝑥_𝑐+1=𝑥₊, 𝑠_𝑘=𝑥_𝑐+1−𝑥_𝑐, 𝑦_𝑘=∇𝑓(𝑥_𝑐+1)

(8)   Go to Step 12

(9) else

(10)  Define 𝜆_new∈[𝜎₁𝜆,𝜎₂𝜆] and 𝜆←𝜆_new

(11)  Go to Step 5

(12) Compute the trial step length 𝛼_𝑐+1=𝛼_𝑐𝑓(𝑥_𝑐)²/𝑓(𝑥_𝑐+1)²