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

Algorithm 6

Spectral 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_𝑐𝑔𝑥_𝑐−𝑥_𝑐.

(4) Set 𝜆←1

(5)   Set 𝑥₊=𝑥_𝑐+𝜆Δ𝑓(𝑥_𝑐)

(6) if  𝑓𝑥₊≤max_{{}0≤𝑗≤min𝑐,𝑀−1}𝑓𝑥_𝑐−𝑗+𝛾𝜆𝑑_𝑐,∇𝑓𝑥_𝑐  then

(7)   Define 𝜆_𝑐=𝜆, 𝑥_𝑐+1=𝑥₊, 𝑠_𝑐=𝑥_𝑐+1−𝑥_𝑐, 𝑦_𝑐=∇𝑓𝑥_𝑐+1−∇𝑓𝑥_𝑐

(8)   Got to Step 11

(9) else

(10)  Set 𝜆_new∈[𝜎₁𝜆,𝜎₂𝜆], 𝜆←𝜆_new  and go to Step 5

(11)  Compute 𝑏_𝑐=𝑠_𝑐,𝑦_𝑐.

(12) if  𝑏_𝑘≤0  then

(13)  𝛼_𝑐+1=𝛼_max.

(14) else

(15)  𝛼_𝑐=𝑠_𝑐,𝑠_𝑐 and 𝛼_𝑐+1=min𝛼_max,max𝛼_min,𝛼_𝑐/𝑏_𝑐