Table 1.

Pseudo-codes for gradient descent method with adaptive step lengths.

Setting αk,1, αmax, c1, c2 and lmax;

Initializing αk,0 = 0, l = 1;

repeat

if ϕ (αk,l) > ϕ 0 + c1αk,lϕ′ (0) or [ϕ(αk,l) ≥ ϕ(αk,l−1) and l > 1]

Choose the αk,* with dichotomy within (αk,l−1, αk,l) to fulfil the conditions (13) and (15), then break;

elseif |ϕ′(αk,l)| ≤ −c2ϕ′(0)

αk,* = αk,l, then break;

elseif ϕ′(αk,l) ≥ 0

Choose the αk,* with dichotomy within (αk,l−1, αk,l) to fulfil the conditions (13) and (15), then break;

else

αk,l+1 = αk,l * 1.1;

l + +;

If l > lmax or αk,l > αmax

error(‘Initial step length is not suitable!’)

end

end

end(repeat)