O restart;
# conjugate direction algorithm:
x(k+1) = x(k) + alpha(k) * d(k)
with(plots):
O # Here's our function -- note that I am not starting with Q and
b, so I will not
# be able to use our closed form expression for alpha(k).
f := (x,y) -> (1/2)*(x-1)^2+3*(y+1/2)^2-4;
1
1 2
f := x, y /
xK1 2 C3 yC
K4
2
2
O # Here's the plot...in the Maple worksheet, you can spin it
around.
(1)
plot3d(f(x,y),x=-2..2,y=-2..2, axes=boxed);
O # The minimizer is clearly at (1, -1/2).
x[0] := [3,2];
Let's start at (3,2).
x0 := 3, 2
O # Here's our first search direction: d(0) = -g(0).
can use subs instead of unapply if you want....
(2)
(NOTE: You
O
O d[0][1] := -unapply(diff(f(x,y),x),(x,y))(3,3);
d[0][2] := -unapply(diff(f(x,y),y),(x,y))(3,3);
d0 := K2
1
d0 := K21
2
(3)
O # Let's just pick alpha(0) out of the air (refining it as
necessary): (You could do a line search to find it, though
there's a formula for it in the case of f being quadratic.
alpha[0] := 0.12;
a 0 := 0.12
(4)
O # Now we can update x to get our next iterate:
# (This block also gives the function value at our new x.)
x[1][1] := x[0][1] + alpha[0]*d[0][1];
x[1][2] := x[0][2] + alpha[0]*d[0][2];
f(x[1][1], x[1][2]);
x1 := 2.76
1
x1 := K0.52
2
K2.450000000
(5)
O # Here's our next search direction: d(1) = -g(1) + beta(0)*g(0).
# To start, here's the gradient:
g[1][1] := unapply(diff(f(x,y),x),(x,y))(x[1][1],x[1][2]);
g[1][2] := unapply(diff(f(x,y),y),(x,y))(x[1][1],x[1][2]);
g1 := 1.76
1
g1 := K0.12
2
(6)
O # Using Fletcher-Reeves (since it's simple and I don't have Q
handy): (d[0]=-g[0])
beta[0] := (g[1][1]^2 + g[1][2]^2)/(d[0][1]^2 + d[0][2]^2);
b 0 := 0.006993258427
(7)
O # So here's our direction:
d[1][1] := -g[1][1] + beta[0]*d[0][1];
d[1][2] := -g[1][2] + beta[0]*d[0][2];
d1 := K1.773986517
1
d1 := K0.0268584270
2
(8)
O #Again, we pick alpha out of the air (or with a formula for
quadratics or with a line search more generally):
alpha[1] := 1;
a 1 := 1
O #Finally, here's x[2] and f(x[2]):
x[2][1] := x[1][1] + alpha[1]*d[1][1];
(9)
O
x[2][2] := x[1][2] + alpha[1]*d[1][2];
f(x[2][1],x[2][2]);
x2 := 0.986013483
1
x2 := K0.5468584270
2
K3.993315052
O # Recall that the minimizer was at (1, -1/2) and the value of f
there is -4. As expected, in 2 variables, with a quadratic,
conjugate direction required 2 steps.
(10)