Problem with Maximize

Question

As $\{p1,p2,p3,p4,p5\}$ represents a probability distribution and $l1,l2$ are the eigenvalues of a stochastic matrix, the function $f$, as defined below, has to be maximized over a certain region.

L[p_List] := (p[[1]] - 0.2)*Log[p[[1]]] + (p[[2]] - 0.2)*
Log[p[[2]]] + (p[[3]] - 0.2)*Log[p[[3]]] + (p[[4]] - 0.2)*
Log[p[[4]]] + (p[[5]] - 0.2)*Log[p[[5]]]

CirculantMatrix[l_List?VectorQ] := 
NestList[RotateRight, RotateRight[l], Length[l] - 1]
CirculantMatrix[l_List?VectorQ, n_Integer] := 
NestList[RotateRight, 
RotateRight[Join[Table[0, {n - Length[l]}], l]], n - 1] /; 
n >= Length[l]

invFou[l_List] := 0.2*{1 + 2 l[[1]] Cos[2 Pi/5] + 2 l[[2]] Cos[4 Pi/5], 
1 + 2 l[[1]] Cos[4 Pi/5] + 2 l[[2]] Cos[2 Pi/5], 
1 + 2 l[[1]] Cos[4 Pi/5] + 2 l[[2]] Cos[2 Pi/5], 
1 + 2 l[[1]] Cos[2 Pi/5] + 2 l[[2]] Cos[4 Pi/5], 
1 + 2 l[[1]] + 2 l[[2]]}

m[l_List] := CirculantMatrix[invFou[l]]

f[p_List, M_List] := L[p.m[M]]/L[p]

We may assume that $l1=0.5$ and $0.37\leq l2 \leq 0.5$. If we try

Maximize[{f[{p1, p2, p3, p4, p5}, {0.5, 
l2}], {p1 + p2 + p3 + p4 + p5 == 1 && p1 >= 0 && p2 >= 0 && 
p3 >= 0 && p4 >= 0 && p5 >= 0 && 0.37 <= l2 <= 0.5}}, {p1, p2, p3, p4, p5, l2}]

we get an error:

NMaximize::nrnum: The function value -0.206184-0.256075 I is not a real number at {l2,p1,p2,p3,p4,p5} = {0.446053,-0.0362605,0.263705,0.324533,0.369047,0.0789762}. >>

Why doesn't Mathematica take the maximum over the specified region?

Answer 1

The docs specify that the domain should (usually) be Reals or Integers. These are keywords. You probably want the "domain" to be specified as a constraint.

Maximize[{
  Abs[f[{p1, p2, p3, p4, p5}, {0.5, l2}]],
  {p1 + p2 + p3 + p4 + p5 == 1 && p1 >= 0 && p2 >= 0 && p3 >= 0 && p4 >= 0 &&
   p5 >= 0 && 0 <= l2 <= 0.5}},
 {p1, p2, p3, p4, p5, l2}]

Or in V10 you can use the new region functions to specify the domain.

Maximize[
 {Abs[f[{p1, p2, p3, p4, p5}, {0.5, l2}]]},
 {p1, p2, p3, p4, p5, l2} \[Element] 
  ImplicitRegion[
   p1 + p2 + p3 + p4 + p5 == 1 && p1 >= 0 && p2 >= 0 && p3 >= 0 && 
    p4 >= 0 && p5 >= 0 && 0 <= l2 <= 0.5, {p1, p2, p3, p4, p5, l2}]]

With a capriciously chosen objective function

f[p_, l_] := p.p - Flatten[{1, l, l}].p

both of the above return

{0.45,
 {p1 -> 0.4, p2 -> 0.149999, p3 -> 0.150002, p4 -> 0.149999, p5 -> 0.150002, l2 -> 0.5}}