Maximize does not return the global maximum. Why?

Question

Bug introduced in between 10.0.1 and 10.4.1, and fixed in 11.1

I have the following code

Maximize[{-(((1-3 p0^2-3 p1^2-3 p2^2) (18 p0 p1-27 p0^2 p1-27 p0 p1^2+18 p0    p2-27 p0^2 p2+18 p1 p2-27 p1^2 p2-27 p0 p2^2-27 p1 p2^2))/(9 Sqrt[3]))+((-2+9 p0^2-9 p0^3+9 p1^2-9 p1^3+9 p2^2-9 p2^3) (-p1 p2-p0 (p1+p2)))/(3 Sqrt[3]),p0>=0&&p0<=1&&p1>=0&&p1<=1&&p2>=0&&p2<=1&&p0+p1+p2==1&&1/27 (-2+9 p0^2-9 p0^3+9 p1^2-9 p1^3+9 p2^2-9 p2^3)<=0},{p0,p1,p2}]

The result I get is zero. But this is wrong. This triple

p0->0.284088318172241,p1->0.4318233636555181,p2->0.284088318172241

satisfies all the constraints and the objective expression is positive. It equals 0.0000921367. Curiously, if I multiply the objective function by $\sqrt{3}$ I get the presumed global maximum

$-{16\over(-50013-6049\sqrt{69})}>0.$

The documentation to Maximize says If f and cons are linear or polynomial, Maximize will always find a global maximum. But $\sqrt{3}$ is just a number in my expression and it should not matter that it is not rational.

I get -((22228 Sqrt[3])/1953125) + (24196 Sqrt[23])/5859375, which equals 0.0000921367 approximately. V11.1 — Michael E2, Mar 27 '17 at 00:20
Hmm, my version is 11.0.1.0. Yes, 0.0000921367 is the answer one should get. — user47662, Mar 27 '17 at 00:21
Quite shocking to be honest that this high in the version numbering an error like this can happen. — user47662, Mar 27 '17 at 00:29
It works fine in V9.0.1. The bug was present in V10.4.1; I can't check the versions in between. — Michael E2, Mar 27 '17 at 00:32
For related problems with Maximize before version 10 see e.g. How do I determine the maximum value for a polynomial, given a range of x values? — Artes, Mar 27 '17 at 07:35

score 2 · Answer 1 · edited Apr 13 '17 at 12:55

Amplifying on comment by Michael E2

$Version

(*  "11.1.0 for Mac OS X x86 (64-bit) (March 16, 2017)"  *)

sys = {-(((1 - 3 p0^2 - 3 p1^2 - 3 p2^2) (18 p0 p1 - 27 p0^2 p1 - 
           27 p0 p1^2 + 18 p0 p2 - 27 p0^2 p2 + 18 p1 p2 - 27 p1^2 p2 - 
           27 p0 p2^2 - 27 p1 p2^2))/(9 Sqrt[3])) + ((-2 + 9 p0^2 - 9 p0^3 + 
         9 p1^2 - 9 p1^3 + 9 p2^2 - 9 p2^3) (-p1 p2 - p0 (p1 + p2)))/(3 Sqrt[
        3]), p0 >= 0 && p0 <= 1 && p1 >= 0 && p1 <= 1 && p2 >= 0 && p2 <= 1 &&
     p0 + p1 + p2 == 1 && 
    1/27 (-2 + 9 p0^2 - 9 p0^3 + 9 p1^2 - 9 p1^3 + 9 p2^2 - 9 p2^3) <= 0};

The exact solution is

max1 = Maximize[sys, {p0, p1, p2}]

(*  {-((22228 Sqrt[3])/1953125) + (24196 Sqrt[23])/
  5859375, {p0 -> 
   1 - Root[{-23 + #1^2 &, -3 + #2^2 &, 
      2613168 #1 + 24048128 #2 - 39197520 #1 #3 - 735721920 #2 #3 + 
        7382812500 #2 #3^2 - 30164062500 #2 #3^3 + 64072265625 #2 #3^4 - 
        71191406250 #2 #3^5 + 35595703125 #2 #3^6 &}, {2, 2, 2}] - 
    Root[{-23 + #1^2 &, -3 + #2^2 &, 
      2613168 #1 + 24048128 #2 - 39197520 #1 #3 - 735721920 #2 #3 + 
        7382812500 #2 #3^2 - 30164062500 #2 #3^3 + 64072265625 #2 #3^4 - 
        71191406250 #2 #3^5 + 35595703125 #2 #3^6 &, 
      72588 #1 - 200052 #2 + 7812500 #2 #3 - 25390625 #2 #3^2 + 
        35156250 #2 #3^3 - 17578125 #2 #3^4 + 7812500 #2 #4 - 
        148437500 #2 #3 #4 + 439453125 #2 #3^2 #4 - 562500000 #2 #3^3 #4 + 
        263671875 #2 #3^4 #4 - 25390625 #2 #4^2 + 439453125 #2 #3 #4^2 - 
        843750000 #2 #3^2 #4^2 + 527343750 #2 #3^3 #4^2 + 35156250 #2 #4^3 - 
        562500000 #2 #3 #4^3 + 527343750 #2 #3^2 #4^3 - 17578125 #2 #4^4 + 
        263671875 #2 #3 #4^4 &}, {2, 2, 2, 1}], 
  p1 -> Root[{-23 + #1^2 &, -3 + #2^2 &, 
     2613168 #1 + 24048128 #2 - 39197520 #1 #3 - 735721920 #2 #3 + 
       7382812500 #2 #3^2 - 30164062500 #2 #3^3 + 64072265625 #2 #3^4 - 
       71191406250 #2 #3^5 + 35595703125 #2 #3^6 &}, {2, 2, 2}], 
  p2 -> Root[{-23 + #1^2 &, -3 + #2^2 &, 
     2613168 #1 + 24048128 #2 - 39197520 #1 #3 - 735721920 #2 #3 + 
       7382812500 #2 #3^2 - 30164062500 #2 #3^3 + 64072265625 #2 #3^4 - 
       71191406250 #2 #3^5 + 35595703125 #2 #3^6 &, 
     72588 #1 - 200052 #2 + 7812500 #2 #3 - 25390625 #2 #3^2 + 
       35156250 #2 #3^3 - 17578125 #2 #3^4 + 7812500 #2 #4 - 
       148437500 #2 #3 #4 + 439453125 #2 #3^2 #4 - 562500000 #2 #3^3 #4 + 
       263671875 #2 #3^4 #4 - 25390625 #2 #4^2 + 439453125 #2 #3 #4^2 - 
       843750000 #2 #3^2 #4^2 + 527343750 #2 #3^3 #4^2 + 35156250 #2 #4^3 - 
       562500000 #2 #3 #4^3 + 527343750 #2 #3^2 #4^3 - 17578125 #2 #4^4 + 
       263671875 #2 #3 #4^4 &}, {2, 2, 2, 1}]}}  *)

Simplifying the forms of p0, p1, and p2

max2 = max1 // RootReduce // ToRadicals

(*  {8/3 Sqrt[2/(837672973 + 100842879 Sqrt[69])], {p0 -> 1/75 (49 - 2 Sqrt[69]), 
  p1 -> 1/75 (13 + Sqrt[69]), p2 -> 1/75 (13 + Sqrt[69])}}  *)

Verifying that all of the conditions are met

verify = sys /. max2[[2]] // Simplify

(*  {(4 (-50013 + 6049 Sqrt[69]))/(5859375 Sqrt[3]), True}  *)

The form of the maximum given in verify is simpler than the ones in max1 or max2

LeafCount /@ First /@ {max1, max2, verify}

(*  {19, 21, 18}  *)

The different forms of the maximum are equal

max1[[1]] == max2[[1]] == verify[[1]] // FullSimplify

(*  True  *)

Akku14 · Answer 2 · 2017-03-27T20:11:34.383

Mathematica does not give you the whole truth.

Maximize does not recognize, that the system is totaly symmetric and there are three equal maxima,

corresponding to the six permutations of {p0,p1,p2} divided by two, because of two p's are equal (I corrected my first version).

Redefine sys, calculate the maximum and take the permutations (using MMA 8.0)

    sys[p0_, p1_, p2_] = sys

    max1 = Maximize[sys[p0, p1, p2], {p0, p1, p2}]

    (*   {-((22228 Sqrt[3])/1953125) + (24196 Sqrt[23])/
         5859375, {p0 -> 1 + 2/75 (-13 - Sqrt[69]), 
          p1 -> 1/75 (13 + Sqrt[69]), p2 -> 1/75 (13 + Sqrt[69])}}    *)

    sys[Sequence @@ #] & /@ Permutations[{p0, p1, p2}] /. 
        max1[[2]] // N

    (*   {{0.0000921367, True}, {0.0000921367, True}, {0.0000921367, 
          True}, {0.0000921367, True}, {0.0000921367, True}, {0.0000921367, 
          True}}    *)

Maximize does not return the global maximum. Why?

2 Answers2