This page https://reference.wolfram.com/language/tutorial/ConstrainedOptimizationComparison.html
says "Even for local optimization, it may still be worth using NMinimize for small problems" and that the Minimize and Maximize commands "...are suitable only for problem with a small number of variables."
What does "small" mean here?
Let us consider a sequence of "toy" problems.
Table[First[Maximize[{-Sum[x[j]^2, {j, 1, n}],
Sum[x[j]*j, {j, 1, n}] == 1 &&
(And @@ Table[x[j] >= 0, {j, 1, n}]) &&
x[1] + x[n] <= 1/n^2}, Table[x[j], {j, 1, n}]]] // Timing, {n, 10, 150, 20}]
{{0.078125, -(131/44375)}, {0.71875, -(382727/ 3464370000)}, {14.0156, -(1510731/63162500000)}, {33.1406, -( 3906799/447762490000)}, {63.8438, -(8049883/ 1959803505000)}, {111.672, -(72098867/ 32036996970000)}, {180.719, -(70489201/ 51688555360000)}, {276.109, -(83439379/93974681250000)}}
Draw a conclusion on your own.
NMaximize results is {{1.79688, -0.00295211}, {31.4688, -0.000110475}, {61.5625, -0.0000239182}, {105.313, -8.72515*10^-6}, {154.797, -4.10749*10^-6}, {216.438, -2.25049*10^-6}, {293.359, -1.36373*10^-6}, {394.734, -8.8789*10^-7}}.
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Commented
Jun 9 at 7:28
n=10 in finite time. What's your version?
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Commented
Jun 9 at 8:56
GreaterEqual over lists? A traditional way to enter the system would be Maximize[{-Sum[x[j]^2, {j, 1, n}], Sum[x[j]*j, {j, 1, n}] == 1 && (And @@ Table[x[j] >= 0, {j, 1, n}]) && x[1] + x[n] <= 1/n^2}, Table[x[j], {j, 1, n}]].
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Commented
Jun 9 at 16:47