@Markiyan Hirnyk 

This is only a partial workaround, DS fails. You have asked 3 local extrema, you cannot know whether 1 is global or not.

@Markiyan Hirnyk 

DS is a very good package, but i doubt that you have never seen a fail.

f:=ln(ln( exp(exp(exp(10^10*(x-1/7)))) + y)):
absminmax(f,{x>=1/7,y>=0,x<=3/7,y<=1});

gives the correct result min=1.
In DS, min=Float(infinity) and  sometimes min=0.

@tomleslie 

Your point of view would apply for almost any exact method versus a numeric one. Should we reject the exact methods?
Note that it is also simple to construct examples where DirectSearch fails.
In this case the exact methods can be extended for (some) functions with infinitely many roots (such as the periodic ones): just use solve with option `allsolutions` and filter the results. Of course  for most cases a symbolic result is impossible, but this is also true for int, solve, factor, is, dsolve ...

@magiblot 

Both use Lagrange multipliers. extrema is more "professional". LagrangeMultipliers is more "didactic"and has some plotting options.

Let's say you are interested in guessing the related sequence

seq( bernoulli(n+2),n=0..100 );

What strategy would you recommend in this case?

@Carl Love 

As I see, we are like Newton and Leibniz :-)

The problem is not the fail of the command but the wrong answer in Maple 2016.
Maybe the unevaluated return in Maple 2017 is acceptable.

@Kitonum 

The ode could be in an interval (0,a), a>0, and f'' unbounded.

@tomleslie 

You are right, I missed the "e-2" at the end.
For me it is more annoying that LSSolve is not able to manage Digits>15.

@Carl Love 

Yes, and it works also for X(n) for n>4  (see a previous comment).
But it does not work for arctan because of combine. E.g.

x1 := arctan(-sqrt(2)-1+sqrt(2-sqrt(2))+sqrt(2+sqrt(2)))

(The tan(k*x1) approach works).

@acer 

Of course. I wonder if such a systematic approach will work for

X:=n->arcsin(((x->sqrt(2+x))@@n)(0)/2);

for n>4. (X(4) = x1).

@Rouben Rostamian  

rij hat  is a versor (= rij/rij) , so the bold formula is OK

@Joe Riel 

Unfortunately Iterator will be anyway limited by the huge dimension of the discrete structure.
It will be not possible to use Permute for n=15 because 15! > 10^12;  20! > 10^18.
We will have to wait for the quantum computers.

@Joe Riel 

Thank you for the answer.
It is also strange for me that the procedure is also very slow for the very simple graph defined by

A:=Matrix(n, (i,j)->`if`(i=j,0,i+j)):
(and here the triangle inequality holds)

but becomes very fast for e.g.

A:=Matrix(n, (i,j) -> evalf(abs(sin(i)-sin(j))));
(at least for n=10).

Very nice, vote up.
As optimizations I'd suggest:

1. Remove  HC(S):= P[k]  and recompute it at the end of MinPart (being a waste of memory).
2. Probably  combinat:-permute  should be called separately (at start) only r times for MinPart([m1,...,mr])
and then only translate the indices.

@Carl Love