@vv That's a pity; I don't know why Maple^® cannot handle real algebraic functions directly and then give a more readable (but still correct) solution.
Let us see what Symbolic Math Toolbox™ returns: 

@vv Thanks.

But how about the poster's question (instead of generalized cases)? 

@vv Thanks. But how do I force Maple to eliminate the dummy variable t3 from the output automatically (instead of by hand later)? Actually, the desired result should resemble .

@Kitonum 

Download inequalities_are_not_supported.mw

@vs140580 Sorry, I just want to reply to CL's "fast or slow" statement. I don't answer your question.

@Carl Love Thanks.

Incidentally, it appears that the seq version is somewhat faster: (Another comparison.) 

Download iRandomPointVector1.mw
 

Download iRandomPointVector2.mw

@vs140580 How large is your n? However, I think that the most convenient way is using seq(seq(seq(…, k in list), j in list), i in list); (or seq(…, l in combinat:-choose(list, 3));).
(By the way, it seems to me that using for loops in Maple is generally not the most efficient way to iterate over some index variable(s).)

@acer Thanks for your observation.
Can anyone reproduce that bug in Maple 2018?

@mmcdara Thanks. If I understand right, for this purpose, I have to replace the problem variables manually first. So, is it impossible to find the exact supermun directly?
Actually, I have some more complex expressions, but Maple's maximize (or minimize) often gives incorrect results without any exception messages (although the Student[MultivariateCalculus][LagrangeMultipliers] function sometimes works). 

@Carl Love Well, you may consider 

k*x*y - (x - y)^2/2; # which is equal to (1 + k)*x*y - (x^2 + y^2)/2

 for some real parameter k. When k = 0, its supremum is just 0, while when k > 0, its supremum is no longer 0 (but it appears that Maple don't know this fact). (As the arithmetic mean-geometric mean inequality says, we have x²＋y²≥2xy, where the coefficient "2" cannot be replaced with a larger number in general.)

@QM I agree you. This example is simple enough, so it's not hard to find (and then fix) such a bug; nonetheless, ensuring the correctness would never be users' responsibility!

@C_R Thanks. That's really strange. I don't know what happened in Maple 2023: sqrt(A^2) doesn't work, but sqrt(A)^2, sqrt(A*A), and sqrt(A)*sqrt(A) all works.

@C_R Thanks for your try. Since the error occurs in `simplify/sqrt/fraction`, can you reproduce this in Maple 2022.2? 
 

Download lasterror.mw

@Preben Alsholm Many thanks. Could you test f~(x,y) and f~(x,` $`,y) additionally? (I think that the latter would run a bit faster, but I cannot confirm this.)

@Scot Gould This works: 

restart;
with(Units):
Automatically loading the Units[Simple] subpackage


A := 1, 2, 3:
B := 4, 5, 6:
AB := :-eval('B' - 'A');
                         AB := 3, 3, 3

Note that the original B - A is equivalent to 

Units:-Simple:-`+`(B, Units:-Simple:-`-`(A));
                               11

B - A;
                               11

:-eval('B' - 'A'); # fix it
                            3, 3, 3