@John May Thanks. I think that I have an extra problem. GraphTheory:-DijkstrasAlgorithm (as well as GraphTheory:-BellmanFordAlgorithm and GraphTheory:-ShortestPath) does give one shortest path, yet sometimes more are needed; is there a ready-to-use function that can find all (shortest) paths? 
For example, by some visual inspection, one may find three other optimal paths in this puzzle: 

I wonder if the GraphTheory package provides such a command?

This is amazing. But when I run your code, the rendered plot does not seem aesthetically compelling: 

Have you set any additional options? 
Besides, I notice that this page claims that 

while your spd₂ is 5036. Any typos? 

@dharr Actually,  assuming x>0 has no effect when Maple solves eq, since solve ignores assumptions on the variables for which it is solving. (The above equation is adapted from the fourth problem in this page.)

This must be a bug, since the returned value of “applyrule(x::anything^n::'nonunit'(anything) = f(n), 1);” is “f(`/n1`)”.
(“applyrule(x::'nonunit'(anything)^p::'nonunit'(anything) = f(p), 1 + x^2 + x^4);” works, but I believe that it is also a bug if it works like this.)

As a side note, the following code also works: 

convert(Xvector, `+`);

@acer Its source code can be found at http://www.mmrc.iss.ac.cn/~lzhi/Research/hybrid/SDPTools/.
@bathudaide There exists another problem: A slightly detailed explanation of several main functions is written in simplified Chinese. (That English paper does not seem to describe their calling format.) 

@bathudaide You may replace all uses of the external FGb package with the built-in Groebner package (although this is not an easy task.). For instance, “FGb:-fgb_gbasis” should be replaced by “Groebner:-Basis”, and “FGb:-fgb_hilbert” should be replaced by “PolynomialIdeals:-HilbertDimension”. The problem is that the source code is not immediately available, so it requires a lot of work to complete the modification.

Some notes:

1. The only way to numerically solve medium-scale SDP problems in Maple is still to call some Python-based solver using Maple's Python package (or to invoke some MatLab-based solver using Maple's MatLab package). 
2. Maple's Optimization module is based on the NAG library, and NAG does provide one SDP solver, but unfortunately, Maple does not yet support this routine. 
3. There exist another third-party SDP solver for Maple: SDPTools. (The semidefinite is so old that I cannot find its code…) However, I do not know how to use it.

Besides, there exists a general evalb function, `property/Evalb`, in Maple, which may run faster than the analogous top-level functions for a lot of non-numeric input (as there is no additional overhead). However, this function only accepts one inequality (so neither `=` nor `<>` is valid). 

@nm As a side note, according to 

showstat(is, 2 .. 3);

is := proc()
       ...
   2       FAIL
       elif _npassed = 1 and type(_passed[1],'relation(numeric)') then
   3       evalb(_passed)
       ...
end

calling `is` will automatically invoke `evalb` if the input is of type 'numeric', so when using `is` it is always unnecessary to check if the input is of type 'numeric' by hand. 
I believe that another safe method is: 

if signum(0, n - m, 0)::{0, 1} then
    "yes"
else
    "no"
fi:

(By the way, even though the documentation of `verify/greater_than` refers to `signum`, the use of “verify(n, m, 'greater_equal')” is, unfortunately, unsafe.) 

@Carl Love Thanks. Strangely, if _EnvTry is not set to 'hard', the result will not be incorrect (but that does not make sense, since coulditbe just outputs FAIL this time …). 

@ecterrab Thanks for the prompt fix! There are no errors now.

@Christian Wolinski Thanks for the correction. I shall submit a new SCR soon.

@Carl Love Many thanks! So this means that neither coulditbe nor RealDomain:-solve is that reliable … I have just submitted a SCR. 

@dharr Thanks. This command is powerful! But it seems that some heavy post-processing will have to be done, since the help page says: 

So the condition p^2*q + 3*p*r - 4*q^2 > 0 cannot be equivalent to condition q＞0. For instance, the returned dec does not consider (𝑥－1)³ (corresponding to the case p = -3, q = 3, and r = -1). (By the way, I think that [p, q, r] (the fourth argument) should be changed into [].) 

@Carl Love This is very useful!
Maple's select function just filters items based on a criterion, and so do Mma's Select and Cases, but there exist two generalized functions in Mma: SequenceCases and SubsetCases (where the former only finds consecutive sequences). Although these two generalized functions are not commonly used, they are still useful sometimes (for instance, the latter may be used to enumerate cycles of some length in arbitrary graphs). However, I cannot find an equivalent function that can filter (sub-)sequences satisfying certain functions (or matching certain patterns) in Maple. Is there an (elegant) way to simulate the two Mma functions in Maple?