@Ronan Thank you for your question, and I welcome any other questions that you have about this code.

When making a substitution with subs into a procedure, the target of the substitution must be an unassigned global variable. Using unevaluation quotes or :- won't work. By convention, variables beginning with an underscore shouldn't be assigned. Note that _M also occurs in the arrow procedure that is the final argument of the subs. That second _M is replaced by one of the two arrow procedures equated to _M in the first argument of subs.

Those globalness restrictions for subs when its last argument is a procedure are not well documented, and I don't know if they're all still true in Maple 2023.

I wasn't the one who deleted some other version of this Question, nor did I even see that version. However, the Moderator who deleted it did not consider it to be spam. There are other reasons that Questions get deleted by Moderators, the most common of which is that it was considered to be a close followup to a recent Question posted by the same author. In those cases it should be posted in the toriginal thread.

As a Moderator, I wish that we had the capability to simply move an out-of-place Question to the correct thread. Unfortunately MaplePrimes lacks that capability.

Are the different "strategies" that are used in your worksheet intended to produce solutions that are mathematically equivalent but for which that equivalence is difficult to verify due to the enormous complexity? Or, are they different ways of approximating the true complete solution and thus not 100% mathematically equivalent?

It goes without saying that it's impossible to Answer your Question if you don't post the code. As a worksheet showing a timed example would likely be best.

@mmcdara Here's an example to show the multi-argument capability of ~, which would be quite awkward  to duplicate with map:

f~([a,b], X*Y, [c,d], [1,2]);
               [f(a, X Y, c, 1), f(b, X Y, d, 2)]

Here is, I believe, the only situation where ~ and map are equivalent: Let L be an object of one of the four basic container types that I mentioned in my last Reply. Let a1, a2, ..., ak be a sequence (possibly empty) of additional arguments, none of which are of any of those container types. Then map(f, L, a1, a2, ..., ak) = f~(L, a1, a2, ..., ak). (Both map and ~ can be overloaded by object methods, which can alter those rules somewhat.)

Regarding map(f, L) versus map(x-> f(x), L): They produce the same result, but the former is more efficient, and can be much more efficient some cases, such as when f is a builtin command. In most cases (whether or not map is involved), (x-> f(x)) can be replaced by simply f, and it's more efficient to do so.

@nm Are you saying that in some earlier version of Maple, solve needed the parametric option to solve that pair of linear equations? I find that hard to believe. Indeed, one can get the solutions symbolically for uninstantiated initial conditions:

sol:= <c1,c2>^+.(<3,-1> +~ exp(4*t))/4/exp(t):
ICsol:= solve(eval([y(t) = sol, D(y)(t) = diff(sol,t)], t= t0), [c1,c2]);
simplify(eval(ICsol, [t0, y(t0), D(y)(t0)]=~ [4, -3, -17]));

The elementwise operator ~ affects only arguments that are specific types of containers: lists, sets, rtables, and tables. For other types of arguments, such as your `*`, it's simply ignored. So, your issue is only with simplify; the simplify~ is correctly passing it's argument to simplify. Note that ~ is not simply an abbreviation for map; it simultaneously applies to all arguments of those container types (tables are handled a bit differently).

I am by no means denying that your point about simplify itself is interesting.

@RukevweOyinloye For the follwing discussion, let

• P = number of profiles,
• G = number of groups,
• W = number of websites.

So for the toy problem, [P,G,W] = [4,2,3], and for the larger problem [256, 16, 103].

Let's consider the size of the coefficient matrix needed to apply the simplex algorithm to the continuous LP (Linear Program) subproblem of the larger ILP (Integer LP). There are P*W binary decision variables. The number of inequality constraints is P, so there are P slack variables, and we add 1 artificial variable, z, so the total number of variables is 1 + P + P*W. There's 1 column for the constants, so the matrix has 2 + P + P*W columns. The number of equality constraints is G*W. There's 1 objective row. So the number of rows is 1 + P + G*W. The number of matrix entries is thus (1 + P + G*W)*(2 + P + P*W) = 50722530 ~ 51 million.

That's a lot of entries; however, the problem is quite sparse. Let's count the nonzero entries in the initial matrix. (A variable's value being zero is irrelevant to this; we're only interested in the variables' coefficients and the constant terms.) Each equality constraint has P/G + 1 terms. Each inequality constraint has W + 2 terms. The objective row has 2 terms. So the total nonzero terms is (P/G + 1)*G*W + (W + 2)*P + 2 =  54898. So the proportion of nonzero entries is 54998/50722630 ~ 0.001, which is very sparse. But special syntax is needed to get the solver to make efficient  use of that sparsity..

@C_R I don't undestand your suggestion from your 3rd paragraph. Why would it be any easier to obtain an older version?

@RukevweOyinloye You mentioned wanting to extend this to 256 profiles in 16 groups. How many websites would there be?

Just a curiosity at this point: That mysterious r shows up in the result of an example on help page ?linalg,eigenvectors with no explanation in the help page text. I guess it's an escaped local of a procedure called by linalg[eigenvectors]. Anyway, there's likely little point in investigating it further as that code has been obsolete for more than 20 years.

This isn't related to your crashed kernel, but I thought that you'd like to know how to do the type of polynomial factoring that you were attempting in the first section of your worksheet:

evala(AFactor(x^2+2));

@RukevweOyinloye I think that all of the errors are due to the first error, the absence of ListTools:-Slice in your version of Maple. Here's a replacement:

Download MinimaxILP.mw

@RukevweOyinloye Here is the fixed program. I changed X from a Matrix back to an Array because an extracted subMatrix (such as the rows belonging to a particular group of profiles) has its rows and columns re-indexed starting at 1. Re-indexing doesn't happen for Arrays. That was the "matrix-indexing idiosyncracy" that I mentioned yesterday.

Download MinimaxILP.mw

@RukevweOyinloye Sorry, there is a bug in my code. Please don't try to assess the code until I fix it. The bug is due to an idiosyncracy of Maple matrix indexing. It has nothing to do with the mathematical formulation of the problem.