How to use parts of expression as variables in Minimize (NMinimize )?

Question

I want to solve some optimization problem involving matrices. A toy example related to the problem looks as follows

Ma = Table[RandomVariate[UniformDistribution[{0, 1}]], {q1, 1, 10}, {q2, 1, 10}];
Minimize[{Norm[Part[Ma, 1;;10-d1, 1;;10-d2]-Part[Ma,d1+1;;10, d2 + 1 ;; 10], 2]^2/(10-d1)/(10-d2),And[1 <= d1, d1 <= 5, d2 >= 1, d2 <= 5],Element[ Alternatives[d2,  d1], Integers]}, {d1, d2}]

Minimize does not recognize 10-d1, 10-d2 as a valid Span specification, though returns the valid solution. Obviously, this problem can be solved without use of Minimize, but for certain reasons it is more convenient to use Minimize or NMinimize

Answer 1

This is just an evaluation order problem as the d1,d2 are treated as symbolic. You can force d1,d2 to be integers using a pattern test:

(* you don't need the table here either *)
Ma = RandomVariate[UniformDistribution[{0, 1}], {10, 10}];

getMa1[d1_?IntegerQ, d2_?IntegerQ] := Part[Ma, 1 ;; 10 - d1, 1 ;; 10 - d2]
getMa2[d1_?IntegerQ, d2_?IntegerQ] := Part[Ma, d1 + 1 ;; 10, d2 + 1 ;; 10]
Minimize[{Norm[getMa1[d1, d2] - getMa2[d1, d2], 2]^2/(10 - d1)/(10 - d2),
  And[1 <= d1, d1 <= 5, d2 >= 1, d2 <= 5], 
  Element[Alternatives[d2, d1], Integers]}, {d1, d2}]