5
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Here is a problem from math folklore. Minimize $$\sum_{j=1}^{j=n} x_j$$ under the constraints $$x_1+x_2\ge 1,\, x_2+x_3\ge 2,\dots,x_n+x_1 \ge n .$$ The Mathematica codes

Minimize[{Sum[x[j], {j, 1, n}],Table[x[j] + x[j + 1] >= j, {j, 1, n - 1}], x[n] + x[1] >= n},Table[x[j], {j, 1, n}]]

and

Minimize[{Sum[x[j],{j, 1, n}], Table[x[j] + x[j + 1] >= j, {j, 1, n - 1}], x[n] + x[1] >= n,Table[x[j] >= 0, {j, 1, n}]}, Table[x[j], {j, 1, n}]]

crack it for values of $n$ of order several hundreds, e.g. for $n=1234$ both codes produce the value $381306$ for the objective function, but slowly.

In fact, this is a linear programming problem. Is it possible to solve it for $n=2019$ in Mathematica (maybe, calling external sources)?

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  • 2
    $\begingroup$ Have you tried LinearProgramming? $\endgroup$
    – Mr Puh
    Oct 15, 2020 at 8:30
  • 1
    $\begingroup$ @MrPuh: Thank you for the idea. No. Up to the documentation "If f and cons are linear or polynomial, Minimize will always find a global minimum ", I think Minimize refers to LinearProgramming itself. $\endgroup$
    – user64494
    Oct 15, 2020 at 8:37
  • $\begingroup$ For $n=2019$ the first code produces $1019595$ and {x[1] -> 505, x[2] -> -504, x[3] -> 506, x[4] -> -503,...,x[2019] -> 1514} in more than hour. $\endgroup$
    – user64494
    Oct 15, 2020 at 8:56
  • $\begingroup$ @MrPuh: Trying to use LinearProgramming, I meet the problem with creating the matrix. $\endgroup$
    – user64494
    Oct 15, 2020 at 8:58
  • $\begingroup$ For $ n=2019$ the second code produces {1020099, {x[1] -> 1009, x[2] -> 0, x[3] -> 2, x[4] -> 1, ..., x[2019] -> 1010}} in 4350.85 sec $\endgroup$
    – user64494
    Oct 15, 2020 at 10:38

3 Answers 3

5
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n = 2019;
c = Table[1, n];
m = SparseArray[{Band[{1, 1}] -> 1, Band[{1, 2}] -> 1, {n, 1} -> 1}, {n, n}];
b = Range[n];
LinearOptimization[N[c], {m, -b}, "PrimalMinimumValue"] // AbsoluteTiming // DecimalForm
(* {0.0275781, 1019595.} *)

Note that the use of N to force machine precision is important, otherwise the result will be exact but much slower. For some reason, the slowdown becomes much more significant at n = 201:

           Exact   Approximate
n = 200    0.35 s     0.002 s
n = 201    8.9 s      0.002 s

Using LinearProgramming gives the same results (and has the same timing issue):

Total @ LinearProgramming[N[c], m, b, -∞] // AbsoluteTiming // DecimalForm
(* {0.0286999, 1019595.} *)

Note the -∞, without it the constraint x ≥ 0 is added.

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  • $\begingroup$ It looks like LinearProgramming has two methods that only work with machine precision numbers and are much faster than the ones that work for arbitrary precision. $\endgroup$ Oct 15, 2020 at 18:14
  • $\begingroup$ +1. Thank you. This is a modified approach by Ulrich Neumann. How about the values of the variables? Your phrase "Note the -∞, without it the constraint x ≥ 0 is added" is unclear to me: the constraints imply (by addition) that the target function is greater than $\frac {n(n+1)} 4$. $\endgroup$
    – user64494
    Oct 16, 2020 at 0:20
  • $\begingroup$ Loking in the documentation, I got " Note the -∞, without it the constraint x ≥ 0 is added". $\endgroup$
    – user64494
    Oct 16, 2020 at 0:33
  • $\begingroup$ Having executed LinearOptimization[N[c], {m, -b}] // AbsoluteTiming, I see the optimal values of the variables. $\endgroup$
    – user64494
    Oct 16, 2020 at 1:05
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Using a different approach: guess the sequence. Then the answer for 2019 returns the right result in 20 microseconds. However, it does not prove that the sequence is the actual answer.

sol = ParallelTable[{n, Minimize[{Sum[x[j], {j, 1, n}], 
      Table[x[j] + x[j + 1] >= j, {j, 1, n - 1}], x[n] + x[1] >= n}, 
     Table[x[j], {j, 1, n}]]}, {n, 2, 50}];

f = FindSequenceFunction[sol[[All, 2, 1]]];
answer[n_] = f[n - 1];
(* 1/8 (-1)^(-1 + n) (-1 + 5 (-1)^(-1 + n) + 2 (-1)^(-1 + n) (-1 + n)) n *)

answer[2019]
(* 1019595 *)
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  • $\begingroup$ Unfortunately, 1/8 (-1)^(-1 + n) (-1 + 5 (-1)^(-1 + n) + 2 (-1)^(-1 + n) (-1 + n)) /. n -> 1234 performs $309$, whereas the correct answer is $381306$. Thank you anyway. $\endgroup$
    – user64494
    Oct 15, 2020 at 9:31
  • $\begingroup$ @user64494 Try answer[1234]... You did not copy paste properly (missing n at the end). $\endgroup$
    – anderstood
    Oct 15, 2020 at 9:32
  • $\begingroup$ Sorry, my bad. 1/8 (-1)^(-1 + n) (-1 + 5 (-1)^(-1 + n) + 2 (-1)^(-1 + n) (-1 + n)) n /. n -> 1234 performs 381306. The answer is of order $\frac {n^2} 4$. $\endgroup$
    – user64494
    Oct 15, 2020 at 9:49
  • $\begingroup$ I'd like to add the formula $$\frac{1}{8} (-1)^{n-1} \left(2 (-1)^{n-1} (n-1)^2+8 (-1)^{n-1} (n-1)+3 (-1)^{n-1}-3\right)$$ in the case of nonnegative variables. $\endgroup$
    – user64494
    Oct 15, 2020 at 10:24
  • $\begingroup$ My suggestion is to apply dynamic programming to prove the formulas. $\endgroup$
    – user64494
    Oct 16, 2020 at 9:09
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NMinimize calculates the case n==2019

 mini[n_] :=NMinimize[{Sum[x[j], {j, 1, n}],Table[x[j] + x[j + 1] >= j, {j, 1, n - 1}], x[n] + x[1] >= n},Table[x[j], {j, 1, n}]][[1]]
 
 mini[2019] // Rationalize // AbsoluteTiming
 (*{0.118309, 1019595}*)

in .11 seconds!

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3
  • $\begingroup$ +1. Nice and useful. $\endgroup$
    – user64494
    Oct 15, 2020 at 10:26
  • $\begingroup$ However, the results of mini2[n_] := NMinimize[{Sum[x[j], {j, 1, n}], Table[x[j] + x[j + 1] >= j, {j, 1, n - 1}], x[n] + x[1] >= n}, Table[x[j], {j, 1, n}]][[2]], e.g. mini2[1234] // Rationalize // AbsoluteTiming which performs {0.130057, {x[1] -> -3.72396*10^14, x[2] -> 3.72396*10^14, ...,x[1234] -> 3.72396*10^14 }} (most of values are as the last) are doubtful. $\endgroup$
    – user64494
    Oct 15, 2020 at 10:35
  • $\begingroup$ Maybe, WorkingPrecision should be increased. $\endgroup$
    – user64494
    Oct 15, 2020 at 10:41

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