solve: Cannot find explicit solution but it have solutions, matlab's bug?

Question

andrew wu 2020-8-14

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评论： andrew wu 2020-8-18

this is my code:

syms x0 x1 x2 x3 x4;
cond1 = (1.00028*x0+1.00045*x1)>=1.00035*(x0+x1);
cond2 = ((1.00028*x0+1.00045*x1) + 1.00063*x2)>=1.00054*(x0+x1+x2);
cond3 = (((1.00028*x0+1.00045*x1) + 1.00063*x2)+1.00082*x3)>=1.00076*(x0+x1+x2+x3);
cond4 = ((((1.00028*x0+1.00045*x1) + 1.00063*x2)+1.00082*x3)+1.00101*x4)>=1.00091*(x0+x1+x2+x3+x4);
cond5 = x0+x1+x2+x3+x4>=0;
condconds = [cond1 cond2 cond3 cond4 cond5];
sol = solve(condconds, [x0 x1 x2 x3 x4])

Does my code have any error or matlab have bugs???

My solution :

x0 1

x1 1

x2 3.44

x3 19.94

x4 38.07

by the way, i just need integer x

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andrew wu 2020-8-15

I dont understand your code, it dont include my data, I changed my question that just need integer solutions

andrew wu 2020-8-15

I run your code but it shows undefined variable DA

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Answer 1

John D'Errico 2020-8-14

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编辑：John D'Errico 2020-8-14

在 MATLAB Online 中打开

Look at what you have. A linear system of inequalities. Solve is designed to solve equalities. The solution that you have may be a solution. However, there are infinitely many solutions. One of those solutions must be the zero solution, since your problem is a homogeneous one, with no constant terms.

But, is your "solution truly a solution? I'm afraid not.

vpa(subs(condconds,[x0,x1,x2,x3,x4],[1,1,3.44,19.94,38.07]),10)
ans =
[ 2.0007 <= 2.00073, 5.4429376 <= 5.4428972, 25.3992888 <= 25.399248, 63.5077395 <= 63.5076987, 0.0 <= 63.45]

So the 2nd, 3rd and 4th constraints are all invalidated by your solution, though the errors are small. So I'll ssume you have merely rounded off what may be a valid solution.

Now let me look at the equality form of your constraint system that Stefan found. Is that actually rank deficient?

svd(double(A))
ans =
       2.2361
   0.00074952
   0.00018996
   0.00011237
   9.3138e-05

NO. Not at all.

Regardless, if we return to the inequality constrained form of your problem, can we find solutions? Probably. A common trick is to use linear programming.

linprog wants <= inequalities, so I'll swap the signs on A.

DA = double(A);
linprog(ones(5,1),-DA,zeros(5,1))
Optimal solution found.
ans =
     0
     0
     0
     0
     0
     
linprog(-ones(5,1),-DA,zeros(5,1))
Problem is unbounded.
ans =
     []

So we do find a solution as the zero vector, as we must, since A is actually non-singular. But by looking in another direction, linprog tells us the problem is unbounded. So there are infinitely many solutions in some direction.

Is there a solution where all of the knowns are no smaller than 1?

format long g
>> X = linprog(rand(5,1),-DA,zeros(5,1),[],[],ones(5,1))
Optimal solution found.
X =
                         1
                         1
          3.88888888889285
          21.5925925926071
          41.2222222222452

Interesting, in that linprog finds one near what you called the solution. As I said though, there will be infinitely many solutions, but perhaps this will be good enough for you.

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andrew wu 2020-8-18

Although you don't completely solve my problem, you give me the function "linprog()" that bring me a lot about thoughts, then I use "intlinprog()" to resolve the problem. your answer is best among the answers. Thank you for your help.

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Answer 2

Stephan 2020-8-14

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在 MATLAB Online 中打开

Congratulations, you have 1 solution of an infinite number of possible solutions - look to the borderline case of equal to:

syms x0 x1 x2 x3 x4;
assume([x0 x1 x2 x3 x4], {'real', 'positive'})
cond1 = (1.00028*x0+1.00045*x1)==1.00035*(x0+x1);
cond2 = ((1.00028*x0+1.00045*x1) + 1.00063*x2)==1.00054*(x0+x1+x2);
cond3 = (((1.00028*x0+1.00045*x1) + 1.00063*x2)+1.00082*x3)==1.00076*(x0+x1+x2+x3);
cond4 = ((((1.00028*x0+1.00045*x1) + 1.00063*x2)+1.00082*x3)+1.00101*x4)==1.00091*(x0+x1+x2+x3+x4);
cond5 = x0+x1+x2+x3+x4==0;
condconds = [cond1 cond2 cond3 cond4 cond5];
[A,~] = equationsToMatrix(condconds);
det_A = vpa(det(A))

leads to

det_A =
 
A =
 
[                              -7/100000,                               1/10000,                                      0,                                      0,                             0]
[                              -13/50000,                             -9/100000,  1266637395196663/14073748835532800000,                                      0,                             0]
[                                -3/6250,                            -31/100000, -1829587348620553/14073748835532800000,                                3/50000,                             0]
[ -8866461766385191/14073748835532800000, -1294784892868923/2814749767106560000,           -78812993479/281474976710656, -1266637395197479/14073748835532800000, 450359962737/4503599627370496]
[                                      1,                                     1,                                      1,                                      1,                             1]
 
 
b =
 
 0
 0
 0
 0
 0
 
 
det_A =
 
0.0000000000000033319999999983153682592275926204

which means, that the determinant of your system is zero in fact. Since solve is looking for a explicit solution it will not find any i guess.

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Stephan 2020-8-15

编辑：Stephan 2020-8-15

My first idea was to delete my answer, but I (and may be other people) can learn from your comment. Instead of deleting my incorrect answer, i vote for yours.

;-)

andrew wu 2020-8-18