Reputation: 1243
Mathematica: help solving a system of non-linear equations with inequality constraints
I have a system of 4 non-linear equations in 4 unknowns. In addition I have 1 inequality constraint that I need a function of the four unknowns to satisfy (and then I need the four unknowns to be non-negative). My problem has 12 or so parameters and I intend to eventually solve the problem for a range of parameters to see how the solution behaves. Not all parameters included in the code below are used in this particular setup (they are used in others). I understand that my system may not have a solution for all parameter values and that I will need to do some work to find the parameter space which works but before I do that I need to know how to solve a system of non-linear equations with inequality constraint. I am new to mathematica and I am including my code below. In the code I first give some parameter values, then I define some coefficients and then I write the 4 equations and 1 inequality inside the FindInstance function (which doesn't work). I have solved these 4 equations for a particular set of parameters with the FindRoot function but I got a solution that didn't satisfy the inequality. Thanks alot. The code in mathematica is below:
values = {{r, \[Delta], \[Sigma], Subscript[i, e], Subscript[i, u],
Subscript[\[Lambda], e ], Subscript[\[Lambda], u ], Subscript[H,
0], Subscript[C, f] , F, \[Tau], Subscript[C, Ren]}, {0.047, 0.05,
0.1632, 5, 0, 0.005, 0.02, 100, 17, 80, 0.3, 8}};
G = Grid[values,
Dividers -> {{None, None, None}, {{Blue}, {Blue}, None}}];
r = values[[2, 1]];
\[Delta] = values[[2, 2]];
\[Sigma] = values[[2, 3]];
Subscript[i, e] = values[[2, 4]];
Subscript[i, u] = values[[2, 5]];
Subscript[\[Lambda], e] = values[[2, 6]];
Subscript[\[Lambda], u] = values[[2, 7]];
Subscript[H, 0] = values[[2, 8]];
Subscript[C, f] = values[[2, 9]];
F = values[[2, 10]];
\[Tau] = values[[2, 11]];
Subscript[C, Ren] = values[[2, 12]];
Subscript[I, e] =
Subscript[i, e]/
r - (Subscript[i, e] - Subscript[i, u]) Subscript[\[Lambda],
e]/(r (r + Subscript[\[Lambda], e] + Subscript[\[Lambda], u]));
Subscript[I, u] =
Subscript[i, u]/
r + (Subscript[i, e] - Subscript[i, u]) Subscript[\[Lambda],
u]/(r (r + Subscript[\[Lambda], e] + Subscript[\[Lambda], u]));
Solve[k (k - 1) + 2 (r - \[Delta])/\[Sigma]^2 k - 2 r/\[Sigma]^2 == 0,
k];
{Subscript[k, 1 ], Subscript[k, 2 ]} = k /. % ;
Clear[k];
Subscript[k, 1]
Subscript[k, 2]
L1 = (H1S^(-Subscript[k, 1])* (F - (c1*F)/r + (
H1S^(Subscript[k, 2])*((F* H1D^(Subscript[k, 1])* (-c1 + r))/r +
H1S^(Subscript[k, 1])*(-(c0* F)/r + (c1*F)/
r + (H1D^(Subscript[k,
2])*(-F*H0D^(Subscript[k, 1])* (c0 - r) +
H0S^(Subscript[k, 1])* (c0*F - H0D *r +
r* Subscript[C, f])))/((H0D^(Subscript[k, 2])*
H0S^(Subscript[k, 1]) -
H0D^(Subscript[k, 1])* H0S^(Subscript[k, 2]))*
r) + (H1D^(Subscript[k,
1])*(F*H0D^(Subscript[k, 2])*(c0 - r) -
H0S^(Subscript[k, 2])* (c0*F - H0D* r +
r* Subscript[C, f])))/((H0D^(Subscript[k, 2])*
H0S^(Subscript[k, 1]) -
H0D^(Subscript[k, 1])* H0S^(Subscript[k, 2]))*r) +
Subscript[C, Ren])))/(
H1D^(Subscript[k, 2])* H1S^(Subscript[k, 1]) -
H1D^(Subscript[k, 1])* H1S^(Subscript[k, 2]))));
M1 = (((F* H1D^(Subscript[k, 1])* (-c1 + r))/r +
H1S^(Subscript[k, 1])*(-(c0* F)/r + (c1*F)/
r + (H1D^(Subscript[k,
2])*(-F*H0D^(Subscript[k, 1])* (c0 - r) +
H0S^(Subscript[k, 1])* (c0*F - H0D *r +
r* Subscript[C, f])))/((H0D^(Subscript[k, 2])*
H0S^(Subscript[k, 1]) -
H0D^(Subscript[k, 1])* H0S^(Subscript[k, 2]))*
r) + (H1D^(Subscript[k,
1])*(F*H0D^(Subscript[k, 2])*(c0 - r) -
H0S^(Subscript[k, 2])* (c0*F - H0D* r +
r* Subscript[C, f])))/((H0D^(Subscript[k, 2])*
H0S^(Subscript[k, 1]) -
H0D^(Subscript[k, 1])* H0S^(Subscript[k, 2]))*r) +
Subscript[C, Ren]))/(
H1D^(Subscript[k, 2])* H1S^(Subscript[k, 1]) -
H1D^(Subscript[k, 1])* H1S^(Subscript[k, 2])));
L0 = ((-F*H0D^(Subscript[k, 2])*(c0 - r) +
H0S^(Subscript[k, 2])*(c0*F - H0D*r +
r*Subscript[C, f]))/((H0D^(Subscript[k, 2])*
H0S^(Subscript[k, 1]) -
H0D^(Subscript[k, 1])* H0S^(Subscript[k, 2]))* r));
M0 = ((-F*H0D^(Subscript[k, 1])*(c0 - r) +
H0S^(Subscript[k, 1])*(c0*F - H0D*r +
r*Subscript[C, f]))/((H0D^(Subscript[k, 2])*
H0S^(Subscript[k, 1]) -
H0D^(Subscript[k, 1])* H0S^(Subscript[k, 2]))* r));
c0 = ((-F*H0D^(Subscript[k, 2])*r*Subscript[k, 1] +
H0S^(Subscript[k, 2])*
r*(-H0D + (H0D + Subscript[C, f])*Subscript[k, 1]))/(
F*(H0D^(Subscript[k, 2]) - H0S^(Subscript[k, 2]))*(-1 + \[Tau])*
Subscript[k, 1]));
c1 = (H1D^(Subscript[k, 1])*H1S^(Subscript[k, 1])*
r*(-1* F*H1S^(-Subscript[k, 1]) +
H1D^(-Subscript[k, 1]) *Subscript[C, Ren] + (
F*H1D^(-Subscript[k, 1])* (1 - \[Tau])*(-80*
H0D^(Subscript[k, 2])*Subscript[k, 1] +
H0S^(Subscript[k,
2])*(- H0D + (15 + H0D)* Subscript[k, 1])))/((-56*
H0D^(Subscript[k, 2]) + 56*H0S^(Subscript[k, 2]))*Subscript[
k, 1]) + (
F*H0S^(-Subscript[k,
1])*(r + ((1 - \[Tau])* (-80* H0D^(Subscript[k, 2])* r*
Subscript[k, 1] +
H0S^(Subscript[k, 2])*
r* (-H0D + (15 + H0D)*Subscript[k, 1])))/((-56*
H0D^(Subscript[k, 2]) + 56* H0S^(Subscript[k, 2]))*
Subscript[k, 1])))/r))/(
F *(H1S^(Subscript[k, 1]) - H1D^(Subscript[k, 1]))*(1 - \[Tau]));
A1 = (F*H1S^(-Subscript[k, 1])*(r + c1*(-1 + \[Tau]))*Subscript[k,
2])/(r *(Subscript[k, 1] - Subscript[k, 2]));
B1 = (F*H1S^(-Subscript[k, 2])*(r + c1*(-1 + \[Tau]))*Subscript[k,
1])/(r *(Subscript[k, 2] - Subscript[k, 1]));
FindInstance[
L1*Subscript[H, 0]^(Subscript[k, 1]) +
M1*Subscript[H, 0]^Subscript[k, 2] + c1*F/r == F &&
L0*H1D^(Subscript[k, 1]) + M0*H1D^Subscript[k, 2] + c0*F/r -
Subscript[C, Ren] ==
H1D - Subscript[C,
f] && (F*H0S^(-Subscript[k, 1])*(r + c0*(-1 + \[Tau]))* Subscript[
k, 2]) == (H0D^(-Subscript[k, 1])*(-H0D*
r + (H0D*r + c0*F *(-1 + \[Tau]) + r* Subscript[C, f])*
Subscript[k, 2])) && (F*
H1S^(-Subscript[k, 2])*(r + c1*(-1 + \[Tau]))*Subscript[k,
1]) == (H0S^(-Subscript[k, 2])*
H1D^(-Subscript[k, 2])*((-c0 + c1)*F*
H0S^(Subscript[k, 2])*(-1 + \[Tau]) +
F*H1D^(Subscript[k, 2])* (c0 + r - c0 *\[Tau]) +
H0S^(Subscript[k, 2])* r* Subscript[C, Ren])* Subscript[k,
1]) && A1*(Subscript[H, 0])^(Subscript[k, 1]) +
B1*(Subscript[H, 0])^(Subscript[k, 2]) + (H1S) - (1 - \[Tau])*c1*
F/r > 0 && H1S > 0 && H0S > 0 && H1D > 0 && H0D > 0, {H1S, H0S,
H1D, H0D}, Reals]
Clear[c0, c1, L0, L1, M0, M1, H1D, H0D, H1S, H0S]
Upvotes: 3
Views: 4178
Answers (1)
Reputation: 3374
I don't believe you can use FindRoot with constraints (inequalities, etc.). For non-linear constrained optimization you are going to want to investigate the following built in functions
- Maximize[...]
- NMaximize[...]
- FindMaximum[...]
- Minimize[...]
- NMinimize[...]
- FindMinimum[...]
I'm not sure which one you would want for this particular problem.
Here is an example of Maximize in action:
Maximize[{(2 x + y - z)/(5 x - 7 y + 3),
0 <= x + y + z <= 1 && 1 <= x - y + z <= 2 && x - y - z == 3}, {x,
y, z}]
which gives the following output:
{5/13, {x -> 2, y -> 0, z -> -1}}
More information on constrained optimization in Mathematica (including the non-linear variety) can be found at the following links:
- http://www.wolfram.com/learningcenter/tutorialcollection/ConstrainedOptimization/ConstrainedOptimization.pdf
- http://reference.wolfram.com/mathematica/tutorial/ConstrainedOptimizationOverview.html
I hope this helps.
Upvotes: 2
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