Numerical Minimization of Large ExpressionCombined numerical minimization and maximizationusing array elements in minimization$L_infty$ norm minimizationNumerical Instability?Simple minimization not evaluatingMinimization and integrationIssue with numerical minimizationMinimization over the integersProducing a smooth curve from numerical minimizationoptimization problem with NDSolve and Import cannot get any results, even without error message

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Numerical Minimization of Large Expression


Combined numerical minimization and maximizationusing array elements in minimization$L_infty$ norm minimizationNumerical Instability?Simple minimization not evaluatingMinimization and integrationIssue with numerical minimizationMinimization over the integersProducing a smooth curve from numerical minimizationoptimization problem with NDSolve and Import cannot get any results, even without error message













1












$begingroup$


I want to minimize a function which contains ArcTan functions. The function is also very complicated.



I have the following code.



LenFact = 1.3745/0.31571;
xC = 0.157392 LenFact; yC = 0.035495 LenFact;

q=x1[t],y1[t],x2[t],y2[t],x3[t],y3[t],x4[t],y4[t],x5[t],y5[t],x6[t],y6[t];

CE = 
 -0.609155 + x1[t]^2 + y1[t]^2,
 -0.0383062 + (-x1[t] + x2[t])^2 + (-y1[t] + y2[t])^2,
 -1.64323 + (-0.685234 + x2[t])^2 + (-0.154534 + y2[t])^2,
 -1.88925 + (-x1[t] + x6[t])^2 + (-y1[t] + y6[t])^2,
 -0.428366 + ArcTan[-x1[t] + x2[t], -y1[t] + y2[t]] - 
 ArcTan[-x1[t] + x6[t], -y1[t] + y6[t]],
 -0.783885 + (-x3[t] + x4[t])^2 + (-y3[t] + y4[t])^2,
 -0.0353049 + (-x4[t] + x5[t])^2 + (-y4[t] + y5[t])^2,
 -0.25649 + ArcTan[-x3[t] + x4[t], -y3[t] + y4[t]] - 
 ArcTan[-x4[t] + x5[t], -y4[t] + y5[t]],
 -0.0228637 + (-x5[t] + x6[t])^2 + (-y5[t] + y6[t])^2,
 -0.0138516 + (-x2[t] + x3[t])^2 + (-y2[t] + y3[t])^2,
 -ArcTan[0.685234 - x2[t], 0.154534 - y2[t]] + 
 ArcTan[-x2[t] + x3[t], -y2[t] + y3[t]]
;

y6Start = 0.15 LenFact; 

VPenalty=3.90351 + 1.13817*10^6 (2.74102 - 1. ArcTan[x1[t], y1[t]])^2 + 
 3006.52 (2.92648 - 
 1. ArcTan[-0.685234 + x3[t], -0.154534 + y3[t]])^2 + 
 1082.47 (2.78921 - 
 1. ArcTan[-0.685234 + x3[t], -0.154534 + y3[t]] + 
 ArcTan[-1. x3[t] + x4[t], -1. y3[t] + y4[t]])^2 + 
 6825.76 (2.48452 - 1. ArcTan[x1[t], y1[t]] + 
 ArcTan[-1. x1[t] + x6[t], -1. y1[t] + y6[t]])^2 + 
 239.418 y1[t] + 29.1166 y2[t] + 6.00899 y3[t] + 7.28423 y4[t] + 
 4.93484 y5[t] + 97.0584 y6[t] + 
 1.90484*10^8 ((-ArcTan[0.685234 - x2[t], 0.154534 - y2[t]] + 
 ArcTan[-x2[t] + x3[t], -y2[t] + y3[t]])^2 + (-0.25649 + 
 ArcTan[-x3[t] + x4[t], -y3[t] + y4[t]] - 
 ArcTan[-x4[t] + x5[t], -y4[t] + y5[t]])^2 + (-0.428366 + 
 ArcTan[-x1[t] + x2[t], -y1[t] + y2[t]] - 
 ArcTan[-x1[t] + x6[t], -y1[t] + y6[t]])^2 + (-0.609155 + 
 x1[t]^2 + 
 y1[t]^2)^2 + (-1.64323 + (-0.685234 + x2[t])^2 + (-0.154534 + 
 y2[t])^2)^2 + (-0.0383062 + (-x1[t] + x2[t])^2 + (-y1[t] + 
 y2[t])^2)^2 + (-0.0138516 + (-x2[t] + x3[t])^2 + (-y2[t] + 
 y3[t])^2)^2 + (-0.783885 + (-x3[t] + x4[t])^2 + (-y3[t] + 
 y4[t])^2)^2 + (-0.0353049 + (-x4[t] + x5[t])^2 + (-y4[t] + 
 y5[t])^2)^2 + (-1.88925 + (-x1[t] + x6[t])^2 + (-y1[t] + 
 y6[t])^2)^2 + (-0.0228637 + (-x5[t] + x6[t])^2 + (-y5[t] + 
 y6[t])^2)^2);

[Theta]1 = ArcTan[x1[t], y1[t]];
[Theta]21 = ArcTan[x2[t] - x1[t], y2[t] - y1[t]];
[Theta]3 = ArcTan[x3[t] - xC, y3[t] - yC];
[Theta]22 = ArcTan[x6[t] - x1[t], y6[t] - y1[t]];
[Theta]41 = ArcTan[x4[t] - x3[t], y4[t] - y3[t]];
[Theta]42 = ArcTan[x5[t] - x4[t], y5[t] - y4[t]];
[Theta]5 = ArcTan[x6[t] - x5[t], y6[t] - y5[t]];

eqbr = NMinimize[VPenalty[[1, 1]], Flatten[CE] == 0, 
 90 [Degree] < [Theta]1 < 180 [Degree], 
 20 [Degree] < [Theta]21 < 95 [Degree], 
 145 [Degree] < [Theta]3 < 180 [Degree], 
 80 [Degree] < [Theta]5 < 100 [Degree], 0 < y6[t] < y6Start, 
 q, Reals, MaxIterations -> 10^4]


It is running for several hours with no result. Can any body suggest any specific way of minimizing such large scale expression? I also could not give initial points. It is always showing syntax error. Is it not possible to give one initial point only for each variable?






mathematical-optimization







share|improve this question











$endgroup$











  • $begingroup$
    I am sorry. I have incorporated CE in the code now.
    $endgroup$
    – Soumyajit Roy
    Mar 16 at 15:43










  • $begingroup$
    And what about y6Start?
    $endgroup$
    – Henrik Schumacher
    Mar 16 at 15:45






  • 2




    $begingroup$
    And after all: What is the context? This looks a bit like discrete bending energy minimization with some angle constraints... Maybe the problem can be rephrased into another one that is simpler to solve.
    $endgroup$
    – Henrik Schumacher
    Mar 16 at 15:47











  • $begingroup$
    y6Start = 0.15 LenFact; This is about finding the static equilibrium of a mechanism. Static equilibrium states can be found by minimizing the potential energy function.
    $endgroup$
    – Soumyajit Roy
    Mar 16 at 15:50















1












$begingroup$


I want to minimize a function which contains ArcTan functions. The function is also very complicated.



I have the following code.



LenFact = 1.3745/0.31571;
xC = 0.157392 LenFact; yC = 0.035495 LenFact;

q=x1[t],y1[t],x2[t],y2[t],x3[t],y3[t],x4[t],y4[t],x5[t],y5[t],x6[t],y6[t];

CE = 
 -0.609155 + x1[t]^2 + y1[t]^2,
 -0.0383062 + (-x1[t] + x2[t])^2 + (-y1[t] + y2[t])^2,
 -1.64323 + (-0.685234 + x2[t])^2 + (-0.154534 + y2[t])^2,
 -1.88925 + (-x1[t] + x6[t])^2 + (-y1[t] + y6[t])^2,
 -0.428366 + ArcTan[-x1[t] + x2[t], -y1[t] + y2[t]] - 
 ArcTan[-x1[t] + x6[t], -y1[t] + y6[t]],
 -0.783885 + (-x3[t] + x4[t])^2 + (-y3[t] + y4[t])^2,
 -0.0353049 + (-x4[t] + x5[t])^2 + (-y4[t] + y5[t])^2,
 -0.25649 + ArcTan[-x3[t] + x4[t], -y3[t] + y4[t]] - 
 ArcTan[-x4[t] + x5[t], -y4[t] + y5[t]],
 -0.0228637 + (-x5[t] + x6[t])^2 + (-y5[t] + y6[t])^2,
 -0.0138516 + (-x2[t] + x3[t])^2 + (-y2[t] + y3[t])^2,
 -ArcTan[0.685234 - x2[t], 0.154534 - y2[t]] + 
 ArcTan[-x2[t] + x3[t], -y2[t] + y3[t]]
;

y6Start = 0.15 LenFact; 

VPenalty=3.90351 + 1.13817*10^6 (2.74102 - 1. ArcTan[x1[t], y1[t]])^2 + 
 3006.52 (2.92648 - 
 1. ArcTan[-0.685234 + x3[t], -0.154534 + y3[t]])^2 + 
 1082.47 (2.78921 - 
 1. ArcTan[-0.685234 + x3[t], -0.154534 + y3[t]] + 
 ArcTan[-1. x3[t] + x4[t], -1. y3[t] + y4[t]])^2 + 
 6825.76 (2.48452 - 1. ArcTan[x1[t], y1[t]] + 
 ArcTan[-1. x1[t] + x6[t], -1. y1[t] + y6[t]])^2 + 
 239.418 y1[t] + 29.1166 y2[t] + 6.00899 y3[t] + 7.28423 y4[t] + 
 4.93484 y5[t] + 97.0584 y6[t] + 
 1.90484*10^8 ((-ArcTan[0.685234 - x2[t], 0.154534 - y2[t]] + 
 ArcTan[-x2[t] + x3[t], -y2[t] + y3[t]])^2 + (-0.25649 + 
 ArcTan[-x3[t] + x4[t], -y3[t] + y4[t]] - 
 ArcTan[-x4[t] + x5[t], -y4[t] + y5[t]])^2 + (-0.428366 + 
 ArcTan[-x1[t] + x2[t], -y1[t] + y2[t]] - 
 ArcTan[-x1[t] + x6[t], -y1[t] + y6[t]])^2 + (-0.609155 + 
 x1[t]^2 + 
 y1[t]^2)^2 + (-1.64323 + (-0.685234 + x2[t])^2 + (-0.154534 + 
 y2[t])^2)^2 + (-0.0383062 + (-x1[t] + x2[t])^2 + (-y1[t] + 
 y2[t])^2)^2 + (-0.0138516 + (-x2[t] + x3[t])^2 + (-y2[t] + 
 y3[t])^2)^2 + (-0.783885 + (-x3[t] + x4[t])^2 + (-y3[t] + 
 y4[t])^2)^2 + (-0.0353049 + (-x4[t] + x5[t])^2 + (-y4[t] + 
 y5[t])^2)^2 + (-1.88925 + (-x1[t] + x6[t])^2 + (-y1[t] + 
 y6[t])^2)^2 + (-0.0228637 + (-x5[t] + x6[t])^2 + (-y5[t] + 
 y6[t])^2)^2);

[Theta]1 = ArcTan[x1[t], y1[t]];
[Theta]21 = ArcTan[x2[t] - x1[t], y2[t] - y1[t]];
[Theta]3 = ArcTan[x3[t] - xC, y3[t] - yC];
[Theta]22 = ArcTan[x6[t] - x1[t], y6[t] - y1[t]];
[Theta]41 = ArcTan[x4[t] - x3[t], y4[t] - y3[t]];
[Theta]42 = ArcTan[x5[t] - x4[t], y5[t] - y4[t]];
[Theta]5 = ArcTan[x6[t] - x5[t], y6[t] - y5[t]];

eqbr = NMinimize[VPenalty[[1, 1]], Flatten[CE] == 0, 
 90 [Degree] < [Theta]1 < 180 [Degree], 
 20 [Degree] < [Theta]21 < 95 [Degree], 
 145 [Degree] < [Theta]3 < 180 [Degree], 
 80 [Degree] < [Theta]5 < 100 [Degree], 0 < y6[t] < y6Start, 
 q, Reals, MaxIterations -> 10^4]


It is running for several hours with no result. Can any body suggest any specific way of minimizing such large scale expression? I also could not give initial points. It is always showing syntax error. Is it not possible to give one initial point only for each variable?






mathematical-optimization







share|improve this question











$endgroup$











  • $begingroup$
    I am sorry. I have incorporated CE in the code now.
    $endgroup$
    – Soumyajit Roy
    Mar 16 at 15:43










  • $begingroup$
    And what about y6Start?
    $endgroup$
    – Henrik Schumacher
    Mar 16 at 15:45






  • 2




    $begingroup$
    And after all: What is the context? This looks a bit like discrete bending energy minimization with some angle constraints... Maybe the problem can be rephrased into another one that is simpler to solve.
    $endgroup$
    – Henrik Schumacher
    Mar 16 at 15:47











  • $begingroup$
    y6Start = 0.15 LenFact; This is about finding the static equilibrium of a mechanism. Static equilibrium states can be found by minimizing the potential energy function.
    $endgroup$
    – Soumyajit Roy
    Mar 16 at 15:50













1












1








1





$begingroup$


I want to minimize a function which contains ArcTan functions. The function is also very complicated.



I have the following code.



LenFact = 1.3745/0.31571;
xC = 0.157392 LenFact; yC = 0.035495 LenFact;

q=x1[t],y1[t],x2[t],y2[t],x3[t],y3[t],x4[t],y4[t],x5[t],y5[t],x6[t],y6[t];

CE = 
 -0.609155 + x1[t]^2 + y1[t]^2,
 -0.0383062 + (-x1[t] + x2[t])^2 + (-y1[t] + y2[t])^2,
 -1.64323 + (-0.685234 + x2[t])^2 + (-0.154534 + y2[t])^2,
 -1.88925 + (-x1[t] + x6[t])^2 + (-y1[t] + y6[t])^2,
 -0.428366 + ArcTan[-x1[t] + x2[t], -y1[t] + y2[t]] - 
 ArcTan[-x1[t] + x6[t], -y1[t] + y6[t]],
 -0.783885 + (-x3[t] + x4[t])^2 + (-y3[t] + y4[t])^2,
 -0.0353049 + (-x4[t] + x5[t])^2 + (-y4[t] + y5[t])^2,
 -0.25649 + ArcTan[-x3[t] + x4[t], -y3[t] + y4[t]] - 
 ArcTan[-x4[t] + x5[t], -y4[t] + y5[t]],
 -0.0228637 + (-x5[t] + x6[t])^2 + (-y5[t] + y6[t])^2,
 -0.0138516 + (-x2[t] + x3[t])^2 + (-y2[t] + y3[t])^2,
 -ArcTan[0.685234 - x2[t], 0.154534 - y2[t]] + 
 ArcTan[-x2[t] + x3[t], -y2[t] + y3[t]]
;

y6Start = 0.15 LenFact; 

VPenalty=3.90351 + 1.13817*10^6 (2.74102 - 1. ArcTan[x1[t], y1[t]])^2 + 
 3006.52 (2.92648 - 
 1. ArcTan[-0.685234 + x3[t], -0.154534 + y3[t]])^2 + 
 1082.47 (2.78921 - 
 1. ArcTan[-0.685234 + x3[t], -0.154534 + y3[t]] + 
 ArcTan[-1. x3[t] + x4[t], -1. y3[t] + y4[t]])^2 + 
 6825.76 (2.48452 - 1. ArcTan[x1[t], y1[t]] + 
 ArcTan[-1. x1[t] + x6[t], -1. y1[t] + y6[t]])^2 + 
 239.418 y1[t] + 29.1166 y2[t] + 6.00899 y3[t] + 7.28423 y4[t] + 
 4.93484 y5[t] + 97.0584 y6[t] + 
 1.90484*10^8 ((-ArcTan[0.685234 - x2[t], 0.154534 - y2[t]] + 
 ArcTan[-x2[t] + x3[t], -y2[t] + y3[t]])^2 + (-0.25649 + 
 ArcTan[-x3[t] + x4[t], -y3[t] + y4[t]] - 
 ArcTan[-x4[t] + x5[t], -y4[t] + y5[t]])^2 + (-0.428366 + 
 ArcTan[-x1[t] + x2[t], -y1[t] + y2[t]] - 
 ArcTan[-x1[t] + x6[t], -y1[t] + y6[t]])^2 + (-0.609155 + 
 x1[t]^2 + 
 y1[t]^2)^2 + (-1.64323 + (-0.685234 + x2[t])^2 + (-0.154534 + 
 y2[t])^2)^2 + (-0.0383062 + (-x1[t] + x2[t])^2 + (-y1[t] + 
 y2[t])^2)^2 + (-0.0138516 + (-x2[t] + x3[t])^2 + (-y2[t] + 
 y3[t])^2)^2 + (-0.783885 + (-x3[t] + x4[t])^2 + (-y3[t] + 
 y4[t])^2)^2 + (-0.0353049 + (-x4[t] + x5[t])^2 + (-y4[t] + 
 y5[t])^2)^2 + (-1.88925 + (-x1[t] + x6[t])^2 + (-y1[t] + 
 y6[t])^2)^2 + (-0.0228637 + (-x5[t] + x6[t])^2 + (-y5[t] + 
 y6[t])^2)^2);

[Theta]1 = ArcTan[x1[t], y1[t]];
[Theta]21 = ArcTan[x2[t] - x1[t], y2[t] - y1[t]];
[Theta]3 = ArcTan[x3[t] - xC, y3[t] - yC];
[Theta]22 = ArcTan[x6[t] - x1[t], y6[t] - y1[t]];
[Theta]41 = ArcTan[x4[t] - x3[t], y4[t] - y3[t]];
[Theta]42 = ArcTan[x5[t] - x4[t], y5[t] - y4[t]];
[Theta]5 = ArcTan[x6[t] - x5[t], y6[t] - y5[t]];

eqbr = NMinimize[VPenalty[[1, 1]], Flatten[CE] == 0, 
 90 [Degree] < [Theta]1 < 180 [Degree], 
 20 [Degree] < [Theta]21 < 95 [Degree], 
 145 [Degree] < [Theta]3 < 180 [Degree], 
 80 [Degree] < [Theta]5 < 100 [Degree], 0 < y6[t] < y6Start, 
 q, Reals, MaxIterations -> 10^4]


It is running for several hours with no result. Can any body suggest any specific way of minimizing such large scale expression? I also could not give initial points. It is always showing syntax error. Is it not possible to give one initial point only for each variable?






mathematical-optimization







share|improve this question











$endgroup$




I want to minimize a function which contains ArcTan functions. The function is also very complicated.



I have the following code.



LenFact = 1.3745/0.31571;
xC = 0.157392 LenFact; yC = 0.035495 LenFact;

q=x1[t],y1[t],x2[t],y2[t],x3[t],y3[t],x4[t],y4[t],x5[t],y5[t],x6[t],y6[t];

CE = 
 -0.609155 + x1[t]^2 + y1[t]^2,
 -0.0383062 + (-x1[t] + x2[t])^2 + (-y1[t] + y2[t])^2,
 -1.64323 + (-0.685234 + x2[t])^2 + (-0.154534 + y2[t])^2,
 -1.88925 + (-x1[t] + x6[t])^2 + (-y1[t] + y6[t])^2,
 -0.428366 + ArcTan[-x1[t] + x2[t], -y1[t] + y2[t]] - 
 ArcTan[-x1[t] + x6[t], -y1[t] + y6[t]],
 -0.783885 + (-x3[t] + x4[t])^2 + (-y3[t] + y4[t])^2,
 -0.0353049 + (-x4[t] + x5[t])^2 + (-y4[t] + y5[t])^2,
 -0.25649 + ArcTan[-x3[t] + x4[t], -y3[t] + y4[t]] - 
 ArcTan[-x4[t] + x5[t], -y4[t] + y5[t]],
 -0.0228637 + (-x5[t] + x6[t])^2 + (-y5[t] + y6[t])^2,
 -0.0138516 + (-x2[t] + x3[t])^2 + (-y2[t] + y3[t])^2,
 -ArcTan[0.685234 - x2[t], 0.154534 - y2[t]] + 
 ArcTan[-x2[t] + x3[t], -y2[t] + y3[t]]
;

y6Start = 0.15 LenFact; 

VPenalty=3.90351 + 1.13817*10^6 (2.74102 - 1. ArcTan[x1[t], y1[t]])^2 + 
 3006.52 (2.92648 - 
 1. ArcTan[-0.685234 + x3[t], -0.154534 + y3[t]])^2 + 
 1082.47 (2.78921 - 
 1. ArcTan[-0.685234 + x3[t], -0.154534 + y3[t]] + 
 ArcTan[-1. x3[t] + x4[t], -1. y3[t] + y4[t]])^2 + 
 6825.76 (2.48452 - 1. ArcTan[x1[t], y1[t]] + 
 ArcTan[-1. x1[t] + x6[t], -1. y1[t] + y6[t]])^2 + 
 239.418 y1[t] + 29.1166 y2[t] + 6.00899 y3[t] + 7.28423 y4[t] + 
 4.93484 y5[t] + 97.0584 y6[t] + 
 1.90484*10^8 ((-ArcTan[0.685234 - x2[t], 0.154534 - y2[t]] + 
 ArcTan[-x2[t] + x3[t], -y2[t] + y3[t]])^2 + (-0.25649 + 
 ArcTan[-x3[t] + x4[t], -y3[t] + y4[t]] - 
 ArcTan[-x4[t] + x5[t], -y4[t] + y5[t]])^2 + (-0.428366 + 
 ArcTan[-x1[t] + x2[t], -y1[t] + y2[t]] - 
 ArcTan[-x1[t] + x6[t], -y1[t] + y6[t]])^2 + (-0.609155 + 
 x1[t]^2 + 
 y1[t]^2)^2 + (-1.64323 + (-0.685234 + x2[t])^2 + (-0.154534 + 
 y2[t])^2)^2 + (-0.0383062 + (-x1[t] + x2[t])^2 + (-y1[t] + 
 y2[t])^2)^2 + (-0.0138516 + (-x2[t] + x3[t])^2 + (-y2[t] + 
 y3[t])^2)^2 + (-0.783885 + (-x3[t] + x4[t])^2 + (-y3[t] + 
 y4[t])^2)^2 + (-0.0353049 + (-x4[t] + x5[t])^2 + (-y4[t] + 
 y5[t])^2)^2 + (-1.88925 + (-x1[t] + x6[t])^2 + (-y1[t] + 
 y6[t])^2)^2 + (-0.0228637 + (-x5[t] + x6[t])^2 + (-y5[t] + 
 y6[t])^2)^2);

[Theta]1 = ArcTan[x1[t], y1[t]];
[Theta]21 = ArcTan[x2[t] - x1[t], y2[t] - y1[t]];
[Theta]3 = ArcTan[x3[t] - xC, y3[t] - yC];
[Theta]22 = ArcTan[x6[t] - x1[t], y6[t] - y1[t]];
[Theta]41 = ArcTan[x4[t] - x3[t], y4[t] - y3[t]];
[Theta]42 = ArcTan[x5[t] - x4[t], y5[t] - y4[t]];
[Theta]5 = ArcTan[x6[t] - x5[t], y6[t] - y5[t]];

eqbr = NMinimize[VPenalty[[1, 1]], Flatten[CE] == 0, 
 90 [Degree] < [Theta]1 < 180 [Degree], 
 20 [Degree] < [Theta]21 < 95 [Degree], 
 145 [Degree] < [Theta]3 < 180 [Degree], 
 80 [Degree] < [Theta]5 < 100 [Degree], 0 < y6[t] < y6Start, 
 q, Reals, MaxIterations -> 10^4]


It is running for several hours with no result. Can any body suggest any specific way of minimizing such large scale expression? I also could not give initial points. It is always showing syntax error. Is it not possible to give one initial point only for each variable?






mathematical-optimization




mathematical-optimization






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edited Mar 16 at 15:47







Soumyajit Roy

















asked Mar 16 at 15:39









Soumyajit RoySoumyajit Roy

1809




1809











  • $begingroup$
    I am sorry. I have incorporated CE in the code now.
    $endgroup$
    – Soumyajit Roy
    Mar 16 at 15:43










  • $begingroup$
    And what about y6Start?
    $endgroup$
    – Henrik Schumacher
    Mar 16 at 15:45






  • 2




    $begingroup$
    And after all: What is the context? This looks a bit like discrete bending energy minimization with some angle constraints... Maybe the problem can be rephrased into another one that is simpler to solve.
    $endgroup$
    – Henrik Schumacher
    Mar 16 at 15:47











  • $begingroup$
    y6Start = 0.15 LenFact; This is about finding the static equilibrium of a mechanism. Static equilibrium states can be found by minimizing the potential energy function.
    $endgroup$
    – Soumyajit Roy
    Mar 16 at 15:50
















  • $begingroup$
    I am sorry. I have incorporated CE in the code now.
    $endgroup$
    – Soumyajit Roy
    Mar 16 at 15:43










  • $begingroup$
    And what about y6Start?
    $endgroup$
    – Henrik Schumacher
    Mar 16 at 15:45






  • 2




    $begingroup$
    And after all: What is the context? This looks a bit like discrete bending energy minimization with some angle constraints... Maybe the problem can be rephrased into another one that is simpler to solve.
    $endgroup$
    – Henrik Schumacher
    Mar 16 at 15:47











  • $begingroup$
    y6Start = 0.15 LenFact; This is about finding the static equilibrium of a mechanism. Static equilibrium states can be found by minimizing the potential energy function.
    $endgroup$
    – Soumyajit Roy
    Mar 16 at 15:50















$begingroup$
I am sorry. I have incorporated CE in the code now.
$endgroup$
– Soumyajit Roy
Mar 16 at 15:43




$begingroup$
I am sorry. I have incorporated CE in the code now.
$endgroup$
– Soumyajit Roy
Mar 16 at 15:43












$begingroup$
And what about y6Start?
$endgroup$
– Henrik Schumacher
Mar 16 at 15:45




$begingroup$
And what about y6Start?
$endgroup$
– Henrik Schumacher
Mar 16 at 15:45




2




2




$begingroup$
And after all: What is the context? This looks a bit like discrete bending energy minimization with some angle constraints... Maybe the problem can be rephrased into another one that is simpler to solve.
$endgroup$
– Henrik Schumacher
Mar 16 at 15:47





$begingroup$
And after all: What is the context? This looks a bit like discrete bending energy minimization with some angle constraints... Maybe the problem can be rephrased into another one that is simpler to solve.
$endgroup$
– Henrik Schumacher
Mar 16 at 15:47













$begingroup$
y6Start = 0.15 LenFact; This is about finding the static equilibrium of a mechanism. Static equilibrium states can be found by minimizing the potential energy function.
$endgroup$
– Soumyajit Roy
Mar 16 at 15:50




$begingroup$
y6Start = 0.15 LenFact; This is about finding the static equilibrium of a mechanism. Static equilibrium states can be found by minimizing the potential energy function.
$endgroup$
– Soumyajit Roy
Mar 16 at 15:50










2 Answers
2






active

oldest

votes


















2












$begingroup$

I corrected typos. The code is fast and gives an answer, albeit with messages.



LenFact = 1.3745/0.31571;
xC = 0.157392 LenFact; yC = 0.035495 LenFact;

q = x1[t], y1[t], x2[t], y2[t], x3[t], y3[t], x4[t], y4[t], x5[t], 
 y5[t], x6[t], 
 y6[t]; CE = -0.609155 + x1[t]^2 + 
 y1[t]^2, -0.0383062 + (-x1[t] + x2[t])^2 + (-y1[t] + 
 y2[t])^2, -1.64323 + (-0.685234 + x2[t])^2 + (-0.154534 + 
 y2[t])^2, -1.88925 + (-x1[t] + x6[t])^2 + (-y1[t] + 
 y6[t])^2, -0.428366 + 
 ArcTan[-x1[t] + x2[t], -y1[t] + y2[t]] - 
 ArcTan[-x1[t] + x6[t], -y1[t] + 
 y6[t]], -0.783885 + (-x3[t] + x4[t])^2 + (-y3[t] + 
 y4[t])^2, -0.0353049 + (-x4[t] + x5[t])^2 + (-y4[t] + 
 y5[t])^2, -0.25649 + ArcTan[-x3[t] + x4[t], -y3[t] + y4[t]] -
 ArcTan[-x4[t] + x5[t], -y4[t] + 
 y5[t]], -0.0228637 + (-x5[t] + x6[t])^2 + (-y5[t] + 
 y6[t])^2, -0.0138516 + (-x2[t] + x3[t])^2 + (-y2[t] + 
 y3[t])^2, -ArcTan[0.685234 - x2[t], 0.154534 - y2[t]] + 
 ArcTan[-x2[t] + x3[t], -y2[t] + y3[t]];

VPenalty = 3.90351 + 
 1.13817*10^6 (2.74102 - 1. ArcTan[x1[t], y1[t]])^2 + 
 3006.52 (2.92648 - 
 1. ArcTan[-0.685234 + x3[t], -0.154534 + y3[t]])^2 + 
 1082.47 (2.78921 - 
 1. ArcTan[-0.685234 + x3[t], -0.154534 + y3[t]] + 
 ArcTan[-1. x3[t] + x4[t], -1. y3[t] + y4[t]])^2 + 
 6825.76 (2.48452 - 1. ArcTan[x1[t], y1[t]] + 
 ArcTan[-1. x1[t] + x6[t], -1. y1[t] + y6[t]])^2 + 
 239.418 y1[t] + 29.1166 y2[t] + 6.00899 y3[t] + 7.28423 y4[t] + 
 4.93484 y5[t] + 97.0584 y6[t] + 
 1.90484*10^8 ((-ArcTan[0.685234 - x2[t], 0.154534 - y2[t]] + 
 ArcTan[-x2[t] + x3[t], -y2[t] + y3[t]])^2 + (-0.25649 + 
 ArcTan[-x3[t] + x4[t], -y3[t] + y4[t]] - 
 ArcTan[-x4[t] + x5[t], -y4[t] + y5[t]])^2 + (-0.428366 + 
 ArcTan[-x1[t] + x2[t], -y1[t] + y2[t]] - 
 ArcTan[-x1[t] + x6[t], -y1[t] + y6[t]])^2 + (-0.609155 + 
 x1[t]^2 + 
 y1[t]^2)^2 + (-1.64323 + (-0.685234 + 
 x2[t])^2 + (-0.154534 + 
 y2[t])^2)^2 + (-0.0383062 + (-x1[t] + 
 x2[t])^2 + (-y1[t] + 
 y2[t])^2)^2 + (-0.0138516 + (-x2[t] + 
 x3[t])^2 + (-y2[t] + 
 y3[t])^2)^2 + (-0.783885 + (-x3[t] + 
 x4[t])^2 + (-y3[t] + 
 y4[t])^2)^2 + (-0.0353049 + (-x4[t] + 
 x5[t])^2 + (-y4[t] + y5[t])^2)^2 + (-1.88925 + (-x1[t] +
 x6[t])^2 + (-y1[t] + 
 y6[t])^2)^2 + (-0.0228637 + (-x5[t] + 
 x6[t])^2 + (-y5[t] + y6[t])^2)^2);

[Theta]1 = ArcTan[x1[t], y1[t]];
[Theta]21 = ArcTan[x2[t] - x1[t], y2[t] - y1[t]];
[Theta]3 = ArcTan[x3[t] - xC, y3[t] - yC];
[Theta]22 = ArcTan[x6[t] - x1[t], y6[t] - y1[t]];
[Theta]41 = ArcTan[x4[t] - x3[t], y4[t] - y3[t]];
[Theta]42 = ArcTan[x5[t] - x4[t], y5[t] - y4[t]];
[Theta]5 = ArcTan[x6[t] - x5[t], y6[t] - y5[t]];

With[y6Start = 0.15 LenFact, 
 eqbr = NMinimize[VPenalty[[1, 1]], 
 Flatten[CE] == Table[0, Length[CE]], Pi/2 < [Theta]1 < Pi, 
 20*Pi/180 < [Theta]21 < 95*Pi/180, 145*Pi/180 < [Theta]3 < Pi, 
 80*Pi/180 < [Theta]5 < 100*Pi/180, 0 < y6[t] < y6Start, q, 
 Reals]]
(*1652.07, x1[t] -> -0.729157, y1[t] -> 0.278358, x2[t] -> -0.573507, 
 y2[t] -> 0.397027, x3[t] -> -0.457944, y3[t] -> 0.374765, 
 x4[t] -> 0.423225, y4[t] -> 0.460944, x5[t] -> 0.60875, 
 y5[t] -> 0.431194, x6[t] -> 0.611298, y6[t] -> 0.582379*)





share|improve this answer









$endgroup$












  • $begingroup$
    Thanks Alex. At least the constraints are matching to a reasonable degree. Although I am still not sure about the obtained value of Vtotal to be the global one.
    $endgroup$
    – Soumyajit Roy
    Mar 16 at 16:30










  • $begingroup$
    Here VPenalty[] is a non-linear function, so the min can be local.
    $endgroup$
    – Alex Trounev
    Mar 16 at 17:10


















2












$begingroup$

The method "RandomSearch" usually gives a result reasonably quickly (other than using FindMinimum). It's less robust, though, than the other methods. The following takes about 30 sec.



eqbr = NMinimize[VPenalty[[1, 1]], Flatten[CE] == 0, 
 90 ° < θ1 < 180 °, 20 ° < θ21 < 95 °, 
 145 ° < θ3 < 180 °, 80 ° < θ5 < 100 °, 0 < y6[t] < y6Start, q, 
 Method -> "RandomSearch"]



NMinimize::incst: NMinimize was unable to generate any initial points satisfying the inequality constraints...




(*
161.292, x1[t] -> -0.718754, y1[t] -> 0.30422, x2[t] -> -0.567146, 
 y2[t] -> 0.427983, x3[t] -> -0.452162, y3[t] -> 0.402877, 
 x4[t] -> 0.424915, y4[t] -> 0.523793, x5[t] -> 0.611471, 
 y5[t] -> 0.501395, x6[t] -> 0.610863, y6[t] -> 0.652602
*)


While NMinimize did not find good starting points, it found a solution within the constraints:



90 ° < θ1 < 180 °, 20 ° < θ21 < 95 °, 145 ° < θ3 < 180 °, 
 80 ° < θ5 < 100 °, 0 < y6[t] < y6Start /. 
 Last@eqbr
(* True, True, True, True, True *)





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    2 Answers
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    2 Answers
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    2












    $begingroup$

    I corrected typos. The code is fast and gives an answer, albeit with messages.



    LenFact = 1.3745/0.31571;
    xC = 0.157392 LenFact; yC = 0.035495 LenFact;

    q = x1[t], y1[t], x2[t], y2[t], x3[t], y3[t], x4[t], y4[t], x5[t], 
     y5[t], x6[t], 
     y6[t]; CE = -0.609155 + x1[t]^2 + 
     y1[t]^2, -0.0383062 + (-x1[t] + x2[t])^2 + (-y1[t] + 
     y2[t])^2, -1.64323 + (-0.685234 + x2[t])^2 + (-0.154534 + 
     y2[t])^2, -1.88925 + (-x1[t] + x6[t])^2 + (-y1[t] + 
     y6[t])^2, -0.428366 + 
     ArcTan[-x1[t] + x2[t], -y1[t] + y2[t]] - 
     ArcTan[-x1[t] + x6[t], -y1[t] + 
     y6[t]], -0.783885 + (-x3[t] + x4[t])^2 + (-y3[t] + 
     y4[t])^2, -0.0353049 + (-x4[t] + x5[t])^2 + (-y4[t] + 
     y5[t])^2, -0.25649 + ArcTan[-x3[t] + x4[t], -y3[t] + y4[t]] -
     ArcTan[-x4[t] + x5[t], -y4[t] + 
     y5[t]], -0.0228637 + (-x5[t] + x6[t])^2 + (-y5[t] + 
     y6[t])^2, -0.0138516 + (-x2[t] + x3[t])^2 + (-y2[t] + 
     y3[t])^2, -ArcTan[0.685234 - x2[t], 0.154534 - y2[t]] + 
     ArcTan[-x2[t] + x3[t], -y2[t] + y3[t]];

    VPenalty = 3.90351 + 
     1.13817*10^6 (2.74102 - 1. ArcTan[x1[t], y1[t]])^2 + 
     3006.52 (2.92648 - 
     1. ArcTan[-0.685234 + x3[t], -0.154534 + y3[t]])^2 + 
     1082.47 (2.78921 - 
     1. ArcTan[-0.685234 + x3[t], -0.154534 + y3[t]] + 
     ArcTan[-1. x3[t] + x4[t], -1. y3[t] + y4[t]])^2 + 
     6825.76 (2.48452 - 1. ArcTan[x1[t], y1[t]] + 
     ArcTan[-1. x1[t] + x6[t], -1. y1[t] + y6[t]])^2 + 
     239.418 y1[t] + 29.1166 y2[t] + 6.00899 y3[t] + 7.28423 y4[t] + 
     4.93484 y5[t] + 97.0584 y6[t] + 
     1.90484*10^8 ((-ArcTan[0.685234 - x2[t], 0.154534 - y2[t]] + 
     ArcTan[-x2[t] + x3[t], -y2[t] + y3[t]])^2 + (-0.25649 + 
     ArcTan[-x3[t] + x4[t], -y3[t] + y4[t]] - 
     ArcTan[-x4[t] + x5[t], -y4[t] + y5[t]])^2 + (-0.428366 + 
     ArcTan[-x1[t] + x2[t], -y1[t] + y2[t]] - 
     ArcTan[-x1[t] + x6[t], -y1[t] + y6[t]])^2 + (-0.609155 + 
     x1[t]^2 + 
     y1[t]^2)^2 + (-1.64323 + (-0.685234 + 
     x2[t])^2 + (-0.154534 + 
     y2[t])^2)^2 + (-0.0383062 + (-x1[t] + 
     x2[t])^2 + (-y1[t] + 
     y2[t])^2)^2 + (-0.0138516 + (-x2[t] + 
     x3[t])^2 + (-y2[t] + 
     y3[t])^2)^2 + (-0.783885 + (-x3[t] + 
     x4[t])^2 + (-y3[t] + 
     y4[t])^2)^2 + (-0.0353049 + (-x4[t] + 
     x5[t])^2 + (-y4[t] + y5[t])^2)^2 + (-1.88925 + (-x1[t] +
     x6[t])^2 + (-y1[t] + 
     y6[t])^2)^2 + (-0.0228637 + (-x5[t] + 
     x6[t])^2 + (-y5[t] + y6[t])^2)^2);

    [Theta]1 = ArcTan[x1[t], y1[t]];
    [Theta]21 = ArcTan[x2[t] - x1[t], y2[t] - y1[t]];
    [Theta]3 = ArcTan[x3[t] - xC, y3[t] - yC];
    [Theta]22 = ArcTan[x6[t] - x1[t], y6[t] - y1[t]];
    [Theta]41 = ArcTan[x4[t] - x3[t], y4[t] - y3[t]];
    [Theta]42 = ArcTan[x5[t] - x4[t], y5[t] - y4[t]];
    [Theta]5 = ArcTan[x6[t] - x5[t], y6[t] - y5[t]];

    With[y6Start = 0.15 LenFact, 
     eqbr = NMinimize[VPenalty[[1, 1]], 
     Flatten[CE] == Table[0, Length[CE]], Pi/2 < [Theta]1 < Pi, 
     20*Pi/180 < [Theta]21 < 95*Pi/180, 145*Pi/180 < [Theta]3 < Pi, 
     80*Pi/180 < [Theta]5 < 100*Pi/180, 0 < y6[t] < y6Start, q, 
     Reals]]
    (*1652.07, x1[t] -> -0.729157, y1[t] -> 0.278358, x2[t] -> -0.573507, 
     y2[t] -> 0.397027, x3[t] -> -0.457944, y3[t] -> 0.374765, 
     x4[t] -> 0.423225, y4[t] -> 0.460944, x5[t] -> 0.60875, 
     y5[t] -> 0.431194, x6[t] -> 0.611298, y6[t] -> 0.582379*)





    share|improve this answer









    $endgroup$












    • $begingroup$
      Thanks Alex. At least the constraints are matching to a reasonable degree. Although I am still not sure about the obtained value of Vtotal to be the global one.
      $endgroup$
      – Soumyajit Roy
      Mar 16 at 16:30










    • $begingroup$
      Here VPenalty[] is a non-linear function, so the min can be local.
      $endgroup$
      – Alex Trounev
      Mar 16 at 17:10















    2












    $begingroup$

    I corrected typos. The code is fast and gives an answer, albeit with messages.



    LenFact = 1.3745/0.31571;
    xC = 0.157392 LenFact; yC = 0.035495 LenFact;

    q = x1[t], y1[t], x2[t], y2[t], x3[t], y3[t], x4[t], y4[t], x5[t], 
     y5[t], x6[t], 
     y6[t]; CE = -0.609155 + x1[t]^2 + 
     y1[t]^2, -0.0383062 + (-x1[t] + x2[t])^2 + (-y1[t] + 
     y2[t])^2, -1.64323 + (-0.685234 + x2[t])^2 + (-0.154534 + 
     y2[t])^2, -1.88925 + (-x1[t] + x6[t])^2 + (-y1[t] + 
     y6[t])^2, -0.428366 + 
     ArcTan[-x1[t] + x2[t], -y1[t] + y2[t]] - 
     ArcTan[-x1[t] + x6[t], -y1[t] + 
     y6[t]], -0.783885 + (-x3[t] + x4[t])^2 + (-y3[t] + 
     y4[t])^2, -0.0353049 + (-x4[t] + x5[t])^2 + (-y4[t] + 
     y5[t])^2, -0.25649 + ArcTan[-x3[t] + x4[t], -y3[t] + y4[t]] -
     ArcTan[-x4[t] + x5[t], -y4[t] + 
     y5[t]], -0.0228637 + (-x5[t] + x6[t])^2 + (-y5[t] + 
     y6[t])^2, -0.0138516 + (-x2[t] + x3[t])^2 + (-y2[t] + 
     y3[t])^2, -ArcTan[0.685234 - x2[t], 0.154534 - y2[t]] + 
     ArcTan[-x2[t] + x3[t], -y2[t] + y3[t]];

    VPenalty = 3.90351 + 
     1.13817*10^6 (2.74102 - 1. ArcTan[x1[t], y1[t]])^2 + 
     3006.52 (2.92648 - 
     1. ArcTan[-0.685234 + x3[t], -0.154534 + y3[t]])^2 + 
     1082.47 (2.78921 - 
     1. ArcTan[-0.685234 + x3[t], -0.154534 + y3[t]] + 
     ArcTan[-1. x3[t] + x4[t], -1. y3[t] + y4[t]])^2 + 
     6825.76 (2.48452 - 1. ArcTan[x1[t], y1[t]] + 
     ArcTan[-1. x1[t] + x6[t], -1. y1[t] + y6[t]])^2 + 
     239.418 y1[t] + 29.1166 y2[t] + 6.00899 y3[t] + 7.28423 y4[t] + 
     4.93484 y5[t] + 97.0584 y6[t] + 
     1.90484*10^8 ((-ArcTan[0.685234 - x2[t], 0.154534 - y2[t]] + 
     ArcTan[-x2[t] + x3[t], -y2[t] + y3[t]])^2 + (-0.25649 + 
     ArcTan[-x3[t] + x4[t], -y3[t] + y4[t]] - 
     ArcTan[-x4[t] + x5[t], -y4[t] + y5[t]])^2 + (-0.428366 + 
     ArcTan[-x1[t] + x2[t], -y1[t] + y2[t]] - 
     ArcTan[-x1[t] + x6[t], -y1[t] + y6[t]])^2 + (-0.609155 + 
     x1[t]^2 + 
     y1[t]^2)^2 + (-1.64323 + (-0.685234 + 
     x2[t])^2 + (-0.154534 + 
     y2[t])^2)^2 + (-0.0383062 + (-x1[t] + 
     x2[t])^2 + (-y1[t] + 
     y2[t])^2)^2 + (-0.0138516 + (-x2[t] + 
     x3[t])^2 + (-y2[t] + 
     y3[t])^2)^2 + (-0.783885 + (-x3[t] + 
     x4[t])^2 + (-y3[t] + 
     y4[t])^2)^2 + (-0.0353049 + (-x4[t] + 
     x5[t])^2 + (-y4[t] + y5[t])^2)^2 + (-1.88925 + (-x1[t] +
     x6[t])^2 + (-y1[t] + 
     y6[t])^2)^2 + (-0.0228637 + (-x5[t] + 
     x6[t])^2 + (-y5[t] + y6[t])^2)^2);

    [Theta]1 = ArcTan[x1[t], y1[t]];
    [Theta]21 = ArcTan[x2[t] - x1[t], y2[t] - y1[t]];
    [Theta]3 = ArcTan[x3[t] - xC, y3[t] - yC];
    [Theta]22 = ArcTan[x6[t] - x1[t], y6[t] - y1[t]];
    [Theta]41 = ArcTan[x4[t] - x3[t], y4[t] - y3[t]];
    [Theta]42 = ArcTan[x5[t] - x4[t], y5[t] - y4[t]];
    [Theta]5 = ArcTan[x6[t] - x5[t], y6[t] - y5[t]];

    With[y6Start = 0.15 LenFact, 
     eqbr = NMinimize[VPenalty[[1, 1]], 
     Flatten[CE] == Table[0, Length[CE]], Pi/2 < [Theta]1 < Pi, 
     20*Pi/180 < [Theta]21 < 95*Pi/180, 145*Pi/180 < [Theta]3 < Pi, 
     80*Pi/180 < [Theta]5 < 100*Pi/180, 0 < y6[t] < y6Start, q, 
     Reals]]
    (*1652.07, x1[t] -> -0.729157, y1[t] -> 0.278358, x2[t] -> -0.573507, 
     y2[t] -> 0.397027, x3[t] -> -0.457944, y3[t] -> 0.374765, 
     x4[t] -> 0.423225, y4[t] -> 0.460944, x5[t] -> 0.60875, 
     y5[t] -> 0.431194, x6[t] -> 0.611298, y6[t] -> 0.582379*)





    share|improve this answer









    $endgroup$












    • $begingroup$
      Thanks Alex. At least the constraints are matching to a reasonable degree. Although I am still not sure about the obtained value of Vtotal to be the global one.
      $endgroup$
      – Soumyajit Roy
      Mar 16 at 16:30










    • $begingroup$
      Here VPenalty[] is a non-linear function, so the min can be local.
      $endgroup$
      – Alex Trounev
      Mar 16 at 17:10













    2












    2








    2





    $begingroup$

    I corrected typos. The code is fast and gives an answer, albeit with messages.



    LenFact = 1.3745/0.31571;
    xC = 0.157392 LenFact; yC = 0.035495 LenFact;

    q = x1[t], y1[t], x2[t], y2[t], x3[t], y3[t], x4[t], y4[t], x5[t], 
     y5[t], x6[t], 
     y6[t]; CE = -0.609155 + x1[t]^2 + 
     y1[t]^2, -0.0383062 + (-x1[t] + x2[t])^2 + (-y1[t] + 
     y2[t])^2, -1.64323 + (-0.685234 + x2[t])^2 + (-0.154534 + 
     y2[t])^2, -1.88925 + (-x1[t] + x6[t])^2 + (-y1[t] + 
     y6[t])^2, -0.428366 + 
     ArcTan[-x1[t] + x2[t], -y1[t] + y2[t]] - 
     ArcTan[-x1[t] + x6[t], -y1[t] + 
     y6[t]], -0.783885 + (-x3[t] + x4[t])^2 + (-y3[t] + 
     y4[t])^2, -0.0353049 + (-x4[t] + x5[t])^2 + (-y4[t] + 
     y5[t])^2, -0.25649 + ArcTan[-x3[t] + x4[t], -y3[t] + y4[t]] -
     ArcTan[-x4[t] + x5[t], -y4[t] + 
     y5[t]], -0.0228637 + (-x5[t] + x6[t])^2 + (-y5[t] + 
     y6[t])^2, -0.0138516 + (-x2[t] + x3[t])^2 + (-y2[t] + 
     y3[t])^2, -ArcTan[0.685234 - x2[t], 0.154534 - y2[t]] + 
     ArcTan[-x2[t] + x3[t], -y2[t] + y3[t]];

    VPenalty = 3.90351 + 
     1.13817*10^6 (2.74102 - 1. ArcTan[x1[t], y1[t]])^2 + 
     3006.52 (2.92648 - 
     1. ArcTan[-0.685234 + x3[t], -0.154534 + y3[t]])^2 + 
     1082.47 (2.78921 - 
     1. ArcTan[-0.685234 + x3[t], -0.154534 + y3[t]] + 
     ArcTan[-1. x3[t] + x4[t], -1. y3[t] + y4[t]])^2 + 
     6825.76 (2.48452 - 1. ArcTan[x1[t], y1[t]] + 
     ArcTan[-1. x1[t] + x6[t], -1. y1[t] + y6[t]])^2 + 
     239.418 y1[t] + 29.1166 y2[t] + 6.00899 y3[t] + 7.28423 y4[t] + 
     4.93484 y5[t] + 97.0584 y6[t] + 
     1.90484*10^8 ((-ArcTan[0.685234 - x2[t], 0.154534 - y2[t]] + 
     ArcTan[-x2[t] + x3[t], -y2[t] + y3[t]])^2 + (-0.25649 + 
     ArcTan[-x3[t] + x4[t], -y3[t] + y4[t]] - 
     ArcTan[-x4[t] + x5[t], -y4[t] + y5[t]])^2 + (-0.428366 + 
     ArcTan[-x1[t] + x2[t], -y1[t] + y2[t]] - 
     ArcTan[-x1[t] + x6[t], -y1[t] + y6[t]])^2 + (-0.609155 + 
     x1[t]^2 + 
     y1[t]^2)^2 + (-1.64323 + (-0.685234 + 
     x2[t])^2 + (-0.154534 + 
     y2[t])^2)^2 + (-0.0383062 + (-x1[t] + 
     x2[t])^2 + (-y1[t] + 
     y2[t])^2)^2 + (-0.0138516 + (-x2[t] + 
     x3[t])^2 + (-y2[t] + 
     y3[t])^2)^2 + (-0.783885 + (-x3[t] + 
     x4[t])^2 + (-y3[t] + 
     y4[t])^2)^2 + (-0.0353049 + (-x4[t] + 
     x5[t])^2 + (-y4[t] + y5[t])^2)^2 + (-1.88925 + (-x1[t] +
     x6[t])^2 + (-y1[t] + 
     y6[t])^2)^2 + (-0.0228637 + (-x5[t] + 
     x6[t])^2 + (-y5[t] + y6[t])^2)^2);

    [Theta]1 = ArcTan[x1[t], y1[t]];
    [Theta]21 = ArcTan[x2[t] - x1[t], y2[t] - y1[t]];
    [Theta]3 = ArcTan[x3[t] - xC, y3[t] - yC];
    [Theta]22 = ArcTan[x6[t] - x1[t], y6[t] - y1[t]];
    [Theta]41 = ArcTan[x4[t] - x3[t], y4[t] - y3[t]];
    [Theta]42 = ArcTan[x5[t] - x4[t], y5[t] - y4[t]];
    [Theta]5 = ArcTan[x6[t] - x5[t], y6[t] - y5[t]];

    With[y6Start = 0.15 LenFact, 
     eqbr = NMinimize[VPenalty[[1, 1]], 
     Flatten[CE] == Table[0, Length[CE]], Pi/2 < [Theta]1 < Pi, 
     20*Pi/180 < [Theta]21 < 95*Pi/180, 145*Pi/180 < [Theta]3 < Pi, 
     80*Pi/180 < [Theta]5 < 100*Pi/180, 0 < y6[t] < y6Start, q, 
     Reals]]
    (*1652.07, x1[t] -> -0.729157, y1[t] -> 0.278358, x2[t] -> -0.573507, 
     y2[t] -> 0.397027, x3[t] -> -0.457944, y3[t] -> 0.374765, 
     x4[t] -> 0.423225, y4[t] -> 0.460944, x5[t] -> 0.60875, 
     y5[t] -> 0.431194, x6[t] -> 0.611298, y6[t] -> 0.582379*)





    share|improve this answer









    $endgroup$



    I corrected typos. The code is fast and gives an answer, albeit with messages.



    LenFact = 1.3745/0.31571;
    xC = 0.157392 LenFact; yC = 0.035495 LenFact;

    q = x1[t], y1[t], x2[t], y2[t], x3[t], y3[t], x4[t], y4[t], x5[t], 
     y5[t], x6[t], 
     y6[t]; CE = -0.609155 + x1[t]^2 + 
     y1[t]^2, -0.0383062 + (-x1[t] + x2[t])^2 + (-y1[t] + 
     y2[t])^2, -1.64323 + (-0.685234 + x2[t])^2 + (-0.154534 + 
     y2[t])^2, -1.88925 + (-x1[t] + x6[t])^2 + (-y1[t] + 
     y6[t])^2, -0.428366 + 
     ArcTan[-x1[t] + x2[t], -y1[t] + y2[t]] - 
     ArcTan[-x1[t] + x6[t], -y1[t] + 
     y6[t]], -0.783885 + (-x3[t] + x4[t])^2 + (-y3[t] + 
     y4[t])^2, -0.0353049 + (-x4[t] + x5[t])^2 + (-y4[t] + 
     y5[t])^2, -0.25649 + ArcTan[-x3[t] + x4[t], -y3[t] + y4[t]] -
     ArcTan[-x4[t] + x5[t], -y4[t] + 
     y5[t]], -0.0228637 + (-x5[t] + x6[t])^2 + (-y5[t] + 
     y6[t])^2, -0.0138516 + (-x2[t] + x3[t])^2 + (-y2[t] + 
     y3[t])^2, -ArcTan[0.685234 - x2[t], 0.154534 - y2[t]] + 
     ArcTan[-x2[t] + x3[t], -y2[t] + y3[t]];

    VPenalty = 3.90351 + 
     1.13817*10^6 (2.74102 - 1. ArcTan[x1[t], y1[t]])^2 + 
     3006.52 (2.92648 - 
     1. ArcTan[-0.685234 + x3[t], -0.154534 + y3[t]])^2 + 
     1082.47 (2.78921 - 
     1. ArcTan[-0.685234 + x3[t], -0.154534 + y3[t]] + 
     ArcTan[-1. x3[t] + x4[t], -1. y3[t] + y4[t]])^2 + 
     6825.76 (2.48452 - 1. ArcTan[x1[t], y1[t]] + 
     ArcTan[-1. x1[t] + x6[t], -1. y1[t] + y6[t]])^2 + 
     239.418 y1[t] + 29.1166 y2[t] + 6.00899 y3[t] + 7.28423 y4[t] + 
     4.93484 y5[t] + 97.0584 y6[t] + 
     1.90484*10^8 ((-ArcTan[0.685234 - x2[t], 0.154534 - y2[t]] + 
     ArcTan[-x2[t] + x3[t], -y2[t] + y3[t]])^2 + (-0.25649 + 
     ArcTan[-x3[t] + x4[t], -y3[t] + y4[t]] - 
     ArcTan[-x4[t] + x5[t], -y4[t] + y5[t]])^2 + (-0.428366 + 
     ArcTan[-x1[t] + x2[t], -y1[t] + y2[t]] - 
     ArcTan[-x1[t] + x6[t], -y1[t] + y6[t]])^2 + (-0.609155 + 
     x1[t]^2 + 
     y1[t]^2)^2 + (-1.64323 + (-0.685234 + 
     x2[t])^2 + (-0.154534 + 
     y2[t])^2)^2 + (-0.0383062 + (-x1[t] + 
     x2[t])^2 + (-y1[t] + 
     y2[t])^2)^2 + (-0.0138516 + (-x2[t] + 
     x3[t])^2 + (-y2[t] + 
     y3[t])^2)^2 + (-0.783885 + (-x3[t] + 
     x4[t])^2 + (-y3[t] + 
     y4[t])^2)^2 + (-0.0353049 + (-x4[t] + 
     x5[t])^2 + (-y4[t] + y5[t])^2)^2 + (-1.88925 + (-x1[t] +
     x6[t])^2 + (-y1[t] + 
     y6[t])^2)^2 + (-0.0228637 + (-x5[t] + 
     x6[t])^2 + (-y5[t] + y6[t])^2)^2);

    [Theta]1 = ArcTan[x1[t], y1[t]];
    [Theta]21 = ArcTan[x2[t] - x1[t], y2[t] - y1[t]];
    [Theta]3 = ArcTan[x3[t] - xC, y3[t] - yC];
    [Theta]22 = ArcTan[x6[t] - x1[t], y6[t] - y1[t]];
    [Theta]41 = ArcTan[x4[t] - x3[t], y4[t] - y3[t]];
    [Theta]42 = ArcTan[x5[t] - x4[t], y5[t] - y4[t]];
    [Theta]5 = ArcTan[x6[t] - x5[t], y6[t] - y5[t]];

    With[y6Start = 0.15 LenFact, 
     eqbr = NMinimize[VPenalty[[1, 1]], 
     Flatten[CE] == Table[0, Length[CE]], Pi/2 < [Theta]1 < Pi, 
     20*Pi/180 < [Theta]21 < 95*Pi/180, 145*Pi/180 < [Theta]3 < Pi, 
     80*Pi/180 < [Theta]5 < 100*Pi/180, 0 < y6[t] < y6Start, q, 
     Reals]]
    (*1652.07, x1[t] -> -0.729157, y1[t] -> 0.278358, x2[t] -> -0.573507, 
     y2[t] -> 0.397027, x3[t] -> -0.457944, y3[t] -> 0.374765, 
     x4[t] -> 0.423225, y4[t] -> 0.460944, x5[t] -> 0.60875, 
     y5[t] -> 0.431194, x6[t] -> 0.611298, y6[t] -> 0.582379*)






    share|improve this answer












    share|improve this answer



    share|improve this answer










    answered Mar 16 at 16:07









    Alex TrounevAlex Trounev

    8,0681621




    8,0681621











    • $begingroup$
      Thanks Alex. At least the constraints are matching to a reasonable degree. Although I am still not sure about the obtained value of Vtotal to be the global one.
      $endgroup$
      – Soumyajit Roy
      Mar 16 at 16:30










    • $begingroup$
      Here VPenalty[] is a non-linear function, so the min can be local.
      $endgroup$
      – Alex Trounev
      Mar 16 at 17:10
















    • $begingroup$
      Thanks Alex. At least the constraints are matching to a reasonable degree. Although I am still not sure about the obtained value of Vtotal to be the global one.
      $endgroup$
      – Soumyajit Roy
      Mar 16 at 16:30










    • $begingroup$
      Here VPenalty[] is a non-linear function, so the min can be local.
      $endgroup$
      – Alex Trounev
      Mar 16 at 17:10















    $begingroup$
    Thanks Alex. At least the constraints are matching to a reasonable degree. Although I am still not sure about the obtained value of Vtotal to be the global one.
    $endgroup$
    – Soumyajit Roy
    Mar 16 at 16:30




    $begingroup$
    Thanks Alex. At least the constraints are matching to a reasonable degree. Although I am still not sure about the obtained value of Vtotal to be the global one.
    $endgroup$
    – Soumyajit Roy
    Mar 16 at 16:30












    $begingroup$
    Here VPenalty[] is a non-linear function, so the min can be local.
    $endgroup$
    – Alex Trounev
    Mar 16 at 17:10




    $begingroup$
    Here VPenalty[] is a non-linear function, so the min can be local.
    $endgroup$
    – Alex Trounev
    Mar 16 at 17:10











    2












    $begingroup$

    The method "RandomSearch" usually gives a result reasonably quickly (other than using FindMinimum). It's less robust, though, than the other methods. The following takes about 30 sec.



    eqbr = NMinimize[VPenalty[[1, 1]], Flatten[CE] == 0, 
     90 ° < θ1 < 180 °, 20 ° < θ21 < 95 °, 
     145 ° < θ3 < 180 °, 80 ° < θ5 < 100 °, 0 < y6[t] < y6Start, q, 
     Method -> "RandomSearch"]



    NMinimize::incst: NMinimize was unable to generate any initial points satisfying the inequality constraints...




    (*
    161.292, x1[t] -> -0.718754, y1[t] -> 0.30422, x2[t] -> -0.567146, 
     y2[t] -> 0.427983, x3[t] -> -0.452162, y3[t] -> 0.402877, 
     x4[t] -> 0.424915, y4[t] -> 0.523793, x5[t] -> 0.611471, 
     y5[t] -> 0.501395, x6[t] -> 0.610863, y6[t] -> 0.652602
    *)


    While NMinimize did not find good starting points, it found a solution within the constraints:



    90 ° < θ1 < 180 °, 20 ° < θ21 < 95 °, 145 ° < θ3 < 180 °, 
     80 ° < θ5 < 100 °, 0 < y6[t] < y6Start /. 
     Last@eqbr
    (* True, True, True, True, True *)





    share|improve this answer









    $endgroup$

















      2












      $begingroup$

      The method "RandomSearch" usually gives a result reasonably quickly (other than using FindMinimum). It's less robust, though, than the other methods. The following takes about 30 sec.



      eqbr = NMinimize[VPenalty[[1, 1]], Flatten[CE] == 0, 
       90 ° < θ1 < 180 °, 20 ° < θ21 < 95 °, 
       145 ° < θ3 < 180 °, 80 ° < θ5 < 100 °, 0 < y6[t] < y6Start, q, 
       Method -> "RandomSearch"]



      NMinimize::incst: NMinimize was unable to generate any initial points satisfying the inequality constraints...




      (*
      161.292, x1[t] -> -0.718754, y1[t] -> 0.30422, x2[t] -> -0.567146, 
       y2[t] -> 0.427983, x3[t] -> -0.452162, y3[t] -> 0.402877, 
       x4[t] -> 0.424915, y4[t] -> 0.523793, x5[t] -> 0.611471, 
       y5[t] -> 0.501395, x6[t] -> 0.610863, y6[t] -> 0.652602
      *)


      While NMinimize did not find good starting points, it found a solution within the constraints:



      90 ° < θ1 < 180 °, 20 ° < θ21 < 95 °, 145 ° < θ3 < 180 °, 
       80 ° < θ5 < 100 °, 0 < y6[t] < y6Start /. 
       Last@eqbr
      (* True, True, True, True, True *)





      share|improve this answer









      $endgroup$















        2












        2








        2





        $begingroup$

        The method "RandomSearch" usually gives a result reasonably quickly (other than using FindMinimum). It's less robust, though, than the other methods. The following takes about 30 sec.



        eqbr = NMinimize[VPenalty[[1, 1]], Flatten[CE] == 0, 
         90 ° < θ1 < 180 °, 20 ° < θ21 < 95 °, 
         145 ° < θ3 < 180 °, 80 ° < θ5 < 100 °, 0 < y6[t] < y6Start, q, 
         Method -> "RandomSearch"]



        NMinimize::incst: NMinimize was unable to generate any initial points satisfying the inequality constraints...




        (*
        161.292, x1[t] -> -0.718754, y1[t] -> 0.30422, x2[t] -> -0.567146, 
         y2[t] -> 0.427983, x3[t] -> -0.452162, y3[t] -> 0.402877, 
         x4[t] -> 0.424915, y4[t] -> 0.523793, x5[t] -> 0.611471, 
         y5[t] -> 0.501395, x6[t] -> 0.610863, y6[t] -> 0.652602
        *)


        While NMinimize did not find good starting points, it found a solution within the constraints:



        90 ° < θ1 < 180 °, 20 ° < θ21 < 95 °, 145 ° < θ3 < 180 °, 
         80 ° < θ5 < 100 °, 0 < y6[t] < y6Start /. 
         Last@eqbr
        (* True, True, True, True, True *)





        share|improve this answer









        $endgroup$



        The method "RandomSearch" usually gives a result reasonably quickly (other than using FindMinimum). It's less robust, though, than the other methods. The following takes about 30 sec.



        eqbr = NMinimize[VPenalty[[1, 1]], Flatten[CE] == 0, 
         90 ° < θ1 < 180 °, 20 ° < θ21 < 95 °, 
         145 ° < θ3 < 180 °, 80 ° < θ5 < 100 °, 0 < y6[t] < y6Start, q, 
         Method -> "RandomSearch"]



        NMinimize::incst: NMinimize was unable to generate any initial points satisfying the inequality constraints...




        (*
        161.292, x1[t] -> -0.718754, y1[t] -> 0.30422, x2[t] -> -0.567146, 
         y2[t] -> 0.427983, x3[t] -> -0.452162, y3[t] -> 0.402877, 
         x4[t] -> 0.424915, y4[t] -> 0.523793, x5[t] -> 0.611471, 
         y5[t] -> 0.501395, x6[t] -> 0.610863, y6[t] -> 0.652602
        *)


        While NMinimize did not find good starting points, it found a solution within the constraints:



        90 ° < θ1 < 180 °, 20 ° < θ21 < 95 °, 145 ° < θ3 < 180 °, 
         80 ° < θ5 < 100 °, 0 < y6[t] < y6Start /. 
         Last@eqbr
        (* True, True, True, True, True *)






        share|improve this answer












        share|improve this answer



        share|improve this answer










        answered Mar 16 at 17:24









        Michael E2Michael E2

        149k12201481




        149k12201481



























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            Daza language Contents Vocabulary Phonology References External links Navigation menudaza1242Daza"Dazaga"eeee178086576

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            FrenchArabicFrenchAir TamajeqChadian ArabicDazaKaadoTamahaqStandardArabicEnglishDazaDinkaDoniFurKadugliKangaKeigaKrongoTemeinTeseTulishiTumtumZaghawaAcheronAmiraDengebuEbangJomangKatlaKoKwalibLaroLogolLumunMoroNdingNgileShirumbaTagoiTegaliTegemTimaTiroTochoToronaUtoroWarnangBejaDomariSudanese ArabicSudanese Sign LanguagesBertiZaghawa Tebu languagesSaharan languagesLanguages of ChadLanguages of NigerToubou people Nilo-Saharan languageDaza peopleChadGouranDjurab DesertBorkouFaya-LargeauTibesti MountainsChadNigerN'guigmiLibyaSudanOmdurmanJeddahSaudi ArabiaTedaga languageTedaToubou peopleSabhaSaharanKanuri languageKanembu languageTebu languagesZaghawa languageBerti languageFrenchArabicliterary languagebilingualHausaZarmaTuaregromanisation Not to be confused with Dazawa language. "Goran language" redirects here. It is not to be confused with Gorani language (Zaza-Gorani) or Gora dialect. Daza Dazaga Native to Chad, Niger, Sudan, Libya Region Chad, BET Region, bahr el Gazel R...
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