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
    4 hours ago










  • $begingroup$
    And what about y6Start?
    $endgroup$
    – Henrik Schumacher
    4 hours ago






  • 1




    $begingroup$
    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
    4 hours ago











  • $begingroup$
    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
    4 hours ago















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
    4 hours ago










  • $begingroup$
    And what about y6Start?
    $endgroup$
    – Henrik Schumacher
    4 hours ago






  • 1




    $begingroup$
    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
    4 hours ago











  • $begingroup$
    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
    4 hours ago













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 4 hours ago







Soumyajit Roy

















asked 4 hours ago









Soumyajit RoySoumyajit Roy

1809




1809











  • $begingroup$
    I am sorry. I have incorporated CE in the code now.
    $endgroup$
    – Soumyajit Roy
    4 hours ago










  • $begingroup$
    And what about y6Start?
    $endgroup$
    – Henrik Schumacher
    4 hours ago






  • 1




    $begingroup$
    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
    4 hours ago











  • $begingroup$
    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
    4 hours ago
















  • $begingroup$
    I am sorry. I have incorporated CE in the code now.
    $endgroup$
    – Soumyajit Roy
    4 hours ago










  • $begingroup$
    And what about y6Start?
    $endgroup$
    – Henrik Schumacher
    4 hours ago






  • 1




    $begingroup$
    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
    4 hours ago











  • $begingroup$
    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
    4 hours ago















$begingroup$
I am sorry. I have incorporated CE in the code now.
$endgroup$
– Soumyajit Roy
4 hours ago




$begingroup$
I am sorry. I have incorporated CE in the code now.
$endgroup$
– Soumyajit Roy
4 hours ago












$begingroup$
And what about y6Start?
$endgroup$
– Henrik Schumacher
4 hours ago




$begingroup$
And what about y6Start?
$endgroup$
– Henrik Schumacher
4 hours ago




1




1




$begingroup$
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
4 hours ago





$begingroup$
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
4 hours ago













$begingroup$
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
4 hours ago




$begingroup$
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
4 hours ago










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
    3 hours ago










  • $begingroup$
    Here VPenalty[] is a non-linear function, so the min can be local.
    $endgroup$
    – Alex Trounev
    2 hours ago


















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












    $begingroup$

    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
      3 hours ago










    • $begingroup$
      Here VPenalty[] is a non-linear function, so the min can be local.
      $endgroup$
      – Alex Trounev
      2 hours ago















    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
      3 hours ago










    • $begingroup$
      Here VPenalty[] is a non-linear function, so the min can be local.
      $endgroup$
      – Alex Trounev
      2 hours ago













    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 3 hours ago









    Alex TrounevAlex Trounev

    7,8831521




    7,8831521











    • $begingroup$
      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
      3 hours ago










    • $begingroup$
      Here VPenalty[] is a non-linear function, so the min can be local.
      $endgroup$
      – Alex Trounev
      2 hours ago
















    • $begingroup$
      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
      3 hours ago










    • $begingroup$
      Here VPenalty[] is a non-linear function, so the min can be local.
      $endgroup$
      – Alex Trounev
      2 hours ago















    $begingroup$
    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
    3 hours ago




    $begingroup$
    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
    3 hours ago












    $begingroup$
    Here VPenalty[] is a non-linear function, so the min can be local.
    $endgroup$
    – Alex Trounev
    2 hours ago




    $begingroup$
    Here VPenalty[] is a non-linear function, so the min can be local.
    $endgroup$
    – Alex Trounev
    2 hours ago











    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 2 hours ago









        Michael E2Michael E2

        149k12200479




        149k12200479



























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