Minimize distance given other distance constraints

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I want to minimize the Euclidean distance between a pair of points $mathbfP_1, mathbfP_2$ by finding a new point $mathbfP_2*$, with the inequality constraints that for each $R_i$ of a set of points $R$, $$midmid mathbfP_2*-R_imidmid_2 leq midmid mathbfP_2-R_imidmid_2 $$



Geometrically, I could express $mathbfR_i$ as the red points, $mathbfP_2$ as the green point and $mathbfP_1$ as the blue point. Circles with radius of the distance to the green point can then be 'indifference' curves, and help to identify the feasible set.



In the left figure the new point would be the intersection of a line from blue point to red point and the circle - however, in the right figure it would be the intersections of the two red circles.



Since I am not sure if there would be other exceptions, it would be great if someone could perhaps point me to better approaches of solving this. Thank you!



enter image description here




optimization nonlinear-optimization constraints




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  • Are you trying to do this computationally? If so, then the inequality constraints you propose are convex, and the resulting system can be handled with a convex optimization solver.
    – David M.
    Jul 22 at 15:08






  • 1




    Thanks, that was exactly the hint I needed - finally managed to implement it in R and it works great :) @DavidM.
    – eok
    Jul 23 at 21:45














up vote
0
down vote

favorite












I want to minimize the Euclidean distance between a pair of points $mathbfP_1, mathbfP_2$ by finding a new point $mathbfP_2*$, with the inequality constraints that for each $R_i$ of a set of points $R$, $$midmid mathbfP_2*-R_imidmid_2 leq midmid mathbfP_2-R_imidmid_2 $$



Geometrically, I could express $mathbfR_i$ as the red points, $mathbfP_2$ as the green point and $mathbfP_1$ as the blue point. Circles with radius of the distance to the green point can then be 'indifference' curves, and help to identify the feasible set.



In the left figure the new point would be the intersection of a line from blue point to red point and the circle - however, in the right figure it would be the intersections of the two red circles.



Since I am not sure if there would be other exceptions, it would be great if someone could perhaps point me to better approaches of solving this. Thank you!



enter image description here




optimization nonlinear-optimization constraints




share|cite|improve this question



















  • Are you trying to do this computationally? If so, then the inequality constraints you propose are convex, and the resulting system can be handled with a convex optimization solver.
    – David M.
    Jul 22 at 15:08






  • 1




    Thanks, that was exactly the hint I needed - finally managed to implement it in R and it works great :) @DavidM.
    – eok
    Jul 23 at 21:45












up vote
0
down vote

favorite









up vote
0
down vote

favorite











I want to minimize the Euclidean distance between a pair of points $mathbfP_1, mathbfP_2$ by finding a new point $mathbfP_2*$, with the inequality constraints that for each $R_i$ of a set of points $R$, $$midmid mathbfP_2*-R_imidmid_2 leq midmid mathbfP_2-R_imidmid_2 $$



Geometrically, I could express $mathbfR_i$ as the red points, $mathbfP_2$ as the green point and $mathbfP_1$ as the blue point. Circles with radius of the distance to the green point can then be 'indifference' curves, and help to identify the feasible set.



In the left figure the new point would be the intersection of a line from blue point to red point and the circle - however, in the right figure it would be the intersections of the two red circles.



Since I am not sure if there would be other exceptions, it would be great if someone could perhaps point me to better approaches of solving this. Thank you!



enter image description here




optimization nonlinear-optimization constraints




share|cite|improve this question











I want to minimize the Euclidean distance between a pair of points $mathbfP_1, mathbfP_2$ by finding a new point $mathbfP_2*$, with the inequality constraints that for each $R_i$ of a set of points $R$, $$midmid mathbfP_2*-R_imidmid_2 leq midmid mathbfP_2-R_imidmid_2 $$



Geometrically, I could express $mathbfR_i$ as the red points, $mathbfP_2$ as the green point and $mathbfP_1$ as the blue point. Circles with radius of the distance to the green point can then be 'indifference' curves, and help to identify the feasible set.



In the left figure the new point would be the intersection of a line from blue point to red point and the circle - however, in the right figure it would be the intersections of the two red circles.



Since I am not sure if there would be other exceptions, it would be great if someone could perhaps point me to better approaches of solving this. Thank you!



enter image description here





optimization nonlinear-optimization constraints





share|cite|improve this question










share|cite|improve this question




share|cite|improve this question









asked Jul 22 at 13:57









eok

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  • Are you trying to do this computationally? If so, then the inequality constraints you propose are convex, and the resulting system can be handled with a convex optimization solver.
    – David M.
    Jul 22 at 15:08






  • 1




    Thanks, that was exactly the hint I needed - finally managed to implement it in R and it works great :) @DavidM.
    – eok
    Jul 23 at 21:45
















  • Are you trying to do this computationally? If so, then the inequality constraints you propose are convex, and the resulting system can be handled with a convex optimization solver.
    – David M.
    Jul 22 at 15:08






  • 1




    Thanks, that was exactly the hint I needed - finally managed to implement it in R and it works great :) @DavidM.
    – eok
    Jul 23 at 21:45















Are you trying to do this computationally? If so, then the inequality constraints you propose are convex, and the resulting system can be handled with a convex optimization solver.
– David M.
Jul 22 at 15:08




Are you trying to do this computationally? If so, then the inequality constraints you propose are convex, and the resulting system can be handled with a convex optimization solver.
– David M.
Jul 22 at 15:08




1




1




Thanks, that was exactly the hint I needed - finally managed to implement it in R and it works great :) @DavidM.
– eok
Jul 23 at 21:45




Thanks, that was exactly the hint I needed - finally managed to implement it in R and it works great :) @DavidM.
– eok
Jul 23 at 21:45















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