I appreciate your assistance! I’m working on locating a point that minimizes the sum of squared distances from a set of circles. The objective function to minimize is:
$$ E(x,y) = \sum_i \big( (x-w_i)^2 + (y-f_i)^2 - m_i^2 \big)^2 $$
With three circles in my scenario, I aim to determine this point. Despite calculating the partial derivatives, which include both ( x ) and ( y ), I’m unsure if an analytical solution exists or if I must resort to numerical methods. Considering a numerical approach, I’ve thought about employing gradient descent but am uncertain how to apply it to both coordinates simultaneously. My initial strategy is:
-Start with random values for ( x ) and ( y ).
-Evaluate the cost function ( E(x,y) ).
-Derive the partial derivatives. Apply a scaling factor ( K ) to the derivatives.
-Adjust ( x ) and ( y ) by adding the scaled derivatives.
-Re-evaluate from step 2, stopping if the cost function increases, or persist until it nearly vanishes.
Could you please explain if a numerical method is the only option for this problem and why an exact solution might not be attainable?