Answers to: PDE as Nonlinear least squares problems with binary variableshttp://www.or-exchange.com/questions/11333/pde-as-nonlinear-least-squares-problems-with-binary-variables<p>Suppose we have an edge-weighted i a heat network, then we want to identify which edges
will be contribute in the heat convection.
I am using heat equation on a network and then solve it via least-squares problem.
I want to solve the heat equation <code>T_t(x,t) = - L_x . T(x,t) + F(x,t)</code> in an edge-weighted graph where <code>L_x = \sum_i x_i e_{ij}</code> is weighted Laplacian matrix of the graph, F() are sources on some nodes and T() is temperature of each node in each time step. Then using Theta method I conclude to the following least squares problem:</p>
<p><code>\min_{x\in \{0,1\}} \|\hat{T}_j(t_i) - T_j(t_i, x)\|_2^2</code></p>
<p>I have done the part which variables(weights) are relaxed to be between [0,1].
I used L1 regularization and active set methods in Matlab's fmincon to approximate the edge weights and then simulate the solution graph using edge-weights.</p>
<p>Next step would be to solve this problem using exact methods, for example:branch and bound, etc.
My question: is there would be right way to go with B&B?(I know it would be expensive to apply exact
methods, but will worth to check and compare the solutions). Any other suggestion would be
really appreciate.</p>
<p>I could not find some references on the discrete methods for nonlinear least squares. Would you lease let me know if you know?</p>enSat, 07 Dec 2019 07:23:36 -0000