Minimum Surface AreaSOCP > SOC inequalities | Standard Forms | Group sparsity | Applications > Minimum Surface Area | Next
The minimum surface area problemFunctional problem statementConsider a surface in ![]() The minimum surface area problem is to find the function ![]() where The above is an infinite-dimensional problem, in the sense that the variable is a function, not a finite-dimensional vector. DiscretizationWe can discretize the square with a square grid, with points To approximate the gradient, we start from the first-order expansion of a function of two variables, valid for some small increment ![]() We obtain that the gradient of ![]() SOCP formulationThe discretized version of our problem is thus ![]() The CVX syntax for this problem can be as follows. CVX syntax
>> % input: left_vals and righ_vals, two row vectors of length K+1 >> h = 1/K; cvx_begin variables F(K+1,K+1) variables T(K,K) minimize( sum(T(:)) ) subject to for j = 1:K, for i = 1:K, norm([K*(F(i+1,j)-F(i,j)); K*(F(i,j+1)-F(i,j)); 1],2) <= T(i,j); end, end F(1,:) == left_vals; F(K+1,:) == right_vals; cvx_end Examples
It is interesting to compare the minimal surface area with one that is obtained by squaring the norms. This corresponds to the QP ![]() |