Robust Least-SquaresWe start from the least-squares problem: ![]() where Now assume that the matrix ![]() The interpretation of this problem is that it is trying to minimize the worst-case value of the residual norm. For fixed ![]() By definition of the largest singular value norm, and given our bound on the size of the uncertainty, we have ![]() Thus, we have a bound on the objective value of the robust problem: ![]() It turns out that the upper bound is attained by some choice of the matrix ![]() Hence the robust least-squares is equivalent to the problem ![]() The above is an SOCP: ![]() As given, this SOCP can be solved using SVD methods. However, problems involving constraints (such as sign constraints on |