A Newton Barrier Method for Minimizing a Sum of Euclidean Norms Subject to Linear Equality Constraints

An algorithm for minimizing a sum of Euclidean Norms subject to linear equality constraints is described. The algorithm is based on a recently developed Newton barrier method for the unconstrained minimization of a sum of Euclidean norms (MSN). The linear equality constraints are handled using an exact L 1 penalty function which is made smooth in the same way as the Euclidean norms. It is shown that the dual problem is to maximize a linear objective function subject to homogeneous linear equality constraints and quadratic inequalities. Hence the suggested method also solves such problems eeciently. In fact such a problem from plastic collapse analysis motivated this work. Numerical results are presented for large sparse problems, demonstrating the extreme eeciency of the method.