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The problem of minimizing a sum of Euclidean norms dates from the 17th century and may be the earliest example of duality in the mathematical programming literature. This nonsmooth optimization problem arises in many di erent kinds of modern scienti c applications. We derive a primal-dual interior-point algorithm for the problem, by applying Newton's method directly to a system of nonlinear equations characterizing primal and dual feasibility and a perturbed complementarity condition. The main work at each step consists of solving a system of linear equations (the Schur complement equations). This Schur complement matrix is not symmetric, unlike in linear programming. We incorporate a Mehrotra-type predictor-corrector scheme and present some experimental results comparing several variations of the algorithm, including, as one option, explicit symmetrization of the Schur complement with a skew corrector term. We also present results obtained from a code implemented to solve large sparse problems, using a symmetrized Schur complement. This has been applied to problems arising in plastic collapse analysis, with hundreds of thousands of variables and millions of nonzeros in the constraint matrix. The algorithm typically nds accurate solutions in less than 50 iterations and determines physically meaningful solutions previously unobtainable.

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This paper treats the problem of computing the collapse state in limit analysis for a solid with a quadratic yield condition, such as, for example, the von Mises condition. After discretization with the finite element method, using divergence-free elements for the plastic flow, the kinematic formulation reduces to the problem of minimizing a sum of Euclidean vector norms, subject to a single linear constraint. This is a nonsmooth minimization problem, since many of the norms in the sum may vanish at the optimal point. Recently an efficient solution algorithm has been developed for this particular convex optimization problem in large sparse form.The approach is applied to test problems in limit analysis in two different plane models: plane strain and plates. In the first case more than 80% of the terms in the objective function are zero in the optimal solution, causing extreme ill conditioning. In the second case all terms are nonzero. In both cases the method works very well, and problems are solved which are larger by at least an order of magnitude than previously reported. The relative accuracy for the solution of the discrete problems, measured by duality gap and feasibility, is typically of the order 10 −8 .

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