An Efficient Primal-Dual Interior-Point Method for Minimizing a Sum of Euclidean Norms

Abstract

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.

Limit analysis of plates using the EFG method and second‐order cone programming

“…Of the several methods that have been developed to treat such a singularity, the primal-dual interiorpoint method proposed by Andersen et. al [19] has been found to be especially efficient. The traditional way of replacing the singular function by a differentiable one is to add a square of a fixed positive number μ to the root, so that the function becomes C T Gd 2 + μ 2 .…”

“…However, this may lead to slow convergence as μ → 0. In [19], the quantity μ is treated as an additional variable and can be determined by a duality estimate. With the use of this method, the optimization problem is solved rapidly and accurately even if there are a large number of variables and/or zero terms in the objective function.…”

“…On the other hand, the objective function in the associated optimization problem is convex, but not everywhere differentiable. One of the most efficient algorithms to overcome this difficulty is the primal-dual interior-point method presented in [19,20] and implemented in commercial codes such as the Mosek software package. The limit analysis problem involving conic constraints can then be solved by this efficient algorithm [16,21,22].…”

“…This results in a truly meshless method and reduces computational effort. Attention is also focused on formulating the plate limit analysis problem as one of minimizing a sum of Euclidean vector norms, which can be solved efficiently by a primal-dual interior-point method [19], such as Second-Order Cone Programming (SOCP). To illustrate the method it is then applied to a series of bending problems, including those for which solutions already exist in the literature.…”

Multifrequency electrical impedance tomography with total variation regularization

“…The PDIPM framework is constructed by combining the dual problem and the primal-dual gap problem, and solve these using a multi-variable Guass Newton method. The final form is: Typically, the primal variables σ are updated using a precise gradient-based algorithm, and the dual variables χ are updated typically using a simple, computationally easier, method, such as the scaling rule (Anderson et al, 1998).…”