### 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 different kinds of modern scientific 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-correctly 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 correctly 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 finds accurate solutions in less than 50 iterations and determines physically meaningful solutions previously unobtainable.

Original language | English (US) |
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Pages (from-to) | 243-262 |

Number of pages | 20 |

Journal | SIAM Journal on Scientific Computing |

Volume | 22 |

Issue number | 1 |

DOIs | |

State | Published - 2000 |

### Keywords

- Duality
- Fermat problem
- Interior-point method
- Newton method
- Nonsmooth optimization
- Primal-dual
- Sum of norms

### ASJC Scopus subject areas

- Computational Mathematics
- Applied Mathematics

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## Cite this

*SIAM Journal on Scientific Computing*,

*22*(1), 243-262. https://doi.org/10.1137/S1064827598343954