### 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) |
---|---|

Pages (from-to) | 243-262 |

Number of pages | 20 |

Journal | SIAM Journal on Scientific Computing |

Volume | 22 |

Issue number | 1 |

DOIs | |

State | Published - 2000 |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

### Cite this

*SIAM Journal on Scientific Computing*,

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

**An efficient primal-dual interior-point method for minimizing a sum of Euclidean norms.** / Andersen, K. D.; Christiansen, E.; Conn, A. R.; Overton, M. L.

Research output: Contribution to journal › Article

*SIAM Journal on Scientific Computing*, vol. 22, no. 1, pp. 243-262. https://doi.org/10.1137/S1064827598343954

}

TY - JOUR

T1 - An efficient primal-dual interior-point method for minimizing a sum of Euclidean norms

AU - Andersen, K. D.

AU - Christiansen, E.

AU - Conn, A. R.

AU - Overton, M. L.

PY - 2000

Y1 - 2000

N2 - 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.

AB - 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.

KW - Duality

KW - Fermat problem

KW - Interior-point method

KW - Newton method

KW - Nonsmooth optimization

KW - Primal-dual

KW - Sum of norms

UR - http://www.scopus.com/inward/record.url?scp=0034457609&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0034457609&partnerID=8YFLogxK

U2 - 10.1137/S1064827598343954

DO - 10.1137/S1064827598343954

M3 - Article

VL - 22

SP - 243

EP - 262

JO - SIAM Journal of Scientific Computing

JF - SIAM Journal of Scientific Computing

SN - 1064-8275

IS - 1

ER -