### Abstract

The p-regularized subproblem (p-RS) is the key content of a regularization technique in computing a Newton-like step for unconstrained optimization. The idea is to incorporate a local quadratic approximation of the objective function with a weighted regularization term (σ/p)σXσ^{p} and then globally minimize it at each iteration. In this paper, we establish a complete theory of the p-RSs for general p>2 that covers previous known results on p=3 or p=4. The theory features necessary and sufficient optimality conditions for the global and also for the local non-global minimizers of (p-RS). It gives a closed-form expression for the global minimum set of (p-RS) and shows that (p-RS), p>2 can have at most one local non-global minimizer. Our theory indicates that (p-RS) have all properties that the trust region subproblems do. In application, (p-RS) can appear in natural formulation for optimization problems. We found two examples. One is to utilize the Tikhonov regularization to stabilize the least square solution for an over-determined linear system; and the other comes from numerical approximations to the generalized Ginzburg–Landau functionals. Moreover, when (p-RS) is appended with m additional linear inequality constraints, denoted by (p-RS_{m}), the problem becomes NP-hard. We show that the partition problem, the k-dispersion-sum problem and the quadratic assignment problem in combinatorial optimization can be equivalently formulated as special types of (p-RS_{m}) with p=4. In the end, we develop an algorithm for solving (p-RS_{m}).

Original language | English |
---|---|

Pages (from-to) | 1059-1077 |

Number of pages | 19 |

Journal | Optimization Methods and Software |

Volume | 32 |

Issue number | 5 |

DOIs | |

Publication status | Published - 2017 Sep 3 |

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### All Science Journal Classification (ASJC) codes

- Software
- Control and Optimization
- Applied Mathematics

### Cite this

*Optimization Methods and Software*,

*32*(5), 1059-1077. https://doi.org/10.1080/10556788.2016.1238917

}

*Optimization Methods and Software*, vol. 32, no. 5, pp. 1059-1077. https://doi.org/10.1080/10556788.2016.1238917

**Theory and application of p-regularized subproblems for p>2.** / Hsia, Yong; Sheu, Ruey Lin; Yuan, Ya Xiang.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Theory and application of p-regularized subproblems for p>2

AU - Hsia, Yong

AU - Sheu, Ruey Lin

AU - Yuan, Ya Xiang

PY - 2017/9/3

Y1 - 2017/9/3

N2 - The p-regularized subproblem (p-RS) is the key content of a regularization technique in computing a Newton-like step for unconstrained optimization. The idea is to incorporate a local quadratic approximation of the objective function with a weighted regularization term (σ/p)σXσp and then globally minimize it at each iteration. In this paper, we establish a complete theory of the p-RSs for general p>2 that covers previous known results on p=3 or p=4. The theory features necessary and sufficient optimality conditions for the global and also for the local non-global minimizers of (p-RS). It gives a closed-form expression for the global minimum set of (p-RS) and shows that (p-RS), p>2 can have at most one local non-global minimizer. Our theory indicates that (p-RS) have all properties that the trust region subproblems do. In application, (p-RS) can appear in natural formulation for optimization problems. We found two examples. One is to utilize the Tikhonov regularization to stabilize the least square solution for an over-determined linear system; and the other comes from numerical approximations to the generalized Ginzburg–Landau functionals. Moreover, when (p-RS) is appended with m additional linear inequality constraints, denoted by (p-RSm), the problem becomes NP-hard. We show that the partition problem, the k-dispersion-sum problem and the quadratic assignment problem in combinatorial optimization can be equivalently formulated as special types of (p-RSm) with p=4. In the end, we develop an algorithm for solving (p-RSm).

AB - The p-regularized subproblem (p-RS) is the key content of a regularization technique in computing a Newton-like step for unconstrained optimization. The idea is to incorporate a local quadratic approximation of the objective function with a weighted regularization term (σ/p)σXσp and then globally minimize it at each iteration. In this paper, we establish a complete theory of the p-RSs for general p>2 that covers previous known results on p=3 or p=4. The theory features necessary and sufficient optimality conditions for the global and also for the local non-global minimizers of (p-RS). It gives a closed-form expression for the global minimum set of (p-RS) and shows that (p-RS), p>2 can have at most one local non-global minimizer. Our theory indicates that (p-RS) have all properties that the trust region subproblems do. In application, (p-RS) can appear in natural formulation for optimization problems. We found two examples. One is to utilize the Tikhonov regularization to stabilize the least square solution for an over-determined linear system; and the other comes from numerical approximations to the generalized Ginzburg–Landau functionals. Moreover, when (p-RS) is appended with m additional linear inequality constraints, denoted by (p-RSm), the problem becomes NP-hard. We show that the partition problem, the k-dispersion-sum problem and the quadratic assignment problem in combinatorial optimization can be equivalently formulated as special types of (p-RSm) with p=4. In the end, we develop an algorithm for solving (p-RSm).

UR - http://www.scopus.com/inward/record.url?scp=84991035189&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84991035189&partnerID=8YFLogxK

U2 - 10.1080/10556788.2016.1238917

DO - 10.1080/10556788.2016.1238917

M3 - Article

AN - SCOPUS:84991035189

VL - 32

SP - 1059

EP - 1077

JO - Optimization Methods and Software

JF - Optimization Methods and Software

SN - 1055-6788

IS - 5

ER -