### Abstract

In this paper, we study a proximal-like algorithm for minimizing a closed proper function f(x) subject to x^{3}0, based on the iterative scheme: x^{k} ε argmin{f(x) + μkd(x, x^{k-1})}, where d( , ) is an entropy-like distance function. The algorithm is well-defined under the assumption that the problem has a nonempty and bounded solution set. If, in addition, f is a differentiable quasi-convex function (or f is a differentiable function which is homogeneous with respect to a solution), we show that the sequence generated by the algorithm is convergent (or bounded), and furthermore, it converges to a solution of the problem (or every accumulation point is a solution of the problem) when the parameter μ^{k} approaches to zero. Preliminary numerical results are also reported, which further verify the theoretical results obtained.

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

Pages (from-to) | 319-333 |

Number of pages | 15 |

Journal | Pacific Journal of Optimization |

Volume | 4 |

Issue number | 2 |

Publication status | Published - 2008 May 1 |

### Fingerprint

### Keywords

- Entropy-like distance
- Homogeneous
- Proximal algorithm
- Quasi-convex

### ASJC Scopus subject areas

- Control and Optimization
- Computational Mathematics
- Applied Mathematics

### Cite this

*Pacific Journal of Optimization*,

*4*(2), 319-333.

**A proximal-like algorithm for a class of nonconvex programming.** / Chen, Jein Shan; Pan, Shaohua.

Research output: Contribution to journal › Article

*Pacific Journal of Optimization*, vol. 4, no. 2, pp. 319-333.

}

TY - JOUR

T1 - A proximal-like algorithm for a class of nonconvex programming

AU - Chen, Jein Shan

AU - Pan, Shaohua

PY - 2008/5/1

Y1 - 2008/5/1

N2 - In this paper, we study a proximal-like algorithm for minimizing a closed proper function f(x) subject to x30, based on the iterative scheme: xk ε argmin{f(x) + μkd(x, xk-1)}, where d( , ) is an entropy-like distance function. The algorithm is well-defined under the assumption that the problem has a nonempty and bounded solution set. If, in addition, f is a differentiable quasi-convex function (or f is a differentiable function which is homogeneous with respect to a solution), we show that the sequence generated by the algorithm is convergent (or bounded), and furthermore, it converges to a solution of the problem (or every accumulation point is a solution of the problem) when the parameter μk approaches to zero. Preliminary numerical results are also reported, which further verify the theoretical results obtained.

AB - In this paper, we study a proximal-like algorithm for minimizing a closed proper function f(x) subject to x30, based on the iterative scheme: xk ε argmin{f(x) + μkd(x, xk-1)}, where d( , ) is an entropy-like distance function. The algorithm is well-defined under the assumption that the problem has a nonempty and bounded solution set. If, in addition, f is a differentiable quasi-convex function (or f is a differentiable function which is homogeneous with respect to a solution), we show that the sequence generated by the algorithm is convergent (or bounded), and furthermore, it converges to a solution of the problem (or every accumulation point is a solution of the problem) when the parameter μk approaches to zero. Preliminary numerical results are also reported, which further verify the theoretical results obtained.

KW - Entropy-like distance

KW - Homogeneous

KW - Proximal algorithm

KW - Quasi-convex

UR - http://www.scopus.com/inward/record.url?scp=79958202210&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=79958202210&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:79958202210

VL - 4

SP - 319

EP - 333

JO - Pacific Journal of Optimization

JF - Pacific Journal of Optimization

SN - 1348-9151

IS - 2

ER -