Objective penalty function method for nonlinear programming with inequality constraints

Zhuolin Yan; Xiaowei Jiang; Siyao Wang; Zhuolin Yan; Xiaowei Jiang; Siyao Wang

doi:10.3934/math.20241602

AIMS Mathematics

2024, Volume 9, Issue 12: 33572-33590. doi: 10.3934/math.20241602

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Research article

Objective penalty function method for nonlinear programming with inequality constraints

School of Mathematics and Statistics, Beihua University, Jilin, Jilin, 132013, China

Received: 01 October 2024 Revised: 17 November 2024 Accepted: 20 November 2024 Published: 27 November 2024
MSC : 90C30, 65K05

This paper presents a novel nonsmooth objective penalty function for inequality constrained optimization problems. A modified flattened aggregate function, which is a smooth approximation of the max-value function, is discussed. Then, the smooth objective penalty function that contains the flattened aggregate function is proposed, and the exactness of the new function is studied. Based on this, an objective penalty function algorithm is proposed and its convergence is proven under mild conditions. Because of the flattened aggregate function, the gradient computation can usually be greatly reduced for problems with many constraints. Numerical experiments are included to illustrate the efficiency of the new algorithm through a series of numerical examples, especially for solving problems with many constraints.
- nonlinear programming,
- inequality constraint,
- exact penalty function,
- objective penalty function,
- flattened aggregate function
Citation: Zhuolin Yan, Xiaowei Jiang, Siyao Wang. Objective penalty function method for nonlinear programming with inequality constraints[J]. AIMS Mathematics, 2024, 9(12): 33572-33590. doi: 10.3934/math.20241602

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Abstract

This paper presents a novel nonsmooth objective penalty function for inequality constrained optimization problems. A modified flattened aggregate function, which is a smooth approximation of the max-value function, is discussed. Then, the smooth objective penalty function that contains the flattened aggregate function is proposed, and the exactness of the new function is studied. Based on this, an objective penalty function algorithm is proposed and its convergence is proven under mild conditions. Because of the flattened aggregate function, the gradient computation can usually be greatly reduced for problems with many constraints. Numerical experiments are included to illustrate the efficiency of the new algorithm through a series of numerical examples, especially for solving problems with many constraints.

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