On Smoothing l₁ Exact Penalty Function for Constrained Optimization Problems

Abstract

This article introduces a smoothing technique to the l₁ exact penalty function. An application of the technique yields a twice continuously differentiable penalty function and a smoothed penalty problem. Under some mild conditions, the optimal solution to the smoothed penalty problem becomes an approximate optimal solution to the original constrained optimization problem. Based on the smoothed penalty problem, we propose an algorithm to solve the constrained optimization problem. Every limit point of the sequence generated by the algorithm is an optimal solution. Several numerical examples are presented to illustrate the performance of the proposed algorithm.

Keywords:

• Constrained optimization problems
• l₁ exact penalty function
• smoothing technique

Notes

1 We can prove it as follows: when t < 0, we have $p_{ϵ}(t)=p(t)=0$; when $0\le t<\frac{ϵ}{m\rho }$, it follows $\underset{\rho \to +\infty }{\lim }\frac{ϵ}{m\rho }=0$ and it implies that $\underset{\rho \to +\infty }{\lim }{p}_{ϵ}(t)=\underset{\rho \to +\infty }{\lim }{p}_{ϵ}(0)=0=p(t)$; when $t\ge \frac{ϵ}{m\rho }$, it follows $\underset{\rho \to +\infty }{\lim }{p}_{ϵ}(t)=\underset{\rho \to +\infty }{\lim }(t+\frac{3{ϵ}^2}{5m^2{\rho }^{2}t}-\frac{3ϵ}{2m\rho })=t=p(t)$.

2 We have the following clarification: $$\underset{ϵ\to 0}{\lim }{p}_{ϵ}(t)=\{\begin{array}{ll}=\underset{ϵ\to 0}{\lim }0=0=p(t)\hfill & t<0,\hfill \\ =\underset{ϵ\to 0}{\lim }(\frac{m^3{\rho }^3{t}^4}{10{ϵ}^3})=0=p(t)\hfill & 0\le t<\frac{ϵ}{m\rho },\hfill \\ =\underset{ϵ\to 0}{\lim }(t+\frac{3{ϵ}^2}{5m^2{\rho }^{2}t}-\frac{3ϵ}{2m\rho })=t=p(t)\hfill & t\ge \frac{ϵ}{m\rho }.\hfill \end{array}$$

Additional information

Funding

This study was partially supported by the Science foundation of Binzhou University with Grants [2015Y10], and GRF: PolyU [15201414] and CityU of Hong Kong SAR Government [101113].