A spline smoothing homotopy method for nonconvex nonlinear programming

Abstract

Homotopy methods are globally convergent under weak conditions and robust; however, the efficiency of a homotopy method is closely related with the construction of the homotopy map and the path tracing algorithm. Different homotopies may behave very different in performance even though they are all theoretically convergent. In this paper, a spline smoothing homotopy method for nonconvex nonlinear programming is developed using cubic spline to smooth the max function of the constraints of nonlinear programming. Some properties of spline smoothing function are discussed and the global convergence of spline smoothing homotopy under the weak normal cone condition is proven. The spline smoothing technique uses a smooth constraint instead of m constraints and acts also as an active set technique. So the spline smoothing homotopy method is more efficient than previous homotopy methods like combined homotopy interior point method, aggregate constraint homotopy method and other probability one homotopy methods. Numerical tests with the comparisons to some other methods show that the new method is very efficient for nonlinear programming with large number of complicated constraints.

Keywords:

• nonlinear programming
• spline function
• homotopy method
• interior-point method

AMS Subject Classifications:

• 90C30
• 49M37

Notes

No potential conflict of interest was reported by the authors.

Additional information

Funding

The research was supported by the National Natural Science Foundation of China [grant number 11171051], [grant number 91230103]; the Fundamental Research Funds for the Central Universities [DC201502050408]; the Doctoral Starting up Foundation of Dalian Nationalities University, China [0701110100].