Computation of maximal turning points to nonlinear equations by nonsmooth optimization

Abstract

We develop a new approach to finding bifurcations of solutions of nonlinear equation systems, which is based on the detection of extreme values of a new type of variational functional associated with the considered problems. The variational functional is obtained constructively by the extended functional method which can be applied to a wide class of parametric problems including nonlinear partial differential equations. Sufficient and necessary conditions for the existence of a maximal turning point by the approach are proved. Based on these, an algorithm of the quasi-direction of steepest ascent is introduced. Simulation experiments are used to illustrate the behaviour of the method and to discuss its advantages and disadvantages in comparison with the alternatives.

Keywords:

• turning point
• steepest ascent direction
• nonlinear system
• nonsmooth optimization
• quadratic programming problem

AMS Subject Classifications:

• 37M20
• 49K35
• 65K10
• 65P30
• 90C47

Disclosure statement

No potential conflict of interest was reported by the authors.

ORCID

Yavdat Il'yasov  http://orcid.org/0000-0002-6310-9164

Aleksandr Ivanov  http://orcid.org/0000-0002-5866-7452

Additional information

Funding

Y. Il'yasov was partially supported by grant RFBR 14-01-00736-p-a. A. Ivanov was partially supported by grant RFBR 13-01-00294-p-a.