Alternative algorithms for solving nonlinear function and functional inequalities

Abstract

Several sophisticated and efficient algorithms are now available in the literature for solving nonlinear function inequalities in a finite number of iterations. This paper addresses an alternative approach to this class of problems. The approach is relatively straightforward and easy to use. Essentially, the nonlinear inequality constrained problem is reformulated as a standard unconstrained optimization problem via a differentiable transcription. Thus, any existing efficient unconstrained optimization software packages can be used to solve the corresponding unconstrained optimization problem. Due to the special structure of the differentiable transcription, a solution of the nonlinear inequality constraints can be obtained after a finite number of iterations in the process of solving the unconstrained optimization problem. The main aim of this paper is, however, to extend the approach to solving nonlinear functional, rather than function, inequalities in a finite number of iterations. For illustration, several examples are computed to demonstrate the feasibility and versatility of the approach.