A model algorithm for composite nondifferentiable optimization problems

Composite functions ϕ(x)=f(x)+h(c(x)), where f and c are smooth and h is convex, encompass many nondifferentiable optimization problems of interest including exact penalty functions in nonlinear programming, nonlinear min-max problems, best nonlinear L 1, L 2 and L ∞ approximation and finding feasible points of nonlinear inequalities. The idea is used of making a linear approximation to c(x) whilst including second order terms in a quadratic approximation to f(x). This is used to determine a composite function ψ which approximates ϕ(x) and a basic algorithm is proposed in which ψ is minimized on each iteration. If the technique of a step restriction (or trust region) is incorporated into the algorithm, then it is shown that global convergence can be proved. It is also described briefly how the above approximations ensure that a second order rate of convergence is achieved by the basic algorithm.