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Nonsmooth optimization by successive quadratic programming

The problem of minimizing h(f(x)) over x a real vector of dimension n, where f is continuously differentiable into the real space of dimension m, and h is convex, is currently an important topic in optimization. We develop an algorithm that solves this problem, and which has several features that distinguish is from other algorithms that are currently being used. It uses a subproblem at each iteration that is defined in terms of approximate, instead of exact derivatives of f. It only requires approximate solution of the subproblem at each iteration. It ensures descent by means of a quadratic term instead of trust regions or line searches.

A generalization of the problem is considered where the real space of dimension m is replaced by the space of continuous functions on a compact set and h is either convex or globally Lipschitz continuous. This leads to an implementable method for minimizing the kind of penalty functions that occur in semi-infinite programming.

Input File: nsmooth.omh