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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.
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.