The HTML character set lacks a lot of mathematical symbols, so I'm forced to improvise. I've also used parenthesis in places I would have preferred subscripts. Introduction Consider the problem of maximizing a linear function constrained to a fractal feasible region. In this paper we deal with the relatively simple case of iterated systems made up of affine functions. If a circle encloses a fractal defined by such mappings, the union of the images of the circle under the mappings will contain the fractal, by definition of the fractal.
Analogous things happen in higher dimensions.
It is helpful to keep this geometric image in mind for what follows. What follows is a method to reduce our optimization problem to one of optimization on a polyhedron. Since approaches to linear optimization on a polyhedron are abundant and well-studied, we will regard our problem as solved at that point. We find a hypersphere containing the fractal, then a polyhedron containing the fractal, and then a nested sequence of polyhedrons containing the fractal. We then use information from the construction of the polyhedrons to obtain a nearby feasible point.