## Chaotic Optimization and the Construction of Fractals: Solution of an Inverse Problem

**Giorgio Mantica***Department of Physics, Atlanta University, Atlanta, GA 30314, USA*

and*School of Physics, Georgia Institute of Technology,**Atlanta, GA 30332-0430, USA*

**Alan Sloan***School of Mathematics, Georgia Institute of Technology,**Atlanta, GA 30332, USA*

and*Iterated Systems Incorporated, 5550 P'tree Parkway, Suite 545,**Atlanta, GA 30092, USA*

#### Abstract

An *inverse problem* in fractal set construction is introduced in this paper, according to the theory of *iterated function systems* (IFS). This theory allows the construction of a class of fractals depending on a finite number of parameters. Finding a set of parameters which reconstructs a given fractal is the goal of the inverse problem. As the solution of the inverse problem generally involves a compression of the information encoded in the fractal, complexity theory is here applied. In particular, we define the *IFS-entropy* to characterize the class of fractals for which the problem can be profitably solved.

The inverse problem can be reduced to the *minimization* of a suitable function in parameter space. We describe a new algorithm to obtain a reliable minimum solution, which originates from the theory of dynamical systems. We suggest that this algorithm should greatly improve *simulated thermal annealing à la* Kirkpatrick-Szu when a metric structure can be given to an optimization problem.