Optimization Strategies for Optimization Procedures and Algorithms

New theory of circle patterns (or packing) is now well developed. Uniformization result, the circle packing theorem [1], is based on analytic functions, while Doyle spirals are analoques of entire functions. Doyle constructed entire immersed hexagonal circle patterns analoques to the exponential map and conjectured that these immersed packings which came to be known as Doyle spirals, are the only entire immersed hexagonal circle packings [2, 3]. Schramm had found that a framework of circle patterns based on the square grid which is called SG patterns, is more tractable than the traditional hexagonal patterns [4]. We present a new approach concerned with a theory of discrete multistage optimization [5, 6]. Optimization strategies for optimization procedures are good tools for modelling. Consequently calculations are not far from empirical observations and experimental results. This clearly demonstrates the broad interest in our field and the role which mathematical modelling and analysis is expected to play with regard to the solution of the challenging problems that we are facing at the beginning of this millennium.