Global search method for solving Malfatti's four-circle problem

We consider Malfatti's problem formulated 200 years ago. In the beginning, Malfatti's problem was supposed to be solved in a geometric construction way. In 1994, it was done by Zalgaller and Los for the original Malfatti's problem using so-called greedy algorithm. There is still a conjecture about solving Malfatti's problem for more than four circles by the greedy algorithm. We generalize Malfatti's problem formulated for the case of three circles inscribed in a triangle for four circles. We examine six cases for inscribed circles in a triangle. The problem has been formulated as the convex maximization problem over a nonconvex set. Global optimality conditions by Strekalovsky have been applied to this problem. For solving numerically Malfatti's problem, we propose an algorithm which converges globally. Subproblems of the proposed algorithm were quadratic programming problems with quadratic constraints. These problem can be solved by Lagrangian methods. For a computational purpose, we consider a triangle with given vertices. Some computational results are provided.