On solving a generalized Heron problem

In this thesis, an investigation is made on the optimal solution of a generalization of the Heron problem using the instruments of convex analysis. In the first part of the thesis, the definition and historical development of the generalized Heron problem is given and its relationship with other problems is explained. In the second chapter, the basic concepts of convex analysis, including the concepts of normal cone and subdifferential, are mentioned, and definitions and theorems related to these concepts are given. In the third chapter, the sources and method used in the writing of the thesis are mentioned. In the fourth chapter of the thesis, the definitions of constrained and unconstrained optimization problems are given and the existence conditions of the optimal solution of the generalized Heron problem are given. In addition, theorems and results regarding the characterization of the optimal solution of the problem are given. In the fifth chapter, what has been done throughout the thesis is summarized in general terms and the studies on the generalized problem are mentioned. Finally, the studies that can be done about the thesis subject are mentioned.