Subgradient Optimization Methods in Integer Programming

The monograph deals with analysis and methods of subgradient optimization in integer programming. In it, various versions of subgradient procedures are described and their drawback, vis-ŕ-vis, the zigzagging trend that may slow their convergence is analyzed. To overcome the drawback, a new zigzag free subgradient procedure is designed and presented. Procedures that can be used to construct both primal and dual solutions within the subgradient schemes are also illustrated. Finally, the methods are applied to a problem of optimal radiation therapy planning. In particular, the problem of minimizing total delivery time of a given radiation dose to a cancer patient is discussed. In fact, as the problem is NP-hard, there exists thus far no exact method to solve it to optimality. Dealing with this important problem further, the work presents a new and efficient algorithm that can find a very good approximate solution by combining exact and heuristic procedures. The methods and analysis are well explained and should be useful to professionals, researchers and graduate students in diverse areas such as Optimization, Computational mathematics, Radiation oncology, and Engineering.