Optimization Problems Based on the Minimization of Sub-modular Functions

The idea of the duality gap of optimization problem being rewritten into an integral form of a function is put forth so that the problem is converted to a minimization problem of a sub-modular function. Then by use of Lovász continuation the regularization is achieved. And the method for the construction of sub-modular function regarding the optimization problem is discussed on the basis of the proximal approach and polyhedron thus it is theoretically proven that there exists an equivalent relation between the optimization problem and the sub-modular minimization problem.