Reverse Bridge Theorem under Constraint Partition

Abstract

Reverse bridge theorem (RBTH) has been proved to be both a necessary and sufficient condition for solving Nonlinear programming problems. In this paper, we first propose three algorithms for finding constraint minimum points of continuous, discrete, and mixed-integer nonlinear programming problems based on the reverse bridge theorem. Moreover, we prove that RBTH under constraint partition is also a necessary and sufficient condition for solving nonlinear programming problems. This property can help us to develop an algorithm using RBTH under constraints. Specifically, the algorithm first partitions mixed-integer nonlinear programming problems (MINLPs) by their constraints into some subproblems in similar forms, then solves each subproblem by using RBTH directly, and finally resolves those unsatisfied global constraints by choosing appropriate penalties. Finally, we prove the soundness and completeness of our algorithm. Experimental results also show that our algorithm is effective and sound.