Fixed-point iterative approach for solving linear Diophantine systems with bounds on the variables

ABSTRACT

Systems of linear Diophantine equations arise from several applications. Scholars have given attention to such systems and come up with several effective solutions. A new approach, called the fixed-point iterative method, was proposed to solve linear Diophantine equations with lower and upper bounds on the variables. Two steps are involved in solving this problem. First, the problem is transformed into a polytope judgment problem . Then, the approach is used to judge the existence of an integer point in the polytope. Compared with the branch-and-bound method, results show that the approach is feasible and effective for solving linear Diophantine systems.

KEYWORDS:

• linear Diophantine equation
• fixed-point iterative method
• lattice basis reduction
• integer programming

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work was supported in part by the NSFC under grant nos. 61803279, 71471091 and 62003231, in part by the Natural Science Foundation of Jiangsu Province under grants no. BK20200989, in part by the Qing Lan Project of Jiangsu, in part by the China Postdoctoral Science Foundation under Grant no. 2020M671596 and 2021M692369.