On Boundedness and Solution Size in Rational Linear Programming and Polyhedral Optimization

Abstract

This paper delves into the theoretical and practical aspects of boundedness and structural properties in rational linear programming (LP) and polyhedral optimization. It provides a comprehensive analysis of conditions under which the optimization of linear functions over rational polyhedra remains bounded and establishes explicit constraints on solution size when optimal solutions exist. By exploring the interplay between polyhedral geometry, integer hulls, and rational LP systems, this study sheds light on fundamental principles that underlie modern optimization techniques. Key findings include equivalence conditions for boundedness between rational polyhedra and their integer hulls, as well as precise bounds on the numerical representation of optimal solutions. These results not only enhance the theoretical understanding of LP and polyhedral optimization but also have significant implications for computational efficiency, algorithm design, and numerical stability in solving real-world optimization problems. The discussion is rooted in rigorous mathematical foundations and extends to practical applications in areas such as mixed-integer programming, computational geometry, and combinatorial optimization.