MATHEMATICAL ENGINEERING TECHNICAL REPORTS Monotonicity in Steepest Ascent Algorithms for Polyhedral L-concave Functions

Hassin (1983) proposed a dual algorithm for the minimum cost flow problem, which iteratively updates dual variables in a steepest ascent manner. This algorithm is generalized to the minimum cost submodular flow problem by Chung and Tcha (1991). In discrete convex analysis, the dual of the minimum cost flow problem is known to be formulated as the maximization of a polyhedral L-concave function. It is recently pointed out that Hassin’s algorithm can be recognized as a steepest ascent algorithm for polyhedral L-concave functions. The objective of this paper is to show some nice properties of the steepest ascent algorithm for polyhedral L-concave functions. We show that the algorithm shares a monotonicity property of Hassin’s algorithm. Moreover, the algorithm finds the “nearest” optimal solution to a given initial solution, and the trajectory of the solutions generated by the algorithm is a “shortest” path from the initial solution to the “nearest” optimal solution. The algorithm and its properties can be extended for polyhedral L-concave functions.

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