Convex hull characterizations of lexicographic orderings

Abstract

Given a p-dimensional nonnegative, integral vector α, this paper characterizes the convex hull of the set S of nonnegative, integral vectors x that is lexicographically less than or equal to α. To obtain a finite number of elements in S, the vectors x are restricted to be component-wise upper-bounded by an integral vector u. We show that a linear number of facets is sufficient to describe the convex hull. For the special case in which every entry of u takes the same value (n − 1) for some integer n ≥ 2, the convex hull of the set of n-ary vectors results. Our facets generalize the known family of cover inequalities for the n = 2 binary case. They allow for advances relative to both the modeling of integer variables using base-n expansions, and the solving of n-ary knapsack problems having weakly super-decreasing coefficients.

Design and implementation of virtual fitting system based on gesture recognition and clothing transfer algorithm

“…Then the hand contour is extracted from the binarization image. The barycenter method was used to optimize the traditional Graham scanning algorithm 31 , and the convex envelope points, also known as fingertip candidate points, were calculated from the previously obtained contour points. The contour analysis method based on curvature feature is further used to identify the fingertip 32 .According to the number of fingertips, and combined with the area ratio method to determine grasping gesture.…”

An Improved Unbounded-DP Algorithm for the Unbounded Knapsack Problem with Bounded Coefficients

“…The second direction utilizes mathematical tools, including number theory, to enhance the speed of basic dynamic programming (BDP) and reduce the pseudo-polynomial time complexity, such as in [26][27][28][29]. The third direction involves adding as few and as general constraints as possible to transform N P-hard problems into P, such as utilizing specific data features [30][31][32][33][34] or fixing the number of variables [35][36][37][38][39].…”

Community Participation Strategy for Sustainable Urban Regeneration in Xiamen, China

“…The second direction utilizes mathematical tools, including number theory, to enhance the speed of the basic dynamic programming(BDP) and reduce the pseudo-polynomial time complexity, such as in [24,25,26,50]. The third direction involves adding as few and as general constraints as possible to transform N P-hard problems into P, such as utilizing specific data features [28,29,30,31,32] or fixing the number of variables [33,34,35,36,37].…”