In the monotone integer dualization problem, we are given two sets of vectors in an integer box such that no vector in the first set is dominated by a vector in the second. The question is to check if the two sets of vectors cover the entire integer box by upward and downward domination, respectively. It is known that the problem is (quasi-)polynomially equivalent to that of enumerating all maximal feasible solutions of a given monotone system of linear/separable/supermodular inequalities over integer vectors. The equivalence is established via showing that the dual family of minimal infeasible vectors has size bounded by a (quasi-)polynomial in the sizes of the family to be generated and the input description. Continuing in this line of work, in this paper, we consider systems of polynomial, second-order cone, and semidefinite inequalities. We give sufficient conditions under which such bounds can be established and highlight some applications.

翻译：在单调整数对偶化问题中，我们给定整数箱中的两个向量集合，使得第一个集合中的任意向量均不被第二个集合中的向量所支配。问题在于检验这两个向量集合是否分别通过向上支配和向下支配覆盖整个整数箱。已知该问题与枚举给定单调线性/可分离/超模不等式组在整数向量上所有最大可行解的问题是（拟）多项式等价的。这种等价性通过证明最小不可行向量的对偶族大小受限于待生成族规模与输入描述的（拟）多项式上界而建立。沿着这一研究方向，本文考虑多项式、二阶锥及半定不等式系统。我们给出能够建立此类上界的充分条件，并重点说明若干应用场景。