For two matroids \(M_1\) and \(M_2\) with the same ground set V and two cost functions \(w_1\) and \(w_2\) on \(2^V\), we consider the problem of finding bases \(X_1\) of \(M_1\) and \(X_2\) of \(M_2\) minimizing \(w_1(X_1)+w_2(X_2)\) subject to a certain cardinality constraint on their intersection \(X_1 \cap X_2\). For this problem, Lendl et al. (Matroid bases with cardinality constraints on the intersection, arXiv:1907.04741v2, 2019) discussed modular cost functions: they reduced the problem to weighted matroid intersection for the case where the cardinality constraint is \(|X_1 \cap X_2|\le k\) or \(|X_1 \cap X_2|\ge k\); and designed a new primal-dual algorithm for the case where the constraint is \(|X_1 \cap X_2|=k\). The aim of this paper is to generalize the problems to have nonlinear convex cost functions, and to comprehend them from the viewpoint of discrete convex analysis. We prove that each generalized problem can be solved via valuated independent assignment, valuated matroid intersection, or \(\mathrm {M}\)-convex submodular flow, to offer a comprehensive understanding of weighted matroid intersection with intersection constraints. We also show the NP-hardness of some variants of these problems, which clarifies the coverage of discrete convex analysis for those problems. Finally, we present applications of our generalized problems in the recoverable robust matroid basis problem, combinatorial optimization problems with interaction costs, and matroid congestion games.

对于两个具有相同地面集合V的拟阵拟定\ {M_1 \}和\（M_2 \）以及\（2 ^ V \）上的两个成本函数\（w_1 \）和\（w_2 \），我们考虑寻找问题碱\（X_1 \）的\（M_1 \）和\（X_2 \）的\（M_2 \）最小化\（W_1（X_1）+ W_2（X_2）\）在其交叉点有一定基数约束受试者\（X_1 \ cap X_2 \）。对于这个问题，Lendl等人。（在交叉点上具有基数约束的Matroid基础，arXiv：1907.04741v2，2019）讨论了模块化成本函数：对于基数约束为\（| X_1 \ cap X_2 | \ le k \ ）或\（| X_1 \ cap X_2 | \ ge k \） ; 并针对约束为\（| X_1 \ cap X_2 | = k \）的情况设计了新的原始对偶算法。本文的目的是推广具有非线性凸成本函数的问题，并从离散凸分析的角度理解这些问题。我们证明可以通过评估独立分配，评估拟阵交点或\（\ mathrm {M} \）来解决每个广义问题-凸次模流，以提供对具有交点约束的加权拟阵交点的全面理解。我们还显示了这些问题的某些变体的NP硬度，从而阐明了这些问题的离散凸分析的范围。最后，我们介绍了广义问题在可恢复的鲁棒拟阵基础问题，具有交互成本的组合优化问题以及拟阵拥塞博弈中的应用。