We introduce $\mathcal{V}$-polyhedral disjunctive cuts (VPCs) for generating valid inequalities from general disjunctions. Cuts are critical to integer programming solvers, but the benefit from many families is only realized when the cuts are applied recursively, causing numerical instability and "tailing off" of cut strength after several rounds. To mitigate these difficulties, the VPC framework offers a practical method for generating strong cuts without resorting to recursion. The framework starts with a disjunction whose terms partition the feasible region into smaller subproblems, then obtains a collection of points and rays from the disjunctive terms, from which we build a linear program whose feasible solutions correspond to valid disjunctive cuts. Though a naïve implementation would result in an exponentially-sized optimization problem, we show how to efficiently construct this linear program, such that it is much smaller than the one from the alternative higher-dimensional cut-generating linear program. This enables us to test strong multiterm disjunctions that arise from the leaf nodes of a partial branch-and-bound tree. In addition to proving useful theoretical properties of the cuts, we evaluate their performance computationally through an implementation in the open-source COIN-OR framework. In the results, VPCs from a strong disjunction significantly improve the gap closed compared to existing cuts in solvers, and they also decrease some instances' solving time when used with branch and bound.
On solving integer bilevel optimization problems with a non-convex quadratic follower objective function using disjunctive cuts
Koç University（博阿兹奇大学）；University of Augsburg（奥格斯堡大学）；University of Michigan（密歇根大学）；Vienna University of Economics and Business（维也纳经济与商业大学）；Johannes Kepler University Linz（林茨约翰·开普勒大学）
Sufficient Conditions for Metric Subregularity of Constraint Systems with Applications to Disjunctive and Ortho-Disjunctive Programs
本文研究一般最优化问题的度量次正则约束规格（MSCQ），重点是非凸设置。我们详细阐述了方向伪正态性和准正态性的概念，最近由Bai等人（SIAM J. Opt.，2019年），它结合了标准的方法，通过伪和准常态与现代工具的方向变分分析。我们专注于应用程序的析取程序，其中（方向）伪正规性的特点是通过一个极值条件。这反过来又产生了有效的工具来验证伪正规性和MSCQ，其中包括但不限于Robinson关于多面体多函数的结果和Gfrerer关于度量次正则性的二阶充分条件。最后，我们完善我们的研究，通过定义一类新的ortho-disjunctive程序，其中包括突出的最优化问题，如数学规划的互补性，消失或切换约束。
Coordinate-wise Polyhedral Method for Eliciting Multivariate Linear Utility and Univariate Nonlinear Utility Functions
School of Mathematics and Statistics, Xi’an Jiaotong University（西安交通大学数学与统计学院）；Department of Systems Engineering and Engineering Management, The Chinese University of Hong Kong（香港中文大学系统科学与工程学系）
