Convex Hulls for Graphs of Quadratic Functions With Unit Coefficients: Even Wheels and Complete Split Graphs

P-split 公式：析取约束下介于 big-M 与凸包之间的一类中间公式

P-split formulations: A class of intermediate formulations between big-M and convex hull for disjunctive constraints

本研究提出介于big-M与凸包之间的P-split析取约束混合整数公式，通过拆分凸约束构造层级松弛结构，344个测试实例验证其兼顾紧致性与计算效率，性能优于两类传统公式。

Between steps: Intermediate relaxations between big-M and convex hull formulations

This work develops a class of relaxations in between the big-M and convex hull formulations of disjunctions, drawing advantages from both. The proposed "P-split" formulations split convex additively separable constraints into P partitions and form the convex hull of the partitioned disjuncts. Parameter P represents the trade-off of model size vs. relaxation strength. We examine the novel formulations and prove that, under certain assumptions, the relaxations form a hierarchy starting from a big-M equivalent and converging to the convex hull. We computationally compare the proposed formulations…

Intersection Cuts for Nonlinear Integer Programming: Convexification Techniques for Structured Sets

We study the generalization of split, k-branch split, and intersection cuts from Mixed Integer Linear Programming to the realm of Mixed Integer Nonlinear Programming. Constructing such cuts requires calculating the convex hull of the difference between a convex set and an open set with a simple geometric structure. We introduce two techniques to give precise characterizations of such convex hulls and use them to construct split, k-branch split, and intersection cuts for several classes of non-polyhedral sets. In particular, we give simple formulas for split cuts for essentially all convex sets…

Two-Term Disjunctions on the Second-Order Cone

Balas introduced disjunctive cuts in the 1970s for mixed-integer linear programs. Several recent papers have attempted to extend this work to mixed-integer conic programs. In this paper we study the structure of the convex hull of a two-term disjunction applied to the second-order cone, and develop a methodology to derive closed-form expressions for convex inequalities describing the resulting convex hull. Our approach is based on first characterizing the structure of undominated valid linear inequalities for the disjunction and then using conic duality to derive a family of convex, possibly n…

关于最小化凸Hölder光滑函数的Active-Hull方法的最优线性最小化预言复杂度

On the Optimal Linear Minimization Oracle Complexity of Active-Hull Methods for Minimizing Convex Hölder Smooth Functions

本文针对凸Hölder光滑函数的最小化问题，证明了Active-Hull方法在高维情形下的最优线性最小化预言复杂度，并提出了无参数的元参数搜索算法来实现该复杂度。

How to Convexify the Intersection of a Second Order Cone and a Nonconvex Quadratic

A recent series of papers has examined the extension of disjunctive-programming techniques to mixed-integer second-order-cone programming. For example, it has been shown---by several authors using different techniques---that the convex hull of the intersection of an ellipsoid, $E$, and a split disjunction, $(l - x_j)(x_j - u) \le 0$ with $l < u$, equals the intersection of $E$ with an additional second-order-cone representable (SOCr) set. In this paper, we study more general intersections of the form $K \cap Q$ and $K \cap Q \cap H$, where $K$ is a SOCr cone, $Q$ is a nonconvex cone defined by…