2014/08/29 by Fatma Kılınç-Karzan, Kılınç-Karzan, Fatma · 4 citations
Computer Science · Engineering · Mathematics · #Advanced Optimization Algorithms Research #Combinatorics #Cone (formal languages) #Conic section #Connection (principal bundle) #Convex function #Discrete mathematics #FOS: Mathematics #Geometry #Inequality #Integer (computer science) #Linear inequality #Mathematical analysis #Mathematics #Optimization and Control (math.OC) #Optimization and Mathematical Programming #Optimization and Variational Analysis #Orthant #Pure mathematics #Regular polygon #Sublinear function #math.OC
paper · pdf · doi:10.48550/arxiv.1408.6922
published in arXiv (Cornell University) (Cornell University)
openalex publication_date 2014/08/29 · arxiv created 2015/04/01 · arxiv updated 2015/04/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/06
We study disjunctive conic sets involving a general regular (closed, convex, full dimensional, and pointed) cone K such as the nonnegative orthant, the Lorentz cone or the positive semidefinite cone. In a unified framework, we introduce K-minimal inequalities and show that under mild assumptions, these inequalities together with the trivial cone-implied inequalities are sufficient to describe the convex hull. We study the properties of K-minimal inequalities by establishing algebraic necessary conditions for an inequality to be K-minimal. This characterization leads to a broader algebraically defined class of K- sublinear inequalities. We establish a close connection between K-sublinear inequalities and the support functions of sets with a particular structure. This connection results in practical ways of showing that a given inequality is K-sublinear and K-minimal. Our framework generalizes some of the results from the mixed integer linear case. It is well known that the minimal inequalities for mixed integer linear programs are generated by sublinear (positively homogeneous, subadditive and convex) functions that are also piecewise linear. This result is easily recovered by our analysis. Whenever possible we highlight the connections to the existing literature. However, our study unveils that such a cut generating function view treating the data associated with each individual variable independently is not possible in the case of general cones other than nonnegative orthant, even when the cone involved is the Lorentz cone.