In this paper, we study properties of general closed convex sets that determine the closed-ness and polyhedrality of the convex hull of integer points contained in it. We first present necessary and sufficient conditions for the convex hull of integer points contained in a general convex set to be closed. This leads to useful results for special class of convex sets such as pointed cones, strictly convex sets, and sets containing integer points in their interior. We then present sufficient conditions for the convex hull of integer points in general convex sets to be polyhedron. These sufficient conditions generalize the sufficient conditions given in Meyer [8]. Under a simple technical condition, we show that these sufficient conditions are also necessary conditions for the convex hull of integer points contained in general convex sets to be polyhedra.

Mixed-integer conic programming is a generalization of mixed-integer linear programming. In this paper, we present an extension of the duality theory for mixed-integer linear programming (see [4], [11]) to the case of mixed-integer conic programming. In particular, we construct a subadditive dual for mixed-integer conic programming problems. Under a simple condition on the primal problem, we are able to prove strong duality.

No abstract

Objective: Attention-deficit/hyperactivity disorder (ADHD) is the most common neurodevelopmental disorder of childhood. Clinical guidelines recommend behavior therapy as the first-line treatment for preschool-age children with ADHD. This study evaluated longitudinal patterns of services received by Medicaid-enrolled children ages 2 to 5 with ADHD in seven southeastern states (Alabama, Florida, Georgia, Louisiana, Mississippi, North Carolina, and South Carolina). Methods: A discrete sequence clustering analysis was used with 2005–2012 Medicaid Analytic eXtract data to profile patient-level utilization for each state, with a focus on receipt of psychological services and medication. The model output was used to assess utilization behaviors longitudinally relative to recommended care guidelines and to characterize sources of variation in utilization patterns by demographic and ecological factors. Results: Five states had a utilization profile with a high probability of receipt of psychological services before medication among children with ADHD, covering 16% of the total study population. Most young children’s ADHD care experience in the seven states (65%) fit utilization profiles characterized by a high probability of receiving any ADHD medication. Black race was significantly associated with higher utilization of psychological services in three states. Conclusions: About 16% of Medicaid-enrolled preschool-age children with ADHD received care during 2005–2012 that appeared to be consistent with 2011 recommended care guidelines. State-level and subpopulation variations in utilization for ADHD-related clinical care were found. The findings indicate that there were major gaps in treatment for ADHD among young children and that the gaps are wider for some states and subpopulations of children.