Scanning integer points with lex-inequalities: a finite cutting plane algorithm for integer programming with linear objective

We consider the integer points in a unimodular cone K ordered by a lexicographic rule defined by a lattice basis. To each integer point x in K we associate a family of inequalities (lex-inequalities) that define the convex hull of the integer points in K that are not lexicographically smaller than x. The family of lex-inequalities contains the Chvátal–Gomory cuts, but does not contain and is not contained in the family of split cuts. This provides a finite cutting plane method to solve the integer program $$\min \{cx: x\in S\cap \mathbb {Z}^n\}$$ min { c x : x ∈ S ∩ Z n } , where $$S\subset \mathbb {R}^n$$ S ⊂ R n is a compact set and $$c\in \mathbb {Z}^n$$ c ∈ Z n . We analyze the number of iterations of our algorithm.

The Hybrid-Scan Estimators: Exploiting WSR-88D Split Cuts to Improve the Quality of Polarimetric-Variable Estimates

Since the dual-polarization upgrade of the Weather Surveillance Radar-1988 Doppler (WSR-88D), the polarimetric variables have become a fundamental tool for better interpretation and forecasting of hazardous weather events. Thus, improving their quality has been an important long-standing effort. In this paper, we introduce the hybrid-scan estimators (HSE), which use the available data in split cuts of operational volume coverage patterns (VCP) to provide better estimates of differential reflectivity, differential phase, and correlation coefficient. The HSE are designed to choose between the data provided by either one of the two scans in split cuts based on their expected statistical performance, resulting in the same or better data quality compared to the conventional estimators. The performance improvement realized with the HSE is characterized with simulations and illustrated with data from WSR-88D. While relatively simple, an operational implementation of the HSE could bring improvements to forecasters’ data interpretation and algorithm performance, both of which rely on dual-polarization radar data.