Mixing sets linked by bidirected paths

Recently there has been considerable research on simple mixed-integer sets, called mixing sets, and closely related sets arising in uncapacitated and constant capacity lot- sizing. This in turn has led to study of more general sets, called network-dual sets, for which it is possible to derive extended formulations whose projection gives the convex hull of the network-dual set. Unfortunately this formulation cannot be used (in general) to optimize in polynomial time. Furthermore the inequalities definining the convex hull of a network-dual set in the original space of variables are known only for some special cases. Here we study two new cases, in which the continuous variables of the network-dual set are linked by a bi- directed path. In the first case, which is motivated by lot-sizing problems with (lost) sales, we provide a description of the convex hull as the intersection of the convex hulls of 2^n mixing sets, where n is the number of continuous variables of the set. However optimization is polynomial as only n + 1 of the sets are required for any given objective function. In the second case, generalizing single arc flow sets, we describe again the convex hull as an intersection of an exponential number of mixing sets and also give a combinatorial polynomial-time separation algorithm.mixing sets, extended formulations, mixed integer programming, lot-sizing with sales

Lot-sizing with stock upper bounds and fixed charges

Here we study the discrete lot-sizing problem with an initial stock variable and an associated variable upper bound constraint. This problem is of interest in its own right, and is also a natural relaxation of the constant capacity lot-sizing problem with upper bounds and fixed charges on the stock variables. We show that the convex hull of solutions of the discrete lot-sizing problem is obtained as the intersection of two simpler sets, one involving just 0-1 variables and the second a mixing set with a variable upper bound constraint. For these two sets we derive both inequality descriptions and polynomial-size extended formulations of their respective convex hulls. Finally we carry out some limited computational tests on single-item constant capacity lot-sizing problems with upper bounds and fixed charges on the stock variables in which we use the extended formulations derived above to strengthen the initial mixed integer programming formulations.mixed integer programming, discrete lot-sizing, stock fixed costs, mixing sets

On largest volume simplices and sub-determinants

We show that the problem of finding the simplex of largest volume in the convex hull of $n$ points in $\mathbb{Q}^d$ can be approximated with a factor of $O(\log d)^{d/2}$ in polynomial time. This improves upon the previously best known approximation guarantee of $d^{(d-1)/2}$ by Khachiyan. On the other hand, we show that there exists a constant $c>1$ such that this problem cannot be approximated with a factor of $c^d$, unless $P=NP$. % This improves over the $1.09$ inapproximability that was previously known. Our hardness result holds even if $n = O(d)$, in which case there exists a \bar c\,^{d}-approximation algorithm that relies on recent sampling techniques, where $\bar c$ is again a constant. We show that similar results hold for the problem of finding the largest absolute value of a subdeterminant of a $d\times n$ matrix

On the convergence of the affine hull of the Chv\'atal-Gomory closures

Given an integral polyhedron P and a rational polyhedron Q living in the same n-dimensional space and containing the same integer points as P, we investigate how many iterations of the Chv\'atal-Gomory closure operator have to be performed on Q to obtain a polyhedron contained in the affine hull of P. We show that if P contains an integer point in its relative interior, then such a number of iterations can be bounded by a function depending only on n. On the other hand, we prove that if P is not full-dimensional and does not contain any integer point in its relative interior, then no finite bound on the number of iterations exists.Comment: 13 pages, 2 figures - the introduction has been extended and an extra chapter has been adde

Characterization of convection-related parameters by Raman lidar: Selected case studies from the convective and orographically-induced precipitation study

An approach to determine the convective available potential energy (CAPE) and the convective inhibition (CIN) based on the use of data from a Raman lidar system is illustrated in this work. The use of Raman lidar data allows to provide high temporal resolution measurements (5 min) of CAPE and CIN and follow their evolution over extended time periods covering the full cycle of convective activity. Lidar-based measurements of CAPE and CIN are obtained from Raman lidar measurements of the temperature and water vapor mixing ratio profiles and the surface measurements of temperature, pressure and dew point temperature provided by a surface weather station. The approach is applied to the data collected by the Raman lidar system BASIL in the frame of COPS. Attention was focused on 15 July and 25-26 July 2007. Lidar-based measurements are in good agreement with simultaneous measurements from radiosondes and with estimates from different mesoscale models

Atmospheric Thermodynamic Profiling through the Use of a Micro-Pulse Raman Lidar System: Introducing the Compact Raman Lidar MARCO

It was for a long time believed that lidar systems based on the use of high-repetition micro-pulse lasers could be effectively used to only stimulate atmospheric elastic backscatter echoes, and thus were only exploited in elastic backscatter lidar systems. Their application to stimulate rotational and roto-vibrational Raman echoes, and consequently, their exploitation in atmospheric thermodynamic profiling, was considered not feasible based on the technical specifications possessed by these laser sources until a few years ago. However, recent technological advances in the design and development of micro-pulse lasers, presently achieving high UV average powers (1–5 W) and small divergences (0.3–0.5 mrad), in combination with the use of large aperture telescopes (0.3–0.4 m diameter primary mirrors), allow one to presently develop micro-pulse laser-based Raman lidars capable of measuring the vertical profiles of atmospheric thermodynamic parameters, namely water vapor and temperature, both in the daytime and night-time. This paper is aimed at demonstrating the feasibility of these measurements and at illustrating and discussing the high achievable performance level, with a specific focus on water vapor profile measurements. The technical solutions identified in the design of the lidar system and their technological implementation within the experimental setup of the lidar prototype are also carefully illustrated and discussed

Characterization of the planetary boundary layer height and structure by Raman lidar: comparison of different approaches

Abstract. The planetary boundary layer (PBL) includes the portion of the atmosphere which is directly influenced by the presence of the earth's surface. Aerosol particles trapped within the PBL can be used as tracers to study the boundary-layer vertical structure and time variability. As a result of this, elastic backscatter signals collected by lidar systems can be used to determine the height and the internal structure of the PBL. The present analysis considers three different methods to estimate the PBL height. The first method is based on the determination of the first-order derivative of the logarithm of the range-corrected elastic lidar signals. Estimates of the PBL height for specific case studies obtained through this approach are compared with simultaneous estimates from the potential temperature profiles measured by radiosondes launched simultaneously to lidar operation. Additional estimates of the boundary layer height are based on the determination of the first-order derivative of the range-corrected rotational Raman lidar signals. This latter approach results to be successfully applicable also in the afternoon–evening decaying phase of the PBL, when the effectiveness of the approach based on the elastic lidar signals may be compromised or altered by the presence of the residual layer. Results from these different approaches are compared and discussed in the paper, with a specific focus on selected case studies collected by the University of Basilicata Raman lidar system BASIL during the Convective and Orographically-induced Precipitation Study (COPS)