An interior point algorithm for minimum sum-of-squares clustering

Copyright @ 2000 SIAM PublicationsAn exact algorithm is proposed for minimum sum-of-squares nonhierarchical clustering, i.e., for partitioning a given set of points from a Euclidean m-space into a given number of clusters in order to minimize the sum of squared distances from all points to the centroid of the cluster to which they belong. This problem is expressed as a constrained hyperbolic program in 0-1 variables. The resolution method combines an interior point algorithm, i.e., a weighted analytic center column generation method, with branch-and-bound. The auxiliary problem of determining the entering column (i.e., the oracle) is an unconstrained hyperbolic program in 0-1 variables with a quadratic numerator and linear denominator. It is solved through a sequence of unconstrained quadratic programs in 0-1 variables. To accelerate resolution, variable neighborhood search heuristics are used both to get a good initial solution and to solve quickly the auxiliary problem as long as global optimality is not reached. Estimated bounds for the dual variables are deduced from the heuristic solution and used in the resolution process as a trust region. Proved minimum sum-of-squares partitions are determined for the rst time for several fairly large data sets from the literature, including Fisher's 150 iris.This research was supported by the Fonds National de la Recherche Scientifique Suisse, NSERC-Canada, and FCAR-Quebec

Integrated Gate and Bus Assignment at Amsterdam Airport Schiphol

Abstract. At an airport a series of assignment problems need to be solved before aircraft can arrive and depart and passengers can embark and disembark. A lot of different parties are involved with this, each of which having to plan their own schedule. Two of the assignment problems that the ’Regie ’ at Amsterdam Airport Schiphol (AAS) is responsible for, are the gate assignment problem (i.e. where to place which aircraft) and the bus assignment problem (i.e. which bus will transport which passen-gers to or from the aircraft). Currently these two problems are solved in a sequential fashion, the output of the gate assignment problem is used as input for the bus assignment problem. We look at integrating these two sequential problems into one larger problem that considers both prob-lems at the same time. This creates the possibility of using information regarding the bus assignment problem while solving the gate assignment problem. We developed a column generation algorithm for this problem and have implemented a prototype. To make the algorithm efficient we used a special technique called stabilized column generation and also col-umn deletion. Computational experiments with real-life data from AAS indicate that our algorithm is able to compute a planning for one day at Schiphol in a reasonable time

Fundamental properties of the Population II fiducial stars HD 122563 and Gmb 1830 from CHARA interferometric observations

We have determined the angular diameters of two metal-poor stars, HD 122563 and Gmb 1830, using CHARA and Palomar Testbed Interferometer observations. For the giant star HD 122563, we derive an angular diameter theta_3D = 0.940 +- 0.011 milliarcseconds (mas) using limb-darkening from 3D convection simulations and for the dwarf star Gmb 1830 (HD 103095) we obtain a 1D limb-darkened angular diameter theta_1D = 0.679 +- 0.007 mas. Coupling the angular diameters with photometry yields effective temperatures with precisions better than 55 K (Teff = 4598 +- 41 K and 4818 +- 54 K --- for the giant and the dwarf star, respectively). Including their distances results in very well-determined luminosities and radii (L = 230 +- 6 L_sun, R = 23.9 +- 1.9 R_sun and L = 0.213 +- 0.002 L_sun, R = 0.664 +- 0.015 R_sun, respectively). We used the CESAM2k stellar structure and evolution code in order to produce models that fit the observational data. We found values of the mixing-length parameter alpha (which describes 1D convection) that depend on the mass of the star. The masses were determined from the models with precisions of <3% and with the well-measured radii excellent constraints on the surface gravity are obtained (log g = 1.60 +- 0.04, 4.59 +- 0.02, respectively). The very small errors on both log g and Teff provide stringent constraints for spectroscopic analyses given the sensitivity of abundances to both of these values. The precise determination of Teff for the two stars brings into question the photometric scales for metal-poor stars.Comment: accepted A&A, 8 dbl-column pages, incl. 7 tables and 4 figure

Bioavailable Trace Metals in Neurological Diseases

Medical treatment in Wilson’s disease includes chelators (d-penicillamine and trientine) or zinc salts that have to be maintain all the lifelong. This pharmacological treatment is categorised into two phases; the first being a de-coppering phase and the second a maintenance one. The best therapeutic approach remains controversial, as only a few non-controlled trials have compared these treatments. During the initial phase, progressive increase of chelators’ doses adjusted to exchangeable copper and urinary copper might help to avoid neurological deterioration. Liver transplantation is indicated in acute fulminant liver failure and decompensated cirrhosis; in cases of neurologic deterioration, it must be individually discussed. During the maintenance phase, the most important challenge is to obtain a good adherence to lifelong medical therapy. Neurodegenerative diseases that lead to a mislocalisation of iron can be caused by a culmination of localised overload (pro-oxidant siderosis) and localised deficiency (metabolic distress). A new therapeutic concept with conservative iron chelation rescues iron-overloaded neurons by scavenging labile iron and, by delivering this chelated metal to endogenous apo-transferrin, allows iron redistribution to avoid systemic loss of iron

On generalized surrogate duality in mixed-integer nonlinear programming

The most important ingredient for solving mixed-integer nonlinear programs (MINLPs) to global -optimality with spatial branch and bound is a tight, computationally tractable relaxation. Due to both theoretical and practical considerations, relaxations of MINLPs are usually required to be convex. Nonetheless, current optimization solvers can often successfully handle a moderate presence of nonconvexities, which opens the door for the use of potentially tighter nonconvex relaxations. In this work, we exploit this fact and make use of a nonconvex relaxation obtained via aggregation of constraints: a surrogate relaxation. These relaxations were actively studied for linear integer programs in the 70s and 80s, but they have been scarcely considered since. We revisit these relaxations in an MINLP setting and show the computational benefits and challenges they can have. Additionally, we study a generalization of such relaxation that allows for multiple aggregations simultaneously and present the first algorithm that is capable of computing the best set of aggregations. We propose a multitude of computational enhancements for improving its practical performance and evaluate the algorithm’s ability to generate strong dual bounds through extensive computational experiments

Large-scale optimization with the primal-dual column generation method

The primal-dual column generation method (PDCGM) is a general-purpose column generation technique that relies on the primal-dual interior point method to solve the restricted master problems. The use of this interior point method variant allows to obtain suboptimal and well-centered dual solutions which naturally stabilizes the column generation. As recently presented in the literature, reductions in the number of calls to the oracle and in the CPU times are typically observed when compared to the standard column generation, which relies on extreme optimal dual solutions. However, these results are based on relatively small problems obtained from linear relaxations of combinatorial applications. In this paper, we investigate the behaviour of the PDCGM in a broader context, namely when solving large-scale convex optimization problems. We have selected applications that arise in important real-life contexts such as data analysis (multiple kernel learning problem), decision-making under uncertainty (two-stage stochastic programming problems) and telecommunication and transportation networks (multicommodity network flow problem). In the numerical experiments, we use publicly available benchmark instances to compare the performance of the PDCGM against recent results for different methods presented in the literature, which were the best available results to date. The analysis of these results suggests that the PDCGM offers an attractive alternative over specialized methods since it remains competitive in terms of number of iterations and CPU times even for large-scale optimization problems.Comment: 28 pages, 1 figure, minor revision, scaled CPU time

Large-scale unit commitment under uncertainty: an updated literature survey

The Unit Commitment problem in energy management aims at finding the optimal production schedule of a set of generation units, while meeting various system-wide constraints. It has always been a large-scale, non-convex, difficult problem, especially in view of the fact that, due to operational requirements, it has to be solved in an unreasonably small time for its size. Recently, growing renewable energy shares have strongly increased the level of uncertainty in the system, making the (ideal) Unit Commitment model a large-scale, non-convex and uncertain (stochastic, robust, chance-constrained) program. We provide a survey of the literature on methods for the Uncertain Unit Commitment problem, in all its variants. We start with a review of the main contributions on solution methods for the deterministic versions of the problem, focussing on those based on mathematical programming techniques that are more relevant for the uncertain versions of the problem. We then present and categorize the approaches to the latter, while providing entry points to the relevant literature on optimization under uncertainty. This is an updated version of the paper "Large-scale Unit Commitment under uncertainty: a literature survey" that appeared in 4OR 13(2), 115--171 (2015); this version has over 170 more citations, most of which appeared in the last three years, proving how fast the literature on uncertain Unit Commitment evolves, and therefore the interest in this subject