Nonlinear Integer Programming

Research efforts of the past fifty years have led to a development of linear integer programming as a mature discipline of mathematical optimization. Such a level of maturity has not been reached when one considers nonlinear systems subject to integrality requirements for the variables. This chapter is dedicated to this topic. The primary goal is a study of a simple version of general nonlinear integer problems, where all constraints are still linear. Our focus is on the computational complexity of the problem, which varies significantly with the type of nonlinear objective function in combination with the underlying combinatorial structure. Numerous boundary cases of complexity emerge, which sometimes surprisingly lead even to polynomial time algorithms. We also cover recent successful approaches for more general classes of problems. Though no positive theoretical efficiency results are available, nor are they likely to ever be available, these seem to be the currently most successful and interesting approaches for solving practical problems. It is our belief that the study of algorithms motivated by theoretical considerations and those motivated by our desire to solve practical instances should and do inform one another. So it is with this viewpoint that we present the subject, and it is in this direction that we hope to spark further research.Comment: 57 pages. To appear in: M. J\"unger, T. Liebling, D. Naddef, G. Nemhauser, W. Pulleyblank, G. Reinelt, G. Rinaldi, and L. Wolsey (eds.), 50 Years of Integer Programming 1958--2008: The Early Years and State-of-the-Art Surveys, Springer-Verlag, 2009, ISBN 354068274

Decisional Conflict and User Acceptance of Multicriteria Decision-Making Aids *

Despite the development of increasingly sophisticated and refined multicriteria decision-making (MCDM) methods, an examination of the experimental evidence indicates that users most often prefer relatively unsophisticated methods. In this paper, we synthesize theories and empirical findings from the psychology of judgment and choice to provide a new theoretical explanation for such user preferences. Our argument centers on the assertion that the MCDM method preferred by decision makers is a function of the degree to which the method tends to introduce decisional conflict. The model we develop relates response mode, decision strategy, and the salience of decisional conflict to user preferences among decision aids. We then show that the model is consistent with empirical results in MCDM studies. Next, the role of decisional conflict in problem formulation aids is briefly discussed. Finally, we outline future research needed to thoroughly test the theoretical mechanisms we have proposed.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/73461/1/j.1540-5915.1991.tb00371.x.pd

Oxy-fuel combustion of coal and biomass blends

The ignition temperature, burnout and NO emissions of blends of a semi-anthracite and a high-volatile bituminous coal with 10 and 20 wt.% of olive waste were studied under oxy-fuel combustion conditions in an entrained flow reactor (EFR). The results obtained under several oxy-fuel atmospheres (21%O2–79%CO2, 30%O2–70%CO2 and 35%O2–65%CO2) were compared with those attained in air. The results indicated that replacing N2 by CO2 in the combustion atmosphere with 21% of O2 caused an increase in the temperature of ignition and a decrease in the burnout value. When the O2 concentration was increased to 30 and 35%, the temperature of ignition was lower and the burnout value was higher than in air conditions. A significant reduction in ignition temperature and a slight increase in the burnout value was observed after the addition of biomass, this trend becoming more noticeable as the biomass concentration was increased. The emissions of NO during oxy-fuel combustion were lower than under air-firing. However, they remained similar under all the oxy-fuel atmospheres with increasing O2 concentrations. Emissions of NO were significantly reduced by the addition of biomass to the bituminous coal, although this effect was less noticeable in the case of the semi-anthracite.This work was carried out with financial support from the Spanish MICINN (Project PS-120000-2005-2) co-financed by the European Regional Development Fund. M.V.G. and L.A. acknowledge funding from the CSIC JAE-Doc and CSIC JAE-Pre programs, respectively, co-financed by the European Social Fund. J.R. acknowledges funding from the Government of the Principado de Asturias (Severo Ochoa program).Peer reviewe

Towards Machine Wald

The past century has seen a steady increase in the need of estimating and predicting complex systems and making (possibly critical) decisions with limited information. Although computers have made possible the numerical evaluation of sophisticated statistical models, these models are still designed \emph{by humans} because there is currently no known recipe or algorithm for dividing the design of a statistical model into a sequence of arithmetic operations. Indeed enabling computers to \emph{think} as \emph{humans} have the ability to do when faced with uncertainty is challenging in several major ways: (1) Finding optimal statistical models remains to be formulated as a well posed problem when information on the system of interest is incomplete and comes in the form of a complex combination of sample data, partial knowledge of constitutive relations and a limited description of the distribution of input random variables. (2) The space of admissible scenarios along with the space of relevant information, assumptions, and/or beliefs, tend to be infinite dimensional, whereas calculus on a computer is necessarily discrete and finite. With this purpose, this paper explores the foundations of a rigorous framework for the scientific computation of optimal statistical estimators/models and reviews their connections with Decision Theory, Machine Learning, Bayesian Inference, Stochastic Optimization, Robust Optimization, Optimal Uncertainty Quantification and Information Based Complexity.Comment: 37 page

A two-phase approach for real-world train unit scheduling

A two-phase approach for the train unit scheduling problem is proposed. The first phase assigns and sequences train trips to train units temporarily ignoring some station infrastructure details. Real-world scenarios such as compatibility among traction types and banned/restricted locations and time allowances for coupling/ decoupling are considered. Its solutions would be near-operable. The second phase focuses on satisfying the remaining station detail requirements, such that the solutions would be fully operable. The first phase is modeled as an integer fixed-charge multicommodity flow (FCMF) problem. A branch-and-price approach is proposed to solve it. Experiments have shown that it is only capable of handling problem instances within about 500 train trips. The train company collaborating in this research operates over 2400 train trips on a typical weekday. Hence, a heuristic has been designed for compacting the problem instance to a much smaller size before the branch-and-price solver is applied. The process is iterative with evolving compaction based on the results from the previous iteration, thereby converging to near-optimal results. The second phase is modeled as a multidimensional matching problem with a mixed integer linear programming (MILP) formulation. A column-and-dependentrow generation method for it is under development

Optimization methods for electric power systems: An overview

Power systems optimization problems are very difficult to solve because power systems are very large, complex, geographically widely distributed and are influenced by many unexpected events. It is therefore necessary to employ most efficient optimization methods to take full advantages in simplifying the formulation and implementation of the problem. This article presents an overview of important mathematical optimization and artificial intelligence (AI) techniques used in power optimization problems. Applications of hybrid AI techniques have also been discussed in this article