16,729 research outputs found
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
A MINLP Solution for Pellet Reactor Modeling
A fluidized bed reactor for phosphate precipitation and removal from wastewater is modeled according to a two-step procedure. The first modeling phase, based on the development of a thermodynamic model for the computation of phosphate conversion, previously presented elsewhere
is not reported here. The second step is related to the reactor modeling in the core of this paper.
The pellet reactor is modeled as a reactor network involving a set of elementary cells representing
ideal flow patterns. All the potential solutions are imbedded into a superstructure and the modeling
problem is expressed as a MINLP problem. The MINLP problem is solved by means of the GAMS package, first for two flow rate values corresponding to two experimental fluidized bed
behaviours, and then for the two flow rates considered simultaneously. In each case, the problem consists in finding an output concentration as close as possible to the experimental output concentration.
Three objective functions are studied. The results are compared with those of Montastruc et al. (2004) who used a different numerical procedure. Whatever the considered case, the solutions found are structurally simpler than the ones of Montastruc et al. (2004). A major assessment
of this study is that the reactor efficiency can easily be deduced, without any precise knowledge of some key parameters such as the density and thickness of the calcium phosphate layer. Finally a last numerical study concerning the superstructure definition shows that too complex a
superstructure does not provide significant refinements on the solution
The transportation problem with exclusionary side constraints.
We consider the so-called Transportation Problem with Exclusionary Side Con- straints (TPESC), which is a generalization of the ordinary transportation problem. We determine the complexity status for each of two special cases of this problem, by proving NP-completeness, and by exhibiting a pseudo-polynomial time algorithm. For the general problem, we show that it cannot be approximated with a constant perfor- mance ratio in polynomial time (unless P=NP). These results settle the complexity status of the TPESC.
Convex Relaxations for Gas Expansion Planning
Expansion of natural gas networks is a critical process involving substantial
capital expenditures with complex decision-support requirements. Given the
non-convex nature of gas transmission constraints, global optimality and
infeasibility guarantees can only be offered by global optimisation approaches.
Unfortunately, state-of-the-art global optimisation solvers are unable to scale
up to real-world size instances. In this study, we present a convex
mixed-integer second-order cone relaxation for the gas expansion planning
problem under steady-state conditions. The underlying model offers tight lower
bounds with high computational efficiency. In addition, the optimal solution of
the relaxation can often be used to derive high-quality solutions to the
original problem, leading to provably tight optimality gaps and, in some cases,
global optimal soluutions. The convex relaxation is based on a few key ideas,
including the introduction of flux direction variables, exact McCormick
relaxations, on/off constraints, and integer cuts. Numerical experiments are
conducted on the traditional Belgian gas network, as well as other real larger
networks. The results demonstrate both the accuracy and computational speed of
the relaxation and its ability to produce high-quality solutions
Static Pricing Problems under Mixed Multinomial Logit Demand
Price differentiation is a common strategy for many transport operators. In
this paper, we study a static multiproduct price optimization problem with
demand given by a continuous mixed multinomial logit model. To solve this new
problem, we design an efficient iterative optimization algorithm that
asymptotically converges to the optimal solution. To this end, a linear
optimization (LO) problem is formulated, based on the trust-region approach, to
find a "good" feasible solution and approximate the problem from below. Another
LO problem is designed using piecewise linear relaxations to approximate the
optimization problem from above. Then, we develop a new branching method to
tighten the optimality gap. Numerical experiments show the effectiveness of our
method on a published, non-trivial, parking choice model
Algorithm Engineering in Robust Optimization
Robust optimization is a young and emerging field of research having received
a considerable increase of interest over the last decade. In this paper, we
argue that the the algorithm engineering methodology fits very well to the
field of robust optimization and yields a rewarding new perspective on both the
current state of research and open research directions.
To this end we go through the algorithm engineering cycle of design and
analysis of concepts, development and implementation of algorithms, and
theoretical and experimental evaluation. We show that many ideas of algorithm
engineering have already been applied in publications on robust optimization.
Most work on robust optimization is devoted to analysis of the concepts and the
development of algorithms, some papers deal with the evaluation of a particular
concept in case studies, and work on comparison of concepts just starts. What
is still a drawback in many papers on robustness is the missing link to include
the results of the experiments again in the design
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Primal-dual variable neighborhood search for the simple plant-location problem
Copyright @ 2007 INFORMSThe variable neighborhood search metaheuristic is applied to the primal simple plant-location problem and to a reduced dual obtained by exploiting the complementary slackness conditions. This leads to (i) heuristic resolution of (metric) instances with uniform fixed costs, up to n = 15,000 users, and m = n potential locations for facilities with an error not exceeding 0.04%; (ii) exact solution of such instances with up to m = n = 7,000; and (iii) exact solutions of instances with variable fixed costs and up to m = n = 15, 000.This work is supported by NSERC Grant 105574-02; NSERC Grant OGP205041; and partly by the Serbian Ministry of Science, Project 1583
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