509 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 Global Stochastic Programming Approach for the Optimal Placement of Gas Detectors with Nonuniform Unavailabilities
PresentationOptimal design of a gas detection systems is challenging because of the numerous sources of uncertainty, including weather and environmental conditions, leak location and characteristics, and process conditions. Rigorous CFD simulations of dispersion scenarios combined with stochastic programming techniques have been successfully applied to the problem of optimal gas detector placement; however, rigorous treatment of sensor failure and nonuniform unavailability has received less attention. To improve reliability of the design, this paper proposes a problem formulation that explicitly considers nonuniform unavailabilities and all backup detection levels. The resulting sensor placement problem is a large-scale mixed-integer nonlinear programming (MINLP) problem that requires a tailored solution approach for efficient solution. We have developed a multitree method which depends on iteratively solving a sequence of upper-bounding master problems and lower-bounding subproblems. The tailored global solution strategy is tested on a real data problem and the encouraging numerical results indicate that our solution framework is promising in solving sensor placement problems
(Global) Optimization: Historical notes and recent developments
Recent developments in (Global) Optimization are surveyed in this paper. We collected and commented quite a large number of recent references which, in our opinion, well represent the vivacity, deepness, and width of scope of current computational approaches and theoretical results about nonconvex optimization problems. Before the presentation of the recent developments, which are subdivided into two parts related to heuristic and exact approaches, respectively, we briefly sketch the origin of the discipline and observe what, from the initial attempts, survived, what was not considered at all as well as a few approaches which have been recently rediscovered, mostly in connection with machine learning
Global solution of non-convex quadratically constrained quadratic programs
International audienceThe class of mixed-integer quadratically constrained quadratic programs (QCQP) consists of minimizing a quadratic function under quadratic constraints where the variables could be integer or continuous. On a previous paper we introduced a method called MIQCR for solving QC-QPs with the following restriction : all quadratic sub-functions of purely continuous variables are already convex. In this paper, we propose an extension of MIQCR which applies to any QCQP. Let (P) be a QCQP. Our approach to solve (P) is first to build an equivalent mixed-integer quadratic problem (P *). This equivalent problem (P *) has a quadratic convex objective function, linear constraints, and additional variables y that are meant to satisfy the additional quadratic constraints y = xx T , where x are the initial variables of problem (P). We then propose to solve (P *) by a branch-and-bound algorithm based on the relaxation of the additional quadratic constraints and of the integrality constraints. This type of branching is known as spatial branch-and-bound. Computational experiences are carried out on a total of 325 instances. The results show that the solution time of most of the considered instances is improved by our method in comparison with the recent implementation of QuadProgBB, and with the solvers Cplex, Couenne, Scip, BARON and GloMIQO
Productivity enhancement through process integration
A hierarchical procedure is developed to determine maximum overall yield of a process
and optimize process changes to achieve such a yield. First, a targeting procedure is
developed to identify an upper bound of the overall yield ahead of detailed design.
Several mass integration strategies are proposed to attain maximum yield. These
strategies include rerouting of raw materials, optimization of reaction yield, rerouting of
product from undesirable outlets to desirable outlets, and recycling of unreacted raw
materials. Path equations are tailored to provide the appropriate level of detail for
modeling process performance as a function of the optimization variables pertaining to
design and operating variables. Interval analysis is used as an inclusion technique that
provides rigorous bounds regardless of the process nonlinearities and without
enumeration. Then, a new approach for identification of cost-effective implementation of
maximum attainable targets for yield is presented. In this approach, a mathematical
program was developed to identify the maximum feasible yield using a combination of
iterative additions of constraints and problem reformulation. Next, cost objectives were
employed to identify a cost-effective solution with the details of design and operating
variables. Constraint convexification was used to improve the quality of the solution towards globability. A trade-off procedure between the saving and expenses for yield
maximization problem is presented. The proposed procedure is systematic, rigorous, and
computationally efficient. A case study was solved to demonstrate the applicability and
usefulness of the developed procedure
Consistent Second-Order Conic Integer Programming for Learning Bayesian Networks
Bayesian Networks (BNs) represent conditional probability relations among a
set of random variables (nodes) in the form of a directed acyclic graph (DAG),
and have found diverse applications in knowledge discovery. We study the
problem of learning the sparse DAG structure of a BN from continuous
observational data. The central problem can be modeled as a mixed-integer
program with an objective function composed of a convex quadratic loss function
and a regularization penalty subject to linear constraints. The optimal
solution to this mathematical program is known to have desirable statistical
properties under certain conditions. However, the state-of-the-art optimization
solvers are not able to obtain provably optimal solutions to the existing
mathematical formulations for medium-size problems within reasonable
computational times. To address this difficulty, we tackle the problem from
both computational and statistical perspectives. On the one hand, we propose a
concrete early stopping criterion to terminate the branch-and-bound process in
order to obtain a near-optimal solution to the mixed-integer program, and
establish the consistency of this approximate solution. On the other hand, we
improve the existing formulations by replacing the linear "big-" constraints
that represent the relationship between the continuous and binary indicator
variables with second-order conic constraints. Our numerical results
demonstrate the effectiveness of the proposed approaches
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