241 research outputs found
Packing While Traveling: Mixed Integer Programming for a Class of Nonlinear Knapsack Problems
Packing and vehicle routing problems play an important role in the area of
supply chain management. In this paper, we introduce a non-linear knapsack
problem that occurs when packing items along a fixed route and taking into
account travel time. We investigate constrained and unconstrained versions of
the problem and show that both are NP-hard. In order to solve the problems, we
provide a pre-processing scheme as well as exact and approximate mixed integer
programming (MIP) solutions. Our experimental results show the effectiveness of
the MIP solutions and in particular point out that the approximate MIP approach
often leads to near optimal results within far less computation time than the
exact approach
Extended Formulations in Mixed-integer Convex Programming
We present a unifying framework for generating extended formulations for the
polyhedral outer approximations used in algorithms for mixed-integer convex
programming (MICP). Extended formulations lead to fewer iterations of outer
approximation algorithms and generally faster solution times. First, we observe
that all MICP instances from the MINLPLIB2 benchmark library are conic
representable with standard symmetric and nonsymmetric cones. Conic
reformulations are shown to be effective extended formulations themselves
because they encode separability structure. For mixed-integer
conic-representable problems, we provide the first outer approximation
algorithm with finite-time convergence guarantees, opening a path for the use
of conic solvers for continuous relaxations. We then connect the popular
modeling framework of disciplined convex programming (DCP) to the existence of
extended formulations independent of conic representability. We present
evidence that our approach can yield significant gains in practice, with the
solution of a number of open instances from the MINLPLIB2 benchmark library.Comment: To be presented at IPCO 201
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
Error bounds for monomial convexification in polynomial optimization
Convex hulls of monomials have been widely studied in the literature, and
monomial convexifications are implemented in global optimization software for
relaxing polynomials. However, there has been no study of the error in the
global optimum from such approaches. We give bounds on the worst-case error for
convexifying a monomial over subsets of . This implies additive error
bounds for relaxing a polynomial optimization problem by convexifying each
monomial separately. Our main error bounds depend primarily on the degree of
the monomial, making them easy to compute. Since monomial convexification
studies depend on the bounds on the associated variables, in the second part,
we conduct an error analysis for a multilinear monomial over two different
types of box constraints. As part of this analysis, we also derive the convex
hull of a multilinear monomial over .Comment: 33 pages, 2 figures, to appear in journa
On the composition of convex envelopes for quadrilinear terms
International audienceWithin the framework of the spatial Branch-and-Bound algorithm for solving Mixed-Integer Nonlinear Programs, different convex relaxations can be obtained for multilinear terms by applying associativity in different ways. The two groupings ((x1x2)x3)x4 and (x1x2x3)x4 of a quadrilinear term, for example, give rise to two different convex relaxations. In [6] we prove that having fewer groupings of longer terms yields tighter convex relaxations. In this paper we give an alternative proof of the same fact and perform a computational study to assess the impact of the tightened convex relaxation in a spatial Branch-and-Bound setting
Upper Bounding in Inner Regions for Global Optimization under Inequality Constraints
International audienceIn deterministic continuous constrained global optimization, upper bounding the objective function generally resorts to local minimization at several nodes/iterations of the branch and bound. We propose in this paper an alternative approach when the constraints are inequalities and the feasible space has a non-null volume. First, we extract an inner region , i.e., an entirely feasible convex polyhedron or box in which all points satisfy the constraints. Second, we select a point inside the extracted inner region and update the upper bound with its cost. We describe in this paper two original inner region extraction algorithms implemented in our interval B&B called IbexOpt. They apply to nonconvex constraints involving mathematical operators like +,x,power,sqrt,exp,log,sin. This upper bounding shows very good performance obtained on medium-sized systems proposed in the COCONUT suite
Solving Electric Market Quadratic Problems by Branch and Fix Coordination Methods
The electric market regulation in Spain (MIBEL) establishes the rules for bilateral and futures contracts in the day-ahead optimal bid problem. Our model allows a price-taker generation company to decide the unit commitment of the thermal units, the economic dispatch of the bilateral and futures contracts between the thermal units and the optimal sale bids for the thermal units observing the MIBEL regulation. The uncertainty of the spot prices is represented through scenario sets. We solve this model on the framework of the Branch and Fix Coordination metodology as a quadratic two-stage stochastic problem. In order to gain computational efficiency, we use scenario clusters and propose to use perspective cuts. Numerical results are reportedPeer Reviewe
Computational optimization of gas compressor stations: MINLP models versus continuous reformulations
A Contractor Based on Convex Interval Taylor
International audienceInterval Taylor has been proposed in the sixties by the interval analysis community for relaxing continuous non-convex constraint systems. However, it generally produces a non-convex relaxation of the solution set. A simple way to build a convex polyhedral relaxation is to select a corner of the studied domain/box as expansion point of the interval Taylor form, instead of the usual midpoint. The idea has been proposed by Neumaier to produce a sharp range of a single function andby Lin and Stadtherr to handle n Ă n (square) systems of equations. This paper presents an interval Newton-like operator, called X-Newton, that iteratively calls this interval convexification based on an endpoint interval Taylor. This general-purpose contractor uses no preconditioning and can handle any system of equality and inequality constraints. It uses Hansen's variant to compute the interval Taylor form and uses two opposite corners of the domain for every constraint. The X-Newton operator can be rapidly encoded, and produces good speedups in constrained global optimization and constraint satisfaction. First experiments compare X-Newton with affine arithmetic
Branch and bound based coordinate search filter algorithm for nonsmooth nonconvex mixed-integer nonlinear programming problems
Publicado em "Computational science and its applications â ICCSA 2014...", ISBN 978-3-319-09128-0. Series "Lecture notes in computer science", ISSN 0302-9743, vol. 8580.A mixed-integer nonlinear programming problem (MINLP) is a problem with continuous and integer variables and at least, one nonlinear function. This kind of problem appears in a wide range of real applications and is very difficult to solve. The difficulties are due to the nonlinearities of the functions in the problem and the integrality restrictions on some variables. When they are nonconvex then they are the most difficult to solve above all. We present a methodology to solve nonsmooth nonconvex MINLP problems based on a branch and bound paradigm and a stochastic strategy. To solve the relaxed subproblems at each node of the branch and bound tree search, an algorithm based on a multistart strategy with a coordinate search filter methodology is implemented. The produced numerical results show the robustness of the proposed methodology.This work has been supported by FCT (Fundação para a CiĂȘncia e aTecnologia) in the scope of the projects: PEst-OE/MAT/UI0013/2014 and PEst-OE/EEI/UI0319/2014
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