1,629 research outputs found
ALPS: A Linear Program Solver
ALPS is a computer program which can be used to solve general linear program (optimization) problems. ALPS was designed for those who have minimal linear programming (LP) knowledge and features a menu-driven scheme to guide the user through the process of creating and solving LP formulations. Once created, the problems can be edited and stored in standard DOS ASCII files to provide portability to various word processors or even other linear programming packages. Unlike many math-oriented LP solvers, ALPS contains an LP parser that reads through the LP formulation and reports several types of errors to the user. ALPS provides a large amount of solution data which is often useful in problem solving. In addition to pure linear programs, ALPS can solve for integer, mixed integer, and binary type problems. Pure linear programs are solved with the revised simplex method. Integer or mixed integer programs are solved initially with the revised simplex, and the completed using the branch-and-bound technique. Binary programs are solved with the method of implicit enumeration. This manual describes how to use ALPS to create, edit, and solve linear programming problems. Instructions for installing ALPS on a PC compatible computer are included in the appendices along with a general introduction to linear programming. A programmers guide is also included for assistance in modifying and maintaining the program
Branching on multi-aggregated variables
open5siopenGamrath, Gerald; Melchiori, Anna; Berthold, Timo; Gleixner, Ambros M.; Salvagnin, DomenicoGamrath, Gerald; Melchiori, Anna; Berthold, Timo; Gleixner, Ambros M.; Salvagnin, Domenic
A New Dantzig-Wolfe Reformulation And Branch-And-Price Algorithm For The Capacitated Lot Sizing Problem With Set Up Times
The textbook Dantzig-Wolfe decomposition for the Capacitated LotSizing Problem (CLSP),as already proposed by Manne in 1958, has animportant structural deficiency. Imposingintegrality constraints onthe variables in the full blown master will not necessarily givetheoptimal IP solution as only production plans which satisfy theWagner-Whitin condition canbe selected. It is well known that theoptimal solution to a capacitated lot sizing problem willnotnecessarily have this Wagner-Whitin property. The columns of thetraditionaldecomposition model include both the integer set up andcontinuous production quantitydecisions. Choosing a specific set upschedule implies also taking the associated Wagner-Whitin productionquantities. We propose the correct Dantzig-Wolfedecompositionreformulation separating the set up and productiondecisions. This formulation gives the samelower bound as Manne'sreformulation and allows for branch-and-price. We use theCapacitatedLot Sizing Problem with Set Up Times to illustrate our approach.Computationalexperiments are presented on data sets available from theliterature. Column generation isspeeded up by a combination of simplexand subgradient optimization for finding the dualprices. The resultsshow that branch-and-price is computationally tractable andcompetitivewith other approaches. Finally, we briefly discuss how thisnew Dantzig-Wolfe reformulationcan be generalized to other mixedinteger programming problems, whereas in theliterature,branch-and-price algorithms are almost exclusivelydeveloped for pure integer programmingproblems.branch-and-price;Lagrange relaxation;Dantzig-Wolfe decomposition;lot sizing;mixed-integer programming
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
Introduction to ABACUS - A branch-and-cut System
The software system ABACUS is an object-oriented framework for the implementation of branch-and-cut and branch-and-price algorithms. This paper shows the basics of its application to combinatorial and mixed integer optimization problems
A Framework for Generalized Benders' Decomposition and Its Application to Multilevel Optimization
We describe a framework for reformulating and solving optimization problems
that generalizes the well-known framework originally introduced by Benders. We
discuss details of the application of the procedures to several classes of
optimization problems that fall under the umbrella of multilevel/multistage
mixed integer linear optimization problems. The application of this abstract
framework to this broad class of problems provides new insights and a broader
interpretation of the core ideas, especially as they relate to duality and the
value functions of optimization problems that arise in this context
A MIP framework for non-convex uniform price day-ahead electricity auctions
It is well-known that a market equilibrium with uniform prices often does not
exist in non-convex day-ahead electricity auctions. We consider the case of the
non-convex, uniform-price Pan-European day-ahead electricity market "PCR"
(Price Coupling of Regions), with non-convexities arising from so-called
complex and block orders. Extending previous results, we propose a new
primal-dual framework for these auctions, which has applications in both
economic analysis and algorithm design. The contribution here is threefold.
First, from the algorithmic point of view, we give a non-trivial exact (i.e.
not approximate) linearization of a non-convex 'minimum income condition' that
must hold for complex orders arising from the Spanish market, avoiding the
introduction of any auxiliary variables, and allowing us to solve market
clearing instances involving most of the bidding products proposed in PCR using
off-the-shelf MIP solvers. Second, from the economic analysis point of view, we
give the first MILP formulations of optimization problems such as the
maximization of the traded volume, or the minimization of opportunity costs of
paradoxically rejected block bids. We first show on a toy example that these
two objectives are distinct from maximizing welfare. We also recover directly a
previously noted property of an alternative market model. Third, we provide
numerical experiments on realistic large-scale instances. They illustrate the
efficiency of the approach, as well as the economics trade-offs that may occur
in practice
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