13,384 research outputs found
Mixed-integer Quadratic Programming is in NP
Mixed-integer quadratic programming is the problem of optimizing a quadratic
function over points in a polyhedral set where some of the components are
restricted to be integral. In this paper, we prove that the decision version of
mixed-integer quadratic programming is in NP, thereby showing that it is
NP-complete. This is established by showing that if the decision version of
mixed-integer quadratic programming is feasible, then there exists a solution
of polynomial size. This result generalizes and unifies classical results that
quadratic programming is in NP and integer linear programming is in NP
On Finding Maximum Cardinality Subset of Vectors with a Constraint on Normalized Squared Length of Vectors Sum
In this paper, we consider the problem of finding a maximum cardinality
subset of vectors, given a constraint on the normalized squared length of
vectors sum. This problem is closely related to Problem 1 from (Eremeev,
Kel'manov, Pyatkin, 2016). The main difference consists in swapping the
constraint with the optimization criterion.
We prove that the problem is NP-hard even in terms of finding a feasible
solution. An exact algorithm for solving this problem is proposed. The
algorithm has a pseudo-polynomial time complexity in the special case of the
problem, where the dimension of the space is bounded from above by a constant
and the input data are integer. A computational experiment is carried out,
where the proposed algorithm is compared to COINBONMIN solver, applied to a
mixed integer quadratic programming formulation of the problem. The results of
the experiment indicate superiority of the proposed algorithm when the
dimension of Euclidean space is low, while the COINBONMIN has an advantage for
larger dimensions.Comment: To appear in Proceedings of the 6th International Conference on
Analysis of Images, Social Networks, and Texts (AIST'2017
The Quadratic Shortest Path Problem:Complexity, Approximability, and Solution Methods
We consider the problem of finding a shortest path in a directed graph with a quadratic objective function (the QSPP). We show that the QSPP cannot be approximated unless P=NP . For the case of a convex objective function, an n-approximation algorithm is presented, where n is the number of nodes in the graph, and APX-hardness is shown. Furthermore, we prove that even if only adjacent arcs play a part in the quadratic objective function, the problem still cannot be approximated unless P=NP. In order to solve the problem we first propose a mixed integer programming formulation, and then devise an efficient exact Branch-and-Bound algorithm for the general QSPP, where lower bounds are computed by considering a reformulation scheme that is solvable through a number of minimum cost flow problems. In our computational experiments we solve to optimality different classes of instances with up to 1000 nodes
On the exact solution of the no-wait flow shop problem with due date constraints
Peer ReviewedThis paper deals with the no-wait flow shop scheduling problem with due date constraints. In the no-wait flow shop problem, waiting time is not allowed between successive operations of jobs. Moreover, the jobs should be completed before their respective due dates; due date constraints are dealt with as hard constraints. The considered performance criterion is makespan. The problem is strongly NP-hard. This paper develops a number of distinct mathematical models for the problem based on different decision variables. Namely, a mixed integer programming model, two quadratic mixed integer programming models, and two constraint programming models are developed. Moreover, a novel graph representation is developed for the problem. This new modeling technique facilitates the investigation of some of the important characteristics of the problem; this results in a number of propositions to rule out a large number of infeasible solutions from the set of all possible permutations. Afterward, the new graph representation and the resulting propositions are incorporated into a new exact algorithm to solve the problem to optimality. To investigate the performance of the mathematical models and to compare them with the developed exact algorithm, a number of test problems are solved and the results are reported. Computational results demonstrate that the developed algorithm is significantly faster than the mathematical models
Enhancing low-rank solutions in semidefinite relaxations of Boolean quadratic problems
Boolean quadratic optimization problems occur in a number of applications. Their mixed integer-continuous nature is challenging, since it is inherently NP-hard. For this motivation,
semidefinite programming relaxations (SDRâs) are proposed in the literature to approximate the solution, which recasts the problem into convex optimization. Nevertheless, SDRâs
do not guarantee the extraction of the correct binary minimizer. In this paper, we present a novel approach to enhance the binary solution recovery. The key of the proposed method is the exploitation of known information on the eigenvalues of the desired solution. As the proposed approach yields a non-convex program, we develop and analyze an iterative descent strategy, whose practical effectiveness is shown via numerical results
Enhancing low-rank solutions in semidefinite relaxations of Boolean quadratic problems
Boolean quadratic optimization problems occur in a number of applications.
Their mixed integer-continuous nature is challenging, since it is inherently
NP-hard. For this motivation, semidefinite programming relaxations (SDR's) are
proposed in the literature to approximate the solution, which recasts the
problem into convex optimization. Nevertheless, SDR's do not guarantee the
extraction of the correct binary minimizer. In this paper, we present a novel
approach to enhance the binary solution recovery. The key of the proposed
method is the exploitation of known information on the eigenvalues of the
desired solution. As the proposed approach yields a non-convex program, we
develop and analyze an iterative descent strategy, whose practical
effectiveness is shown via numerical results
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A review of portfolio planning: Models and systems
In this chapter, we first provide an overview of a number of portfolio planning models
which have been proposed and investigated over the last forty years. We revisit the
mean-variance (M-V) model of Markowitz and the construction of the risk-return
efficient frontier. A piecewise linear approximation of the problem through a
reformulation involving diagonalisation of the quadratic form into a variable
separable function is also considered. A few other models, such as, the Mean
Absolute Deviation (MAD), the Weighted Goal Programming (WGP) and the
Minimax (MM) model which use alternative metrics for risk are also introduced,
compared and contrasted. Recently asymmetric measures of risk have gained in
importance; we consider a generic representation and a number of alternative
symmetric and asymmetric measures of risk which find use in the evaluation of
portfolios. There are a number of modelling and computational considerations which
have been introduced into practical portfolio planning problems. These include: (a)
buy-in thresholds for assets, (b) restriction on the number of assets (cardinality
constraints), (c) transaction roundlot restrictions. Practical portfolio models may also
include (d) dedication of cashflow streams, and, (e) immunization which involves
duration matching and convexity constraints. The modelling issues in respect of these
features are discussed. Many of these features lead to discrete restrictions involving
zero-one and general integer variables which make the resulting model a quadratic
mixed-integer programming model (QMIP). The QMIP is a NP-hard problem; the
algorithms and solution methods for this class of problems are also discussed. The
issues of preparing the analytic data (financial datamarts) for this family of portfolio
planning problems are examined. We finally present computational results which
provide some indication of the state-of-the-art in the solution of portfolio optimisation
problems
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
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