15,556 research outputs found
Portfolio selection problems in practice: a comparison between linear and quadratic optimization models
Several portfolio selection models take into account practical limitations on
the number of assets to include and on their weights in the portfolio. We
present here a study of the Limited Asset Markowitz (LAM), of the Limited Asset
Mean Absolute Deviation (LAMAD) and of the Limited Asset Conditional
Value-at-Risk (LACVaR) models, where the assets are limited with the
introduction of quantity and cardinality constraints. We propose a completely
new approach for solving the LAM model, based on reformulation as a Standard
Quadratic Program and on some recent theoretical results. With this approach we
obtain optimal solutions both for some well-known financial data sets used by
several other authors, and for some unsolved large size portfolio problems. We
also test our method on five new data sets involving real-world capital market
indices from major stock markets. Our computational experience shows that,
rather unexpectedly, it is easier to solve the quadratic LAM model with our
algorithm, than to solve the linear LACVaR and LAMAD models with CPLEX, one of
the best commercial codes for mixed integer linear programming (MILP) problems.
Finally, on the new data sets we have also compared, using out-of-sample
analysis, the performance of the portfolios obtained by the Limited Asset
models with the performance provided by the unconstrained models and with that
of the official capital market indices
On the complexity of nonlinear mixed-integer optimization
This is a survey on the computational complexity of nonlinear mixed-integer
optimization. It highlights a selection of important topics, ranging from
incomputability results that arise from number theory and logic, to recently
obtained fully polynomial time approximation schemes in fixed dimension, and to
strongly polynomial-time algorithms for special cases.Comment: 26 pages, 5 figures; to appear in: Mixed-Integer Nonlinear
Optimization, IMA Volumes, Springer-Verla
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
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
Using a conic bundle method to accelerate both phases of a quadratic convex reformulation
We present algorithm MIQCR-CB that is an advancement of method
MIQCR~(Billionnet, Elloumi and Lambert, 2012). MIQCR is a method for solving
mixed-integer quadratic programs and works in two phases: the first phase
determines an equivalent quadratic formulation with a convex objective function
by solving a semidefinite problem , and, in the second phase, the
equivalent formulation is solved by a standard solver. As the reformulation
relies on the solution of a large-scale semidefinite program, it is not
tractable by existing semidefinite solvers, already for medium sized problems.
To surmount this difficulty, we present in MIQCR-CB a subgradient algorithm
within a Lagrangian duality framework for solving that substantially
speeds up the first phase. Moreover, this algorithm leads to a reformulated
problem of smaller size than the one obtained by the original MIQCR method
which results in a shorter time for solving the second phase.
We present extensive computational results to show the efficiency of our
algorithm
Recommended from our members
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
- …