4,907 research outputs found
A Regularized Newton Method for Computing Ground States of Bose-Einstein condensates
In this paper, we propose a regularized Newton method for computing ground
states of Bose-Einstein condensates (BECs), which can be formulated as an
energy minimization problem with a spherical constraint. The energy functional
and constraint are discretized by either the finite difference, or sine or
Fourier pseudospectral discretization schemes and thus the original infinite
dimensional nonconvex minimization problem is approximated by a finite
dimensional constrained nonconvex minimization problem. Then an initial
solution is first constructed by using a feasible gradient type method, which
is an explicit scheme and maintains the spherical constraint automatically. To
accelerate the convergence of the gradient type method, we approximate the
energy functional by its second-order Taylor expansion with a regularized term
at each Newton iteration and adopt a cascadic multigrid technique for selecting
initial data. It leads to a standard trust-region subproblem and we solve it
again by the feasible gradient type method. The convergence of the regularized
Newton method is established by adjusting the regularization parameter as the
standard trust-region strategy. Extensive numerical experiments on challenging
examples, including a BEC in three dimensions with an optical lattice potential
and rotating BECs in two dimensions with rapid rotation and strongly repulsive
interaction, show that our method is efficient, accurate and robust.Comment: 25 pages, 6 figure
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
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