3,886 research outputs found
A distributionally robust index tracking model with the CVaR penalty: tractable reformulation
We propose a distributionally robust index tracking model with the
conditional value-at-risk (CVaR) penalty. The model combines the idea of
distributionally robust optimization for data uncertainty and the CVaR penalty
to avoid large tracking errors. The probability ambiguity is described through
a confidence region based on the first-order and second-order moments of the
random vector involved. We reformulate the model in the form of a min-max-min
optimization into an equivalent nonsmooth minimization problem. We further give
an approximate discretization scheme of the possible continuous random vector
of the nonsmooth minimization problem, whose objective function involves the
maximum of numerous but finite nonsmooth functions. The convergence of the
discretization scheme to the equivalent nonsmooth reformulation is shown under
mild conditions. A smoothing projected gradient (SPG) method is employed to
solve the discretization scheme. Any accumulation point is shown to be a global
minimizer of the discretization scheme. Numerical results on the NASDAQ index
dataset from January 2008 to July 2023 demonstrate the effectiveness of our
proposed model and the efficiency of the SPG method, compared with several
state-of-the-art models and corresponding methods for solving them
Large-scale Binary Quadratic Optimization Using Semidefinite Relaxation and Applications
In computer vision, many problems such as image segmentation, pixel
labelling, and scene parsing can be formulated as binary quadratic programs
(BQPs). For submodular problems, cuts based methods can be employed to
efficiently solve large-scale problems. However, general nonsubmodular problems
are significantly more challenging to solve. Finding a solution when the
problem is of large size to be of practical interest, however, typically
requires relaxation. Two standard relaxation methods are widely used for
solving general BQPs--spectral methods and semidefinite programming (SDP), each
with their own advantages and disadvantages. Spectral relaxation is simple and
easy to implement, but its bound is loose. Semidefinite relaxation has a
tighter bound, but its computational complexity is high, especially for large
scale problems. In this work, we present a new SDP formulation for BQPs, with
two desirable properties. First, it has a similar relaxation bound to
conventional SDP formulations. Second, compared with conventional SDP methods,
the new SDP formulation leads to a significantly more efficient and scalable
dual optimization approach, which has the same degree of complexity as spectral
methods. We then propose two solvers, namely, quasi-Newton and smoothing Newton
methods, for the dual problem. Both of them are significantly more efficiently
than standard interior-point methods. In practice, the smoothing Newton solver
is faster than the quasi-Newton solver for dense or medium-sized problems,
while the quasi-Newton solver is preferable for large sparse/structured
problems. Our experiments on a few computer vision applications including
clustering, image segmentation, co-segmentation and registration show the
potential of our SDP formulation for solving large-scale BQPs.Comment: Fixed some typos. 18 pages. Accepted to IEEE Transactions on Pattern
Analysis and Machine Intelligenc
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