593 research outputs found
On the convergence analysis of DCA
In this paper, we propose a clean and general proof framework to establish
the convergence analysis of the Difference-of-Convex (DC) programming algorithm
(DCA) for both standard DC program and convex constrained DC program. We first
discuss suitable assumptions for the well-definiteness of DCA. Then, we focus
on the convergence analysis of DCA, in particular, the global convergence of
the sequence generated by DCA under the Lojasiewicz subgradient
inequality and the Kurdyka-Lojasiewicz property respectively. Moreover, the
convergence rate for the sequences and are also
investigated. We hope that the proof framework presented in this article will
be a useful tool to conveniently establish the convergence analysis for many
variants of DCA and new DCA-type algorithms
An Accelerated DC Programming Approach with Exact Line Search for The Symmetric Eigenvalue Complementarity Problem
In this paper, we are interested in developing an accelerated
Difference-of-Convex (DC) programming algorithm based on the exact line search
for efficiently solving the Symmetric Eigenvalue Complementarity Problem
(SEiCP) and Symmetric Quadratic Eigenvalue Complementarity Problem (SQEiCP). We
first proved that any SEiCP is equivalent to SEiCP with symmetric positive
definite matrices only. Then, we established DC programming formulations for
two equivalent formulations of SEiCP (namely, the logarithmic formulation and
the quadratic formulation), and proposed the accelerated DC algorithm (BDCA) by
combining the classical DCA with inexpensive exact line search by finding real
roots of a binomial for acceleration. We demonstrated the equivalence between
SQEiCP and SEiCP, and extended BDCA to SQEiCP. Numerical simulations of the
proposed BDCA and DCA against KNITRO, FILTERED and MATLAB FMINCON for SEiCP and
SQEiCP on both synthetic datasets and Matrix Market NEP Repository are
reported. BDCA demonstrated dramatic acceleration to the convergence of DCA to
get better numerical solutions, and outperformed KNITRO, FILTERED, and FMINCON
solvers in terms of the average CPU time and average solution precision,
especially for large-scale cases.Comment: 24 page
Higher-order Moment Portfolio Optimization via The Difference-of-Convex Programming and Sums-of-Squares
We are interested in developing a Difference-of-Convex (DC) programming
approach based on Difference-of-Convex-Sums-of-Squares (DC-SOS) decomposition
techniques for high-order moment (Mean-Variance-Skewness-Kurtosis) portfolio
optimization model. This problem can be formulated as a nonconvex quartic
multivariate polynomial optimization, then a DC programming formulation based
on the recently developed DC-SOS decomposition is investigated. We can use a
well-known DC algorithm, namely DCA, for its numerical solution. Moreover, an
acceleration technique for DCA, namely Boosted-DCA (BDCA), based on an inexact
line search (Armijo-type line search) to accelerate the convergence of DCA for
smooth and nonsmooth DC program with convex constraints is proposed. This
technique is applied to DCA based on DC-SOS decomposition, and DCA based on
universal DC decomposition. Numerical simulations of DCA and Boosted-DCA on
synthetic and real datasets are reported. Comparisons with some non-dc
programming based optimization solvers (KNITRO, FILTERSD, IPOPT and MATLAB
fmincon) demonstrate that our Boosted-DC algorithms can achieve same numerical
results with good performance comparable to these efficient methods on solving
the high-order moment portfolio optimization model.Comment: 42 pages, 13 figure
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