Using Positive Spanning Sets to Achieve d-Stationarity with the Boosted DC Algorithm

The Difference of Convex functions Algorithm (DCA) is widely used for minimizing the difference of two convex functions. A recently proposed accelerated version, termed BDCA for Boosted DC Algorithm, incorporates a line search step to achieve a larger decrease of the objective value at each iteratio...

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Bibliographic Details
Published in:Vietnam journal of mathematics Vol. 48; no. 2; pp. 363 - 376
Main Authors: Artacho, F. J. Aragón, Campoy, R., Vuong, P. T.
Format: Journal Article
Language:English
Published: Singapore Springer Nature Singapore 01.06.2020
Springer Nature B.V
Springer Singapore
Subjects:
Algorithms
Clustering
Convex analysis
Critical point
Iterative methods
Mathematics
Mathematics and Statistics
Optimization
Original
Original Article
Vector spaces
Boosted difference of convex functions algorithm
65K05
Difference of convex functions
65K10
Positive spanning sets
90C26
47N10
d-Stationary points
Derivative-free optimization
ISSN:2305-221X, 2305-2228, 2305-2228
Online Access:Get full text
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Summary:The Difference of Convex functions Algorithm (DCA) is widely used for minimizing the difference of two convex functions. A recently proposed accelerated version, termed BDCA for Boosted DC Algorithm, incorporates a line search step to achieve a larger decrease of the objective value at each iteration. Thanks to this step, BDCA usually converges much faster than DCA in practice. The solutions found by DCA are guaranteed to be critical points of the problem, but these may not be local minima. Although BDCA tends to improve the objective value of the solutions it finds, these are frequently just critical points as well. In this paper we combine BDCA with a simple Derivative-Free Optimization (DFO) algorithm to force the d-stationarity (lack of descent direction) at the point obtained. The potential of this approach is illustrated through some computational experiments on a Minimum-Sum-of-Squares clustering problem. Our numerical results demonstrate that the new method provides better solutions while still remains faster than DCA in the majority of test cases.
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ISSN:2305-221X
2305-2228
2305-2228
DOI:10.1007/s10013-020-00400-8