TY - JOUR
T1 - The 2-Coordinate Descent Method for Solving Double-Sided Simplex Constrained Minimization Problems
AU - Beck, Amir
N1 - Funding Information:
Acknowledgements I would like to thank the three anonymous reviewers for their useful comments and additional references which helped to improve the presentation of the paper. This work was partially supported by ISF grant #25312 and by BSF grant 2008100.
PY - 2014/9
Y1 - 2014/9
N2 - This paper considers the problem of minimizing a continuously differentiable function with a Lipschitz continuous gradient subject to a single linear equality constraint and additional bound constraints on the decision variables. We introduce and analyze several variants of a 2-coordinate descent method: a block descent method that performs an optimization step with respect to only two variables at each iteration. Based on two new optimality measures, we establish convergence to stationarity points for general nonconvex objective functions. In the convex case, when all the variables are lower bounded but not upper bounded, we show that the sequence of function values converges at a sublinear rate. Several illustrative numerical examples demonstrate the effectiveness of the method.
AB - This paper considers the problem of minimizing a continuously differentiable function with a Lipschitz continuous gradient subject to a single linear equality constraint and additional bound constraints on the decision variables. We introduce and analyze several variants of a 2-coordinate descent method: a block descent method that performs an optimization step with respect to only two variables at each iteration. Based on two new optimality measures, we establish convergence to stationarity points for general nonconvex objective functions. In the convex case, when all the variables are lower bounded but not upper bounded, we show that the sequence of function values converges at a sublinear rate. Several illustrative numerical examples demonstrate the effectiveness of the method.
KW - Block descent method
KW - Nonconvex optimization
KW - Rate of convergence
KW - Simplex-type constraints
UR - http://www.scopus.com/inward/record.url?scp=84905718861&partnerID=8YFLogxK
U2 - 10.1007/s10957-013-0491-5
DO - 10.1007/s10957-013-0491-5
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AN - SCOPUS:84905718861
SN - 0022-3239
VL - 162
SP - 892
EP - 919
JO - Journal of Optimization Theory and Applications
JF - Journal of Optimization Theory and Applications
IS - 3
ER -