TY - JOUR
T1 - A sequential ascending parameter method for solving constrained minimization problems
AU - Beck, Amir
AU - Ben-Tal, Aharon
AU - Tetruashvili, Luba
PY - 2012
Y1 - 2012
N2 - In this paper, a method for solving constrained convex optimization problems is introduced. The problem is cast equivalently as a parametric unconstrained one, the (single) parameter being the optimal value of the original problem. At each stage of the algorithm the parameter is updated, and the resulting subproblem is only approximately solved. A linear rate of convergence of the parameter sequence is established. Using an optimal gradient method due to Nesterov [Dokl. Akad. Nauk SSSR, 269 (1983), pp. 543-547] to solve the arising subproblems, it is proved that the resulting gradient-based algorithm requires an overall of O(log(1/e)/√e) inner iterations to obtain an e-optimal and feasible solution. An image deblurring problem is solved, demonstrating the capability of the algorithm to solve large-scale problems within reasonable accuracy.
AB - In this paper, a method for solving constrained convex optimization problems is introduced. The problem is cast equivalently as a parametric unconstrained one, the (single) parameter being the optimal value of the original problem. At each stage of the algorithm the parameter is updated, and the resulting subproblem is only approximately solved. A linear rate of convergence of the parameter sequence is established. Using an optimal gradient method due to Nesterov [Dokl. Akad. Nauk SSSR, 269 (1983), pp. 543-547] to solve the arising subproblems, it is proved that the resulting gradient-based algorithm requires an overall of O(log(1/e)/√e) inner iterations to obtain an e-optimal and feasible solution. An image deblurring problem is solved, demonstrating the capability of the algorithm to solve large-scale problems within reasonable accuracy.
KW - Convex optimization
KW - Image deblurring
KW - Linear rate of convergence
KW - Optimal gradient methods
UR - http://www.scopus.com/inward/record.url?scp=84861581153&partnerID=8YFLogxK
U2 - 10.1137/100800580
DO - 10.1137/100800580
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AN - SCOPUS:84861581153
SN - 1052-6234
VL - 22
SP - 244
EP - 260
JO - SIAM Journal on Optimization
JF - SIAM Journal on Optimization
IS - 1
ER -