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 -