TY - JOUR
T1 - The regularized feasible directions method for nonconvex optimization
AU - Beck, Amir
AU - Hallak, Nadav
N1 - Publisher Copyright:
© 2022 Elsevier B.V.
PY - 2022/9
Y1 - 2022/9
N2 - This paper develops and studies a feasible directions approach for the minimization of a continuous function over linear constraints in which the update directions belong to a predetermined finite set spanning the feasible set. These directions are recurrently investigated in a cyclic semi-random order, where the stepsize of the update is determined via univariate optimization. We establish that any accumulation point of this optimization procedure is a stationary point of the problem, meaning that the directional derivative in any feasible direction is nonnegative. To assess and establish a rate of convergence, we develop a new optimality measure that acts as a proxy for the stationarity condition, and substantiate its role by showing that it is coherent with first-order conditions in specific scenarios. Finally we prove that our method enjoys a sublinear rate of convergence of this optimality measure in expectation.
AB - This paper develops and studies a feasible directions approach for the minimization of a continuous function over linear constraints in which the update directions belong to a predetermined finite set spanning the feasible set. These directions are recurrently investigated in a cyclic semi-random order, where the stepsize of the update is determined via univariate optimization. We establish that any accumulation point of this optimization procedure is a stationary point of the problem, meaning that the directional derivative in any feasible direction is nonnegative. To assess and establish a rate of convergence, we develop a new optimality measure that acts as a proxy for the stationarity condition, and substantiate its role by showing that it is coherent with first-order conditions in specific scenarios. Finally we prove that our method enjoys a sublinear rate of convergence of this optimality measure in expectation.
KW - Constrained optimization
KW - Convergence analysis
KW - Feasible directions
KW - Nonconvex optimization
KW - Nonsmooth optimization
UR - http://www.scopus.com/inward/record.url?scp=85135867698&partnerID=8YFLogxK
U2 - 10.1016/j.orl.2022.07.005
DO - 10.1016/j.orl.2022.07.005
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AN - SCOPUS:85135867698
SN - 0167-6377
VL - 50
SP - 517
EP - 523
JO - Operations Research Letters
JF - Operations Research Letters
IS - 5
ER -