TY - JOUR
T1 - Nonconvex nonsmooth optimization via convex–nonconvex majorization–minimization
AU - Lanza, A.
AU - Morigi, S.
AU - Selesnick, I.
AU - Sgallari, F.
N1 - Funding Information:
We would like to thank the referees for comments that lead to improvements of the presentation. Research by IS was supported by the NSF (USA) under Grant No. CCF-1525398. Research by SM and FS was supported in part by the National Group for Scientific Computation (GNCS-INDAM), Research Projects 2015.
Publisher Copyright:
© 2016, Springer-Verlag Berlin Heidelberg.
PY - 2017/6/1
Y1 - 2017/6/1
N2 - The class of majorization–minimization algorithms is based on the principle of successively minimizing upper bounds of the objective function. Each upper bound, or surrogate function, is locally tight at the current estimate, and each minimization step decreases the value of the objective function. We present a majorization–minimization approach based on a novel convex–nonconvex upper bounding strategy for the solution of a certain class of nonconvex nonsmooth optimization problems. We propose an efficient algorithm for minimizing the (convex) surrogate function based on the alternating direction method of multipliers. A preliminary convergence analysis for the proposed approach is provided. Numerical experiments show the effectiveness of the proposed method for the solution of nonconvex nonsmooth minimization problems.
AB - The class of majorization–minimization algorithms is based on the principle of successively minimizing upper bounds of the objective function. Each upper bound, or surrogate function, is locally tight at the current estimate, and each minimization step decreases the value of the objective function. We present a majorization–minimization approach based on a novel convex–nonconvex upper bounding strategy for the solution of a certain class of nonconvex nonsmooth optimization problems. We propose an efficient algorithm for minimizing the (convex) surrogate function based on the alternating direction method of multipliers. A preliminary convergence analysis for the proposed approach is provided. Numerical experiments show the effectiveness of the proposed method for the solution of nonconvex nonsmooth minimization problems.
KW - 49J52 Nonsmooth analysis
KW - 65K05 Mathematical programming
KW - 65K10 Optimization and variational techniques
KW - 90C26 Nonconvex programming, global optimization
UR - http://www.scopus.com/inward/record.url?scp=84991047719&partnerID=8YFLogxK
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U2 - 10.1007/s00211-016-0842-x
DO - 10.1007/s00211-016-0842-x
M3 - Article
AN - SCOPUS:84991047719
SN - 0029-599X
VL - 136
SP - 343
EP - 381
JO - Numerische Mathematik
JF - Numerische Mathematik
IS - 2
ER -