A proximal-gradient inertial algorithm with Tikhonov regularization: strong convergence to the minimal norm solution
We investigate the strong convergence properties of a proximal-gradient inertial algorithm with two Tikhonov regularization terms in connection with the minimization problem of the sum of a convex lower semi-continuous function f and a smooth convex function g. For the appropriate setting of the par...
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| Published in | Optimization methods & software Vol. 40; no. 4; pp. 947 - 976 |
|---|---|
| Main Author | |
| Format | Journal Article |
| Language | English |
| Published |
Taylor & Francis
04.07.2025
|
| Subjects | |
| Online Access | Get full text |
| ISSN | 1055-6788 1029-4937 |
| DOI | 10.1080/10556788.2025.2517172 |
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| Abstract | We investigate the strong convergence properties of a proximal-gradient inertial algorithm with two Tikhonov regularization terms in connection with the minimization problem of the sum of a convex lower semi-continuous function f and a smooth convex function g. For the appropriate setting of the parameters, we provide the strong convergence of the generated sequence
$ (x_k){_{k\ge 0}} $
(
x
k
)
k
≥
0
to the minimum norm minimizer of our objective function f + g. Further, we obtain fast convergence to zero of the objective function values in a generated sequence but also for the discrete velocity and the sub-gradient of the objective function. We also show that for another setting of the parameters the optimal rate of order
$ \mathcal {O}(k^{-2}) $
O
(
k
−
2
)
for the potential energy
$ (f+g)(x_k)-\min (f+g) $
(
f
+
g
)
(
x
k
)
−
min
(
f
+
g
)
can be obtained. |
|---|---|
| AbstractList | We investigate the strong convergence properties of a proximal-gradient inertial algorithm with two Tikhonov regularization terms in connection with the minimization problem of the sum of a convex lower semi-continuous function f and a smooth convex function g. For the appropriate setting of the parameters, we provide the strong convergence of the generated sequence
$ (x_k){_{k\ge 0}} $
(
x
k
)
k
≥
0
to the minimum norm minimizer of our objective function f + g. Further, we obtain fast convergence to zero of the objective function values in a generated sequence but also for the discrete velocity and the sub-gradient of the objective function. We also show that for another setting of the parameters the optimal rate of order
$ \mathcal {O}(k^{-2}) $
O
(
k
−
2
)
for the potential energy
$ (f+g)(x_k)-\min (f+g) $
(
f
+
g
)
(
x
k
)
−
min
(
f
+
g
)
can be obtained. |
| Author | László, Szilárd Csaba |
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| Cites_doi | 10.1007/s10589-024-00620-5 10.1006/jdeq.2001.4034 10.1016/j.jmaa.2016.12.017 10.1007/s00186-024-00867-y 10.1007/s10957-015-0746-4 10.1016/j.jmaa.2023.127689 10.1137/20M1382027 10.1137/080716542 10.1007/s10957-018-1369-3 10.1007/s00245-023-09997-x 10.1007/s00211-015-0751-4 10.1007/s00013-010-0181-6 10.1007/978-1-4419-8853-9 10.1006/jdeq.1996.0104 10.1137/15M1046095 10.1007/978-1-4419-9467-7 10.1007/s10107-020-01591-1 10.1007/s10107-020-01528-8 10.1016/j.cnsns.2025.108924 10.1016/j.jde.2008.08.007 10.1007/s10589-023-00536-6 10.1016/j.jde.2023.03.014 10.1016/j.jde.2021.12.005 10.1007/s00245-024-10163-0 10.1515/anona-2020-0143 |
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| References | e_1_3_4_4_1 e_1_3_4_3_1 e_1_3_4_2_1 e_1_3_4_9_1 e_1_3_4_8_1 e_1_3_4_7_1 e_1_3_4_20_1 e_1_3_4_6_1 e_1_3_4_5_1 e_1_3_4_23_1 e_1_3_4_24_1 e_1_3_4_21_1 e_1_3_4_22_1 e_1_3_4_28_1 e_1_3_4_25_1 e_1_3_4_26_1 e_1_3_4_30_1 e_1_3_4_12_1 e_1_3_4_13_1 e_1_3_4_10_1 Tikhonov A.N. (e_1_3_4_31_1) 1977 e_1_3_4_11_1 e_1_3_4_16_1 e_1_3_4_17_1 e_1_3_4_14_1 e_1_3_4_15_1 e_1_3_4_18_1 e_1_3_4_19_1 Nesterov Y. (e_1_3_4_27_1) 1983; 27 Stolz O. (e_1_3_4_29_1) 1885 |
| References_xml | – ident: e_1_3_4_26_1 doi: 10.1007/s10589-024-00620-5 – ident: e_1_3_4_4_1 doi: 10.1006/jdeq.2001.4034 – ident: e_1_3_4_8_1 doi: 10.1016/j.jmaa.2016.12.017 – ident: e_1_3_4_5_1 doi: 10.1007/s00186-024-00867-y – ident: e_1_3_4_10_1 – ident: e_1_3_4_19_1 doi: 10.1007/s10957-015-0746-4 – ident: e_1_3_4_18_1 doi: 10.1016/j.jmaa.2023.127689 – ident: e_1_3_4_2_1 doi: 10.1137/20M1382027 – ident: e_1_3_4_15_1 doi: 10.1137/080716542 – ident: e_1_3_4_9_1 doi: 10.1007/s10957-018-1369-3 – ident: e_1_3_4_13_1 doi: 10.1007/s00245-023-09997-x – ident: e_1_3_4_7_1 doi: 10.1007/s00211-015-0751-4 – ident: e_1_3_4_21_1 doi: 10.1007/s00013-010-0181-6 – ident: e_1_3_4_28_1 doi: 10.1007/978-1-4419-8853-9 – ident: e_1_3_4_3_1 doi: 10.1006/jdeq.1996.0104 – volume-title: Solutions of Ill-Posed Problems year: 1977 ident: e_1_3_4_31_1 – ident: e_1_3_4_6_1 doi: 10.1137/15M1046095 – ident: e_1_3_4_14_1 doi: 10.1007/978-1-4419-9467-7 – volume: 27 start-page: 372 year: 1983 ident: e_1_3_4_27_1 article-title: A method of solving a convex programming problem with convergence rate O(1/k2) publication-title: Soviet Math. Dokl. – ident: e_1_3_4_12_1 doi: 10.1007/s10107-020-01591-1 – ident: e_1_3_4_17_1 doi: 10.1007/s10107-020-01528-8 – ident: e_1_3_4_25_1 doi: 10.1016/j.cnsns.2025.108924 – ident: e_1_3_4_20_1 doi: 10.1016/j.jde.2008.08.007 – ident: e_1_3_4_22_1 doi: 10.1007/s10589-023-00536-6 – ident: e_1_3_4_24_1 doi: 10.1016/j.jde.2023.03.014 – ident: e_1_3_4_11_1 doi: 10.1016/j.jde.2021.12.005 – volume-title: Vorlesungen über allgemeine Arithmetik: nach den Neueren Ansichten year: 1885 ident: e_1_3_4_29_1 – ident: e_1_3_4_23_1 doi: 10.1007/s00245-024-10163-0 – ident: e_1_3_4_16_1 doi: 10.1515/anona-2020-0143 – ident: e_1_3_4_30_1 |
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| SubjectTerms | convex optimization Inertial algorithm optimal rate proximal-gradient algorithm strong convergence Tikhonov regularization |
| Title | A proximal-gradient inertial algorithm with Tikhonov regularization: strong convergence to the minimal norm solution |
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