A Newton-CG based barrier-augmented Lagrangian method for general nonconvex conic optimization
In this paper we consider finding an approximate second-order stationary point (SOSP) of general nonconvex conic optimization that minimizes a twice differentiable function subject to nonlinear equality constraints and also a convex conic constraint. In particular, we propose a Newton-conjugate grad...
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| Published in | Computational optimization and applications Vol. 89; no. 3; pp. 843 - 894 |
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| Main Authors | , , |
| Format | Journal Article |
| Language | English |
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01.12.2024
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| Abstract | In this paper we consider finding an approximate second-order stationary point (SOSP) of general nonconvex conic optimization that minimizes a twice differentiable function subject to nonlinear equality constraints and also a convex conic constraint. In particular, we propose a Newton-conjugate gradient (Newton-CG) based barrier-augmented Lagrangian method for finding an approximate SOSP of this problem. Under some mild assumptions, we show that our method enjoys a total inner iteration complexity of
O
~
(
ϵ
-
11
/
2
)
and an operation complexity of
O
~
(
ϵ
-
11
/
2
min
{
n
,
ϵ
-
5
/
4
}
)
for finding an
(
ϵ
,
ϵ
)
-SOSP of general nonconvex conic optimization with high probability. Moreover, under a constraint qualification, these complexity bounds are improved to
O
~
(
ϵ
-
7
/
2
)
and
O
~
(
ϵ
-
7
/
2
min
{
n
,
ϵ
-
3
/
4
}
)
, respectively. To the best of our knowledge, this is the first study on the complexity of finding an approximate SOSP of general nonconvex conic optimization. Preliminary numerical results are presented to demonstrate superiority of the proposed method over first-order methods in terms of solution quality. |
|---|---|
| AbstractList | In this paper we consider finding an approximate second-order stationary point (SOSP) of general nonconvex conic optimization that minimizes a twice differentiable function subject to nonlinear equality constraints and also a convex conic constraint. In particular, we propose a Newton-conjugate gradient (Newton-CG) based barrier-augmented Lagrangian method for finding an approximate SOSP of this problem. Under some mild assumptions, we show that our method enjoys a total inner iteration complexity of
O
~
(
ϵ
-
11
/
2
)
and an operation complexity of
O
~
(
ϵ
-
11
/
2
min
{
n
,
ϵ
-
5
/
4
}
)
for finding an
(
ϵ
,
ϵ
)
-SOSP of general nonconvex conic optimization with high probability. Moreover, under a constraint qualification, these complexity bounds are improved to
O
~
(
ϵ
-
7
/
2
)
and
O
~
(
ϵ
-
7
/
2
min
{
n
,
ϵ
-
3
/
4
}
)
, respectively. To the best of our knowledge, this is the first study on the complexity of finding an approximate SOSP of general nonconvex conic optimization. Preliminary numerical results are presented to demonstrate superiority of the proposed method over first-order methods in terms of solution quality. In this paper we consider finding an approximate second-order stationary point (SOSP) of general nonconvex conic optimization that minimizes a twice differentiable function subject to nonlinear equality constraints and also a convex conic constraint. In particular, we propose a Newton-conjugate gradient (Newton-CG) based barrier-augmented Lagrangian method for finding an approximate SOSP of this problem. Under some mild assumptions, we show that our method enjoys a total inner iteration complexity of O(& varepsilon;(-11/2)) and an operation complexity of O(& varepsilon;(-11/2)min{n,& varepsilon;(-5/4)}) for finding an (& varepsilon;,root & varepsilon;)-SOSP of general nonconvex conic optimization with high probability. Moreover, under a constraint qualification, these complexity bounds are improved to O(& varepsilon;(-7/2)) and O(& varepsilon;(-7/2m)in{n,& varepsilon;(-3/4)}), respectively. To the best of our knowledge, this is the first study on the complexity of finding an approximate SOSP of general nonconvex conic optimization. Preliminary numerical results are presented to demonstrate superiority of the proposed method over first-order methods in terms of solution quality. |
| Author | Lu, Zhaosong He, Chuan Huang, Heng |
| Author_xml | – sequence: 1 givenname: Chuan surname: He fullname: He, Chuan organization: Department of Computer Science and Engineering, University of Minnesota, Department of Mathematics, Linköping University – sequence: 2 givenname: Heng surname: Huang fullname: Huang, Heng organization: Department of Computer Science, University of Maryland – sequence: 3 givenname: Zhaosong orcidid: 0000-0003-3277-7853 surname: Lu fullname: Lu, Zhaosong email: zhaosong@umn.edu organization: Department of Industrial and Systems Engineering, University of Minnesota |
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| Keywords | Augmented Lagrangian method 68Q25 Operation complexity 90C26 Newton-conjugate gradient method Iteration complexity Barrier method 90C30 Nonconvex conic optimization 49M05 49M15 Second-order stationary point 90C60 |
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| Snippet | In this paper we consider finding an approximate second-order stationary point (SOSP) of general nonconvex conic optimization that minimizes a twice... |
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| SubjectTerms | Convex and Discrete Geometry Management Science Mathematics Mathematics and Statistics Operations Research Operations Research/Decision Theory Optimization Statistics |
| Title | A Newton-CG based barrier-augmented Lagrangian method for general nonconvex conic optimization |
| URI | https://link.springer.com/article/10.1007/s10589-024-00603-6 https://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-207432 |
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