An Augmented Lagrangian Method for Optimization Problems with Structured Geometric Constraints

Jia, Xiaoxi; Kanzow, Christian; Mehlitz, Patrick; Wachsmuth, Gerd

doi:10.48550/arxiv.2105.08317

Cited by 4 publications

(6 citation statements)

References 45 publications

(85 reference statements)

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“…The latter amounts to a truncated singular value decomposition (SVD). It appears that the best known result for PGD applied to (R) is that its accumulation points are Mordukhovich stationary [29,Thm. 3.4].…”

Section: Introductionmentioning

confidence: 99%

Finding stationary points on bounded-rank matrices: a geometric hurdle and a smooth remedy

2022

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We consider the problem of provably finding a stationary point of a smooth function to be minimized on the variety of bounded-rank matrices. This turns out to be unexpectedly delicate. We trace the difficulty back to a geometric obstacle: On a nonsmooth set, there may be sequences of points along which standard measures of stationarity tend to zero, but whose limit points are not stationary. We name such events apocalypses, as they can cause optimization algorithms to converge to non-stationary points. We illustrate this explicitly for an existing optimization algorithm on bounded-rank matrices. To provably find stationary points, we modify a trust-region method on a standard smooth parameterization of the variety. The method relies on the known fact that second-order stationary points on the parameter space map to stationary points on the variety. Our geometric observations and proposed algorithm generalize beyond boundedrank matrices. We give a geometric characterization of apocalypses on general constraint sets, which implies that Clarke-regular sets do not admit apocalypses. Such sets include smooth manifolds, manifolds with boundaries, and convex sets. Our trust-region method supports parameterization by any complete Riemannian manifold.

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Section: Introductionmentioning

confidence: 99%

Finding stationary points on bounded-rank matrices: a geometric hurdle and a smooth remedy

2022

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“…The optimal function value f opt of all these examples is known. The details of the corresponding results obtained by our method are given in [43]. Here, we summarize the main observations.…”

Section: Maxcut Problemsmentioning

confidence: 81%

“…Since the feasible set of (6.4) is larger than the one of (6.3), we have the inequalities f ALM ≤ f opt ≤ f SDP . The corresponding details for the solution of the SDP-relaxation are provided in [43] for the rudy collection.…”

Section: Maxcut Problemsmentioning

confidence: 99%

An augmented Lagrangian method for optimization problems with structured geometric constraints

et al. 2022

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This paper is devoted to the theoretical and numerical investigation of an augmented Lagrangian method for the solution of optimization problems with geometric constraints. Specifically, we study situations where parts of the constraints are nonconvex and possibly complicated, but allow for a fast computation of projections onto this nonconvex set. Typical problem classes which satisfy this requirement are optimization problems with disjunctive constraints (like complementarity or cardinality constraints) as well as optimization problems over sets of matrices which have to satisfy additional rank constraints. The key idea behind our method is to keep these complicated constraints explicitly in the constraints and to penalize only the remaining constraints by an augmented Lagrangian function. The resulting subproblems are then solved with the aid of a problem-tailored nonmonotone projected gradient method. The corresponding convergence theory allows for an inexact solution of these subproblems. Nevertheless, the overall algorithm computes so-called Mordukhovich-stationary points of the original problem under a mild asymptotic regularity condition, which is generally weaker than most of the respective available problem-tailored constraint qualifications. Extensive numerical experiments addressing complementarity- and cardinality-constrained optimization problems as well as a semidefinite reformulation of MAXCUT problems visualize the power of our approach.

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mentioning

confidence: 99%

Finding stationary points on bounded-rank matrices: A geometric hurdle and a smooth remedy

Levin,

Kileel,

Boumal

2021

Preprint

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We consider the problem of provably finding a stationary point of a smooth function to be minimized on the variety of bounded-rank matrices. While optimization on low-rank matrices has been extensively studied, existing algorithms do not provide such a basic guarantee. We trace this back to a geometric obstacle: On a nonsmooth set, there may be sequences of points along which standard measures of stationarity tend to zero, but whose limit point is not stationary. We name such events apocalypses, as they can cause optimization algorithms to converge to non-stationary points. We illustrate this on an existing algorithm using an explicit apocalypse on the bounded-rank matrix variety. To provably find stationary points, we modify a trust-region method on a standard smooth parameterization of the variety. The method relies on the known fact that second-order stationary points on the parameter space map to stationary points on the variety. Our geometric observations and proposed algorithm generalize beyond bounded-rank matrices. We give a geometric characterization of apocalypses on general constraint sets, which implies that Clarke-regular sets do not admit apocalypses. Such sets include smooth manifolds, manifolds with boundaries, and convex sets. Our trust-region method supports parameterization by any complete Riemannian manifold.

show abstract

An Augmented Lagrangian Method for Optimization Problems with Structured Geometric Constraints

Cited by 4 publications

References 45 publications

Finding stationary points on bounded-rank matrices: a geometric hurdle and a smooth remedy

Finding stationary points on bounded-rank matrices: a geometric hurdle and a smooth remedy

An augmented Lagrangian method for optimization problems with structured geometric constraints

Finding stationary points on bounded-rank matrices: A geometric hurdle and a smooth remedy

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