Reformulation of the M-Stationarity Conditions as a System of Discontinuous Equations and Its Solution by a Semismooth Newton Method

Harder, Felix; Mehlitz, Patrick; Wachsmuth, Gerd

doi:10.1137/20m1321413

“…Another interest in Mstationarity arises from the fact that this system can often be solved directly in order to identify reasonable feasible points of (Q), see e.g. [27,29]. In infinite-dimensional optimization, particularly, in optimal control, M-stationarity has turned out to be of limited practical use since the limiting normal cone to nonconvex sets in function spaces is uncomfortably large due to convexification effects arising when taking weak limits, see e.g.…”

Section: Approximate Stationarity and Uniform Qualification Conditionmentioning

confidence: 99%

Optimality conditions, approximate stationarity, and applications – a story beyond lipschitzness

Kruger

¹

,

Mehlitz

²

2022

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Approximate necessary optimality conditions in terms of Fréchet subgradients and normals for a rather general optimization problem with a potentially non-Lipschitzian objective function are established with the aid of Ekeland's variational principle, the fuzzy Fréchet subdifferential sum rule, and a novel notion of lower semicontinuity relative to a set-valued mapping or set. Feasible points satisfying these optimality conditions are referred to as approximately stationary. As applications, we derive a new general version of the extremal principle. Furthermore, we study approximate stationarity conditions for an optimization problem with a composite objective function and geometric constraints, a qualification condition guaranteeing that approximately stationary points of such a problem are M-stationary, and a multiplier-penalty-method which naturally computes approximately stationary points of the underlying problem. Finally, necessary optimality conditions for an optimal control problem with a non-Lipschitzian sparsity-promoting term in the objective function are established.

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“…Example 6.2. We consider the optimal control of a discretized obstacle problem as investigated in [32,Section 7.4]. Using w := (x, y, z), in our notation, the problem is given by min w f (w) := 1 2 x 2 − e y + 1 2 y 2 s.t.…”

Section: Mpcc Examplesmentioning

confidence: 99%

“…The authors present a technique which computes a suitable stationary point of these subproblems in such a way that the entire method generates M-stationary accumulation points for the original MPCC. Let us also mention that [32] suggests to solve (a discontinuous reformulation of) the M-stationarity system associated with an MPCC by means of a semismooth Newton-type method. Naturally, this approach should be robust with respect to (w.r.t.)…”

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confidence: 99%

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An Augmented Lagrangian Method for Optimization Problems with Structured Geometric Constraints

Jia¹,

Kanzow²,

Mehlitz³

et al. 2021

Preprint

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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 problemtailored 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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“…Another interest in M-stationarity arises from the fact that this system can often be solved directly in order to identify reasonable feasible points of (Q), see e.g. Guo et al (2015); Harder et al (2021). In infinite-dimensional optimization, particularly, in optimal control, M-stationarity has turned out to be of limited practical use since the limiting normal cone to nonconvex sets in function spaces is uncomfortably large due to convexification effects arising when taking weak limits, see e.g.…”

Section: Geometrically-constrained Optimization Problems With Composi...mentioning

confidence: 99%

Optimality conditions, approximate stationarity, and applications -- a story beyond Lipschitzness

Kruger¹,

Mehlitz²

2021

Preprint

Self Cite

0

View full text Add to dashboard Cite

Approximate necessary optimality conditions in terms of Fréchet subgradients and normals for a rather general optimization problem with a potentially non-Lipschitzian objective function are established with the aid of Ekeland's variational principle, the fuzzy Fréchet subdifferential sum rule, and a novel notion of lower semicontinuity relative to a set-valued mapping or set. Feasible points satisfying these optimality conditions are referred to as approximately stationary. As applications, we derive a new general version of the extremal principle. Furthermore, we study approximate stationarity conditions for an optimization problem with a composite objective function and geometric constraints, a qualification condition guaranteeing that approximately stationary points of such a problem are M-stationary, and a multiplier-penalty-method which naturally computes approximately stationary points of the underlying problem. Finally, necessary optimality conditions for an optimal control problem with a non-Lipschitzian sparsity-promoting term in the objective function are established.

show abstract

Reformulation of the M-Stationarity Conditions as a System of Discontinuous Equations and Its Solution by a Semismooth Newton Method

Cited by 8 publications

References 30 publications

Optimality conditions, approximate stationarity, and applications – a story beyond lipschitzness

Optimality conditions, approximate stationarity, and applications – a story beyond lipschitzness

An Augmented Lagrangian Method for Optimization Problems with Structured Geometric Constraints

Optimality conditions, approximate stationarity, and applications -- a story beyond Lipschitzness

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