A Strong Metric Subregularity Analysis of Nonsmooth Mappings Via Steepest Displacement Rate

Uderzo, Amos

doi:10.1007/s10957-016-0952-8

Cited by 11 publications

(17 citation statements)

References 42 publications

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“…[17,Section 2.1]. For the most recent developments in research on metric subregularity, we refer the readers to [2,6,15,18,19,22,23,26,27,31,33,34].…”

mentioning

confidence: 99%

The Radius of Metric Subregularity

Dontchev

Gfrerer

Kruger

et al. 2019

Set-Valued Var. Anal

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There is a basic paradigm, called here the radius of well-posedness, which quantifies the "distance" from a given well-posed problem to the set of ill-posed problems of the same kind. In variational analysis, well-posedness is often understood as a regularity property, which is usually employed to measure the effect of perturbations and approximations of a problem on its solutions. In this paper we focus on evaluating the radius of the property of metric subregularity which, in contrast to its siblings, metric regularity, strong regularity and strong subregularity, exhibits a more complicated behavior under various perturbations. We consider three kinds of perturbations: by Lipschitz continuous functions, by semismooth functions, and by smooth functions, obtaining different expressions/bounds for the radius of subregularity, which involve generalized derivatives of set-valued mappings. We also obtain different expressions when using either Frobenius or Euclidean norm to measure the radius. As an application, we evaluate the radius of subregularity of a general constraint system. Examples illustrate the theoretical findings.

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“…[17,Section 2.1]. For the most recent developments in research on metric subregularity, we refer the readers to [2,6,15,18,19,22,23,26,27,31,33,34].…”

mentioning

confidence: 99%

The Radius of Metric Subregularity

Dontchev

Gfrerer

Kruger

et al. 2019

Set-Valued Var. Anal

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“…By applying the continuity of the mappings x  → g j (x(⋅), ⋅) once again one gets that there exists 2 > 0 such that for any j ∈ J(x) and ∈ [0, 2 ] one has g j (x(t) + Δx(t), t) > 0 for any t ∈ T j (x), g j (x(t j ) + Δx(t j ), t j ) > 2 (x, u)∕3, while g j (x(t) + Δx(t), t) < 2 (x, u)∕3 for any t ∉ T j (x). Hence and from equality (48) it follows that…”

Section: Linear Evolution Equationsmentioning

confidence: 93%

Exact penalty functions for optimal control problems II: Exact penalization of terminal and pointwise state constraints

Долгополик

2020

Optim Control Appl Methods

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The second part of our study is devoted to an analysis of the exactness of penalty functions for optimal control problems with terminal and pointwise state constraints. We demonstrate that with the use of the exact penalty function method one can reduce fixed-endpoint problems for linear time-varying systems and linear evolution equations with convex constraints on the control inputs to completely equivalent free-endpoint optimal control problems, if the terminal state belongs to the relative interior of the reachable set. In the nonlinear case, we prove that a local reduction of fixed-endpoint and variable-endpoint problems to equivalent free-endpoint ones is possible under the assumption that the linearized system is completely controllable, and point out some general properties of nonlinear systems under which a global reduction to equivalent free-endpoint problems can be achieved. In the case of problems with pointwise state inequality constraints, we prove that such problems for linear time-varying systems and linear evolution equations with convex state constraints can be reduced to equivalent problems without state constraints, provided one uses the L ∞ penalty term, and Slater's condition holds true, while for nonlinear systems a local reduction is possible, if a natural constraint qualification is satisfied. Finally, we show that the exact L p -penalization of state constraints with finite p is possible for convex problems, if Lagrange multipliers corresponding to the state constraints belong to L p ′ , where p ′ is the conjugate exponent of p, and for general nonlinear problems, if the cost functional does not depend on the control inputs explicitly.

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“…Let us provide simple sufficient conditions for the global exactness of the penalty function Φ λ . To this end, we need to recall the definition of the rate of steepest descent of a function defined on a metric space . Let

g : X \to double-struckR \cup false{+ \infty false}

be a given function, K ⊂ X be a nonempty set, and x ∈ K be such that g ( x )<+ ∞ .…”

Section: Exact Penalty Functions In Metric Spacesmentioning

confidence: 99%

“…To this end, we need to recall the definition of the rate of steepest descent of a function defined on a metric space. [50][51][52] Let g ∶ X → ℝ ∪ {+∞} be a given function, K ⊂ X be a nonempty set, and x ∈ K be such that g(x) < +∞. The quantity…”

Section: Definitionmentioning

confidence: 99%

Exact penalty functions for optimal control problems I: Main theorem and free‐endpoint problems

Долгополик

Fominyh

2019

Optim Control Appl Methods

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Summary In this two‐part study, we develop a general approach to the design and analysis of exact penalty functions for various optimal control problems, including problems with terminal and state constraints, problems involving differential inclusions, and optimal control problems for linear evolution equations. This approach allows one to simplify an optimal control problem by removing some (or all) constraints of this problem with the use of an exact penalty function, thus allowing one to reduce optimal control problems to equivalent variational problems and apply numerical methods for solving, eg, problems without state constraints, to problems including such constraints, etc. In the first part of our study, we strengthen some existing results on exact penalty functions for optimisation problems in infinite dimensional spaces and utilise them to study exact penalty functions for free‐endpoint optimal control problems, which reduce these problems to equivalent variational ones. We also prove several auxiliary results on integral functionals and Nemytskii operators that are helpful for verifying the assumptions under which the proposed penalty functions are exact.

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A Strong Metric Subregularity Analysis of Nonsmooth Mappings Via Steepest Displacement Rate

Cited by 11 publications

References 42 publications

The Radius of Metric Subregularity

The Radius of Metric Subregularity

Exact penalty functions for optimal control problems II: Exact penalization of terminal and pointwise state constraints

Exact penalty functions for optimal control problems I: Main theorem and free‐endpoint problems

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