On the nonuniqueness and instability of solutions of tracking-type optimal control problems

Christof, Constantin; Hafemeyer, Dominik

doi:10.3934/mcrf.2021028

Cited by 4 publications

(5 citation statements)

References 29 publications

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“…He proved the following: The functional J is convex for all z ∈ H if and only if G is affine. This result was extended in [16] to quite general tracking type functionals and applied to control-to-state mappings G associated with a nonsmooth elliptic equation, a Signorini type variational inequality, and an evolutionary obstacle problem.…”

Section: The Problemmentioning

confidence: 99%

“…Recently, in [24] this question was answered for a semilinear elliptic boundary control problem by constructing a problem with two different optimal solutions. The reader is also referred to [16], where the non-uniqueness of minimizers is established for abstract tracking type problems with quite general state equations generating non affine-linear control-to-state mappings.…”

Section: Introductionmentioning

confidence: 99%

“…While [16] and [24] prove the existence of problems with non-unique minimizers, they do not construct a concrete example. In our paper, we proceed in a different way and provide a concrete example with two different local minimizers.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Stability for Semilinear Parabolic Optimal Control Problems with Respect to Initial Data

Casas

Tröltzsch

2022

Appl Math Optim

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A distributed optimal control problem for a semilinear parabolic partial differential equation is investigated. The stability of locally optimal solutions with respect to perturbations of the initial data is studied. Based on different types of sufficient optimality conditions for a local solution of the unperturbed problem, Lipschitz or Hölder stability with respect to perturbations are proved. Moreover, a particular example with semilinear equation, constant initial data, and standard quadratic tracking type objective functional is constructed that has at least two different locally optimal solutions. By the perturbation analysis, the existence of a problem with non-constant initial data is shown that also has at least two different locally optimal solutions.

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Section: The Problemmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Stability for Semilinear Parabolic Optimal Control Problems with Respect to Initial Data

Casas

Tröltzsch

2022

Appl Math Optim

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“…The above non-uniqueness result has since been extended to more abstract settings by means of techniques using convexity properties of Chebyshev sets (Christof and Hafemeyer 2022), and has also been explored for finite-dimensional control systems in Trélat (2020). We discuss the latter further in Section 15.…”

Section: Theorem 91 (Pighin 2020) Supposementioning

confidence: 99%

Turnpike in optimal control of PDEs, ResNets, and beyond

Geshkovski

Zuazua²

2022

Acta Numerica

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The turnpike property in contemporary macroeconomics asserts that if an economic planner seeks to move an economy from one level of capital to another, then the most efficient path, as long as the planner has enough time, is to rapidly move stock to a level close to the optimal stationary or constant path, then allow for capital to develop along that path until the desired term is nearly reached, at which point the stock ought to be moved to the final target. Motivated in part by its nature as a resource allocation strategy, over the past decade, the turnpike property has also been shown to hold for several classes of partial differential equations arising in mechanics. When formalized mathematically, the turnpike theory corroborates insights from economics: for an optimal control problem set in a finite-time horizon, optimal controls and corresponding states are close (often exponentially) most of the time, except near the initial and final times, to the optimal control and the corresponding state for the associated stationary optimal control problem. In particular, the former are mostly constant over time. This fact provides a rigorous meaning to the asymptotic simplification that some optimal control problems appear to enjoy over long time intervals, allowing the consideration of the corresponding stationary problem for computing and applications. We review a slice of the theory developed over the past decade – the controllability of the underlying system is an important ingredient, and can even be used to devise simple turnpike-like strategies which are nearly optimal – and present several novel applications, including, among many others, the characterization of Hamilton–Jacobi–Bellman asymptotics, and stability estimates in deep learning via residual neural networks.

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“…For this choice, (P) becomes a standard L p -tracking-type problem as often considered in the field of optimal control, cf. [12] and the references therein. A further interesting example is the case…”

Section: Note That In the Casementioning

confidence: 99%

On the Omnipresence of Spurious Local Minima in Certain Neural Network Training Problems

Christof¹,

Kowalczyk²

2022

Preprint

Self Cite

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We study the loss landscape of training problems for deep artificial neural networks with a one-dimensional real output whose activation functions contain an affine segment and whose hidden layers have width at least two. It is shown that such problems possess a continuum of spurious (i.e., not globally optimal) local minima for all target functions that are not affine. In contrast to previous works, our analysis covers all sampling and parameterization regimes, general differentiable loss functions, arbitrary continuous nonpolynomial activation functions, and both the finite-and infinite-dimensional setting. It is further shown that the appearance of the spurious local minima in the considered training problems is a direct consequence of the universal approximation theorem and that the underlying mechanisms also cause, e.g., L p -best approximation problems to be ill-posed in the sense of Hadamard for all networks that do not have a dense image. The latter result also holds without the assumption of local affine linearity and without any conditions on the hidden layers. The paper concludes with a numerical experiment which demonstrates that spurious local minima can indeed affect the convergence behavior of gradient-based solution algorithms in practice.

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On the nonuniqueness and instability of solutions of tracking-type optimal control problems

Cited by 4 publications

References 29 publications

Stability for Semilinear Parabolic Optimal Control Problems with Respect to Initial Data

Stability for Semilinear Parabolic Optimal Control Problems with Respect to Initial Data

Turnpike in optimal control of PDEs, ResNets, and beyond

On the Omnipresence of Spurious Local Minima in Certain Neural Network Training Problems

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