On the Penalization Approach to Optimal Control Problems

Demyanov, V. F.; Giannessi, Franco; Karelin, Vladimir V.

doi:10.1016/s1474-6670(17)39605-2

Cited by 3 publications

(17 citation statements)

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“…The main results on exact penalty functions for various variational problems from other works [13][14][15][16][17][18][19] are based on the assumptions that the objective function is Lipschitz continuous on a rather complicated and possibly unbounded set, and a penalty function attains a global minimum in the space of piecewise continuous functions for any sufficiently large value of the penalty parameter, and it is, once again, unclear how to verify these assumptions in any particular case. The same remark is true for the main results of the papers [9][10][11][12]20,21 devoted to exact penalty functions for optimal control problems. To the best of authors' knowledge, the only verifiable sufficient conditions for the global exactness of an exact penalty function in the infinite dimensional setting were obtained in the work of Gugat and Zuazua, 26 where the exact penalisation of the terminal constraint for optimal control problems involving linear evolution equations was considered.…”

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confidence: 65%

“…From the second inequality in (9) and the fact that is bounded below, it obviously follows that, for any ≥ 0, the penalty function Φ is bounded below on A, and the set S (c) is bounded in L d s (0, T) × L m q (0, T). By applying (11) and the first inequality in (9), one obtains that, for any (…”

Section: Proposition 1 Let and F Be Continuous And One Of The Follomentioning

confidence: 99%

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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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confidence: 65%

Section: Proposition 1 Let and F Be Continuous And One Of The Follomentioning

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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“…Recall that one says that the Mangasarian-Fromovitz constraint qualifications (MFCQ) holds at a point x T ∈ R d , if the gradients ∇g k (x T ), k ∈ J, are linearly independent, and there exists h ∈ R d such that ⟨∇g k (x T ), h⟩ = 0 for any k ∈ J, and ⟨∇g i (x T ), h⟩ < 0 for any i ∈ I(x T ). Let us show that the complete controllability of the linearized system along with MFCQ guarantee the local exactness of the penalty function Φ for problem (39) Theorem 12. Let U = L m q (0, T), q ≥ p, and (x*, u*) be a locally optimal solution of problem (39).…”

Section: Variable-endpoint Problemsmentioning

confidence: 99%

“…Let us show that the complete controllability of the linearized system along with MFCQ guarantee the local exactness of the penalty function Φ for problem (39) Theorem 12. Let U = L m q (0, T), q ≥ p, and (x*, u*) be a locally optimal solution of problem (39). Suppose also that Assumptions 1, 2, and 3 of Theorem 11 are satisfied, is locally Lipschitz continuous, the functions g i , i ∈ I ∪ J are continuously differentiable in a neighbourhood of x*(T), and MFCQ holds true at the point x*(T).…”

Section: Variable-endpoint Problemsmentioning

confidence: 99%

“…41 However, the main results of these papers are based on the assumptions that the penalty function attains a global minimum in the space of piecewise continuous functions for any sufficiently large value of the penalty parameter, and the cost functional is Lipschitz continuous on a possibly unbounded and rather complicated set. It is unclear how to verify these assumptions in particular cases, which makes it very difficult to apply the main results of papers [37][38][39][40][41] to real problems. To the best of author's knowledge, the only verifiable sufficient conditions for the global exactness of penalty functions for optimal control problems were obtained by Gugat 42 for an optimal control of the wave equation, by Gugat and Zuazua 43 for optimal control problems for general linear evolution equations, and by Jayswal and Preeti 44 for an optimal control problem for a system described by a partial differential equation state inequality constraints.…”

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confidence: 99%

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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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An adaptive exact penalty method for nonsmooth optimal control problems with nonsmooth nonconvex state and control constraints

Долгополик¹

2023

Preprint

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A class of exact penalty-type local search methods for optimal control problems with nonsmooth cost functional, nonsmooth (but continuous) dynamics, and nonsmooth state and control constraints is presented, in which the the penalty parameter and several line search parameters are adaptively adjusted during the optimisation process. This class of methods is applicable to problems having a known DC (Differenceof-Convex functions) structure and in its core is based on the classical DCA method, combined with the steering exact penalty rules for updating the penalty parameter and an adaptive nonmonotone line search procedure. Under the assumption that all auxiliary subproblems are solved only approximately (that is, with finite precision), we prove the correctness of the proposed family of methods and present its detailed convergence analysis. The performance of several different versions of the method is illustrated by means of a numerical example, in which the methods are applied to a semi-academic optimal control problem with a nonsmooth nonconvex state constraint.

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On the Penalization Approach to Optimal Control Problems

Cited by 3 publications

References 5 publications

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

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

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

An adaptive exact penalty method for nonsmooth optimal control problems with nonsmooth nonconvex state and control constraints

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