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

Долгополик, М. В.; Fominyh, Alexander

doi:10.1002/oca.2530

Cited by 20 publications

(25 citation statements)

References 68 publications

(236 reference statements)

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“…In the light of this remark, one can interpret this assumption as nonlocal LICQ or as an assumption on nonlocal metric regularity of the constraints of the problem (P). Let us also note that nonlocal CQ and nonlocal metric regularity play a central role in the theory of exact penalty functions in the infinite dimensional case [6,21,23,24,48].…”

Section: Properties Of the Augmented Lagrangianmentioning

confidence: 99%

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Exact augmented Lagrangians for constrained optimization problems in Hilbert spaces I: Theory

Долгополик¹

2022

Preprint

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In this two-part study, we develop a general theory of the so-called exact augmented Lagrangians for constrained optimization problems in Hilbert spaces. In contrast to traditional nonsmooth exact penalty functions, these augmented Lagrangians are continuously differentiable for smooth problems and do not suffer from the Maratos effect, which makes them especially appealing for applications in numerical optimization. Our aim is to present a detailed study of various theoretical properties of exact augmented Lagrangians and discuss several applications of these functions to constrained variational problems, problems with PDE constraints, and optimal control problems.The first paper is devoted to a theoretical analysis of an exact augmented Lagrangian for optimization problems in Hilbert spaces. We obtain several useful estimates of this augmented Lagrangian and its gradient, and present several types of sufficient conditions for KKT-points of a constrained problem corresponding to locally/globally optimal solutions to be local/global minimizers of the exact augmented Lagrangian.

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Section: Properties Of the Augmented Lagrangianmentioning

confidence: 99%

“…From the definition of Q(x)[•] (see equality ( 5) on page 8), equalities (24), and the fact that AA * is the identity map it follows that…”

Section: Global Exactnessmentioning

confidence: 99%

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Exact augmented Lagrangians for constrained optimization problems in Hilbert spaces I: Theory

Долгополик¹

2022

Preprint

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“…The proof of Theorem 1 is given in the first part of our study. 45 Remark 1. Let us note that Theorem 1 is valid even in the case when the set Ω ∖Ω is empty.…”

Section: Exact Penalty Functions In Metric Spacesmentioning

confidence: 99%

“…The main goal of this two-part study is to develop a general theory of exact penalty functions for optimal control problems containing sufficient conditions for the complete or local exactness of penalty functions that can be readily verified in various particular cases. In the first part of our study 45 we obtained simple sufficient conditions for the exactness of penalty functions for free-endpoint optimal control problems. This result allows one to apply numerical method for solving variational problems to free-endpoint optimal control problems.…”

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confidence: 99%

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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A Local Search Scheme for the Inequality-Constrained Optimal Control Problem

Strekalovsky

2021

Mathematical Optimization Theory and Operations Research

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

Cited by 20 publications

References 68 publications

Exact augmented Lagrangians for constrained optimization problems in Hilbert spaces I: Theory

Exact augmented Lagrangians for constrained optimization problems in Hilbert spaces I: Theory

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

A Local Search Scheme for the Inequality-Constrained Optimal Control Problem

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