A new class of exact penalty functions and penalty algorithms

Wang, Changyu; Ma, Cheng; Zhou, Jinchuan

doi:10.1007/s10898-013-0111-9

Cited by 27 publications

(35 citation statements)

References 20 publications

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“…[22][23][24] Furthermore, one can solve the equivalent variational problems via smooth optimisation methods (both continuous and based on discretisation) by applying smoothing approximations of nonsmooth penalty functions [39][40][41][42][43] or by replacing these problems with an equivalent problem of minimising the smooth penalty function proposed by Huyer and Neumaier. [43][44][45]49 Thus, our results pave the way for a comparative analysis of various nonsmooth optimisation methods for solving optimal control problems, as well as for a comparative analysis of smooth and nonsmooth approaches to the solution of optimal control problems. Moreover, our results can be extended to the case of nonsmooth optimal control problems and utilised to develop new numerical methods for solving such problems.…”

Section: Discussionmentioning

confidence: 76%

“…[39][40][41][42][43] or the smooth penalty function proposed by Huyer and Neumaier. 44 This penalty function was analysed in detail in the works of Dolgopolik 43 and Wang et al 45 and applied to discretised optimal control problems in papers. [46][47][48] In the work of Dolgopolik, 49 it was shown that Huyer and Neumaier's penalty function is exact if and only if a corresponding standard nonsmooth penalty function is exact.…”

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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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Section: Discussionmentioning

confidence: 76%

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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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“…Let K ⊂ A be a given set. Recall that the set A consists of all those pairs (x, u) ∈ X for which u ∈ U, and x is a solution oḟx = f (x, u, t) with x(0) = x 0 (see equality (29)). Roughly speaking, the property () is satisfied on K iff for any (x, u) ∈ K and any reachable end-pointx T ∈ (x 0 , T) lying sufficiently close to x(T) one can reachx T by slightly changing the control input u in such a way that the corresponding trajectory stays in a sufficiently small neighbourhood of x(⋅) (more precisely, the magnitude of change of u and x must be proportional to |x(T) −x T |).…”

Section: Definitionmentioning

confidence: 99%

“…An exact penalty method for optimal control problems with delay was proposed by Wong and Teo, 23 and such method for some nonsmooth optimal control problems was studied in the works of Outrata et al [24][25][26] A continuous numerical method for optimal control problems based on the direct minimization of an exact penalty function was considered in recent paper. 27 Finally, closely related methods based on Huyer and Neumaier's exact penalty function 11,28,29 were developed for optimal control problems with state inequality constraints 30,31 and optimal feedback control problems. 32 Despite the abundance of publications on exact penalty methods for optimal control problems, relatively little attention has been paid to an actual analysis of the exactness of penalty functions for such 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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“…The game theory is also efficient for the resource allocation with resource competition, and the Nash Equilibria are often needed to be verified first [46,47].…”

Section: Related Workmentioning

confidence: 99%

Dynamic Resource Allocation for Load Balancing in Fog Environment

Qing

et al. 2018

Wireless Communications and Mobile Computing

162

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Fog computing is emerging as a powerful and popular computing paradigm to perform IoT (Internet of Things) applications, which is an extension to the cloud computing paradigm to make it possible to execute the IoT applications in the network of edge. The IoT applications could choose fog or cloud computing nodes for responding to the resource requirements, and load balancing is one of the key factors to achieve resource efficiency and avoid bottlenecks, overload, and low load. However, it is still a challenge to realize the load balance for the computing nodes in the fog environment during the execution of IoT applications. In view of this challenge, a dynamic resource allocation method, named DRAM, for load balancing in fog environment is proposed in this paper. Technically, a system framework for fog computing and the load-balance analysis for various types of computing nodes are presented first. Then, a corresponding resource allocation method in the fog environment is designed through static resource allocation and dynamic service migration to achieve the load balance for the fog computing systems. Experimental evaluation and comparison analysis are conducted to validate the efficiency and effectiveness of DRAM.

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A new class of exact penalty functions and penalty algorithms

Cited by 27 publications

References 20 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

Dynamic Resource Allocation for Load Balancing in Fog Environment

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