Penalty method for nonsmooth minimax control problems with interdependent variables

Горелик, В. А.; Тараканов, А. Ф.

doi:10.1007/bf01070370

Cited by 7 publications

(2 citation statements)

References 3 publications

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“…If the bounds do not depend on the projection dimension, i.e., hold uniformly w.r.t, l, n, then conditions (66)- (68) are included in (1), (2), (47) and in the second pair of condition (65). We try to take the control sequence 0L ,~ 0 satisfying the condition It does not require any other assumptions regarding H. Additionally, we have to reduce the class of problems considered, since we cannot verify whether the operators' inequalities, defining the domain of integration V, are satisfied.…”

Section: O~tflt(k(it)+ K~(it) )mentioning

confidence: 99%

Chapter 4 Optimization methods in the case of uncontrolled factors in Hilbert space

1994

J Math Sci

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11.1. In the previous chapter, various minimax problems were investigated, where either the maximum or minimum were to be computed over a finite-dimensional set. Several ways of transforming these problems to problem 5 were suggested (if the exterior minimum is finite-dimensional, then the transformation is carried out in the same way as at the end of Subsec.10.5.). In the general case of infinite-dimensional minimax problems without the assumption of concavity w.r.t, a maximizing variable, one discovers, following the recommended scheme, that an integral appears w.r.t, a measure defined on a set of an infiniteZdimensional space. The question of the finite-dimensionM approximation with regard to such Optimization problems involving integrals has not yet been properly worked out, and the more So, the methods of search for the optimum. Thus the possibility of constructing simple, though not highly precise, numerical algorithms for the solution of the problem seems to be somewhat important. Such algorithms can be elaborated within CMIPAS. Below we shall demonstrate this possibility, using a generalization of problem 1 (Sec. 1) for the infinite-dimensional case as an example. Further development of this example (in Subsec. 2) leads to a minimax problem whose solution may also be obtained within CMIPAS. We prove the convergence of the corresponding algorithm in Sec. 3. The suggested algorithm is applicable to any minimax problem in L2~ which is convex w.r.t, a minimizing variable. Methods for solving convex-concave minimax problems (search for a saddle point) in Hilbert space will be considered in the next section (Subsecs. 12.3-5).Problem 11 (a generalization of problem 1). Find the value I ° and the solution u ° E U' of the infimum:inf I(u),u 6 U V where u = { . e u ' c r . ( X ) l a ( . ) ( x ) ___ 0 a . s . V , e X}, V = {v E V' C L2(Y)IH(v)(y) < 0 a.s. Vy E Y};V is #-measurable; U', V' are weakly compact sets; g(.)(y) and G(.)(z) are continuous functionals on Y' and U', respectively, £a.a. y 6 Y C R m, x E X C Rm'; J(u, .) are #-integrable VuE U'. Suppose that the functional J can be represented as follows:where ~(.,-,-) is a function of three variables; g(-), h(.), and f(.,-) are operators from U', V', and U' x V' into L2, respectively. The measure # is a probability measure defined on V'. When Vu 6 U' J(u, .) is integrable w.r.t, a finite-additive measure (a weak distribution [121]), we do not require the measure # to be a-additive. If # is finite-additive, then we assume that 3 c < + o c : J ( . , . ) > -c .In stochastic optimization models, the measure # is not available in explicit form, but one can observe the corresponding independent values v 6 V'. However, such an observation seems to be problematic in the case of an infinite-dimensional space. Therefore, let us consider this possibility with regard to only finitedimensional projections of the measure #, and let us use (in the algorithm) independent values of random variables, distributed in accordance with the measures #n defined on R n, n = 1, 2, .... ...

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Section: O~tflt(k(it)+ K~(it) )mentioning

confidence: 99%

Chapter 4 Optimization methods in the case of uncontrolled factors in Hilbert space

1994

J Math Sci

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“…), relatively little effort has been put into development of numerical methods for solving such problems for ODE systems. Smooth penalty methods for some minimax optimal control problems were proposed by Gorelik and Tarakanov [32,33]. An efficient numerical method for optimal control of piecewise smooth systems was recently developed by Nurkanović and Diehl [59], while a method based on discretisation of nonsmooth optimal control problems with state constraints with the use of the so-called pseudospectral knotting technique was presented by Ross and Fahroo [70].…”

Section: Introductionmentioning

confidence: 99%

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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Penalty method and maximum principle for nonsmooth variable-structure control problems

Горелик¹,

Тараканов²

1992

Cybern Syst Anal

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Penalty method for nonsmooth minimax control problems with interdependent variables

Cited by 7 publications

References 3 publications

Chapter 4 Optimization methods in the case of uncontrolled factors in Hilbert space

Chapter 4 Optimization methods in the case of uncontrolled factors in Hilbert space

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

Penalty method and maximum principle for nonsmooth variable-structure control problems

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