Optimization in Bifurcation Problems using a Continuation Method

Kernevez, J. P.; Doedel, Eusebius J.

doi:10.1007/978-3-0348-7241-6_16

Cited by 10 publications

(23 citation statements)

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“…It is notably difficult to locate such a root without an a priori approximation. To overcome this difficulty, we propose a modification to the successive continuation algorithm introduced by Kernévez and Doedel [16] and described by us in [18] in the context of a nonlinear function similar in form to (4.1) but with q " 0.…”

Section: Regularmentioning

confidence: 99%

“…As shown first by Kernévez and Doedel [16,9], parameter continuation techniques may be effectively deployed as a core element of a search strategy for optima along a constraint manifold. This is accomplished by seeking simultaneous solutions to the original set of equations and a set of additional adjoint conditions, linear and homogeneous in a set of unknown Lagrange multipliers.…”

mentioning

confidence: 99%

“…This is accomplished by seeking simultaneous solutions to the original set of equations and a set of additional adjoint conditions, linear and homogeneous in a set of unknown Lagrange multipliers. The technique introduced in [16] demonstrates how local optima may be found at the terminal points of sequences of continuation runs, with each successive run initialized by the final solution found in the previous run. Remarkably, by the linearity and homogeneity of the adjoint conditions, the initial runs may be conveniently initialized with vanishing Lagrange multipliers from solutions to the original set of equations.…”

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confidence: 99%

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Optimization with equality and inequality constraints using parameter continuation

Dankowicz

2020

Applied Mathematics and Computation

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We generalize the successive continuation paradigm introduced by Kernévez and Doedel [16] for locating locally optimal solutions of constrained optimization problems to the case of simultaneous equality and inequality constraints. The analysis shows that potential optima may be found at the end of a sequence of easily-initialized separate stages of continuation, without the need to seed the first stage of continuation with nonzero values for the corresponding Lagrange multipliers. A key enabler of the proposed generalization is the use of complementarity functions to define relaxed complementary conditions, followed by the use of continuation to arrive at the limit required by the Karush-Kuhn-Tucker theory. As a result, a successful search for optima is found to be possible also from an infeasible initial solution guess. The discussion shows that the proposed paradigm is compatible with the staged construction approach of the coco software package. This is evidenced by a modified form of the coco core used to produce the numerical results reported here. These illustrate the efficacy of the continuation approach in locating stationary solutions of an objective function along families of two-point boundary value problems and in optimal control problems.

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Section: Regularmentioning

confidence: 99%

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confidence: 99%

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confidence: 99%

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Optimization with equality and inequality constraints using parameter continuation

Dankowicz

2020

Applied Mathematics and Computation

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“…Since the dimension n opt of the optimization manifold equals 2, the successive continuation approach proposed by Kernévez and Doedel [11] requires multiple stages (in contrast to the motivating example in Section 2.1, where n opt = 1): one initially optimizes only with respect to one variable, following a curve in the optimization manifold, keeping n opt − 1 variables fixed. At each successive stage of continuation one releases one further optimization variable, until all variables are free.…”

Section: A Duffing Oscillator With Delayed Pd Controlmentioning

confidence: 99%

Optimization along families of periodic and quasiperiodic orbits in dynamical systems with delay

2019

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This paper generalizes a previously-conceived, continuation-based optimization technique for scalar objective functions on constraint manifolds to cases of periodic and quasiperiodic solutions of delay-differential equations. A Lagrange formalism is used to construct adjoint conditions that are linear and homogenous in the unknown Lagrange multipliers. As a consequence, it is shown how critical points on the constraint manifold can be found through several stages of continuation along a sequence of connected one-dimensional manifolds of solutions to increasing subsets of the necessary optimality conditions. Due to the presence of delayed and advanced arguments in the original and adjoint differential equations, care must be taken to determine the degree of smoothness of the Lagrange multipliers with respect to time. Such considerations naturally lead to a formulation in terms of multi-segment boundary-value problems (BVPs), including the possibility that the number of segments may change, or that their order may permute, during continuation. The methodology is illustrated using the software package COCO on periodic orbits of both linear and nonlinear delaydifferential equations, keeping in mind that closed-form solutions are not typically available even in the linear case. Finally, we demonstrate optimization on a family of quasiperiodic invariant tori in an example unfolding of a Hopf bifurcation with delay and parametric forcing. The quasiperiodic case is a further original contribution to the literature on optimization constrained by partial differential BVPs.

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“…1) Optimization algorithm: We use a continuation-based framework developed in [23], [24] and implemented in the software package COCO [25], [26] to find candidate solutions to these optimal control problems. A brief introduction to this framework is given as follows.…”

Section: Preliminary Numerical Resultsmentioning

confidence: 99%

A Unified Analytical Framework for Optimal Control Problems on Networks With Input Homogeneity

Dankowicz

2021

IEEE Trans. Control Netw. Syst.

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We consider optimal control problems on networks with input homogeneity and show a reduction to a universal set of reference problems without differential constraints whose stationary points may be found analytically. Under suitable non-degeneracy conditions, the derivation shows that input homogeneity results in constant candidate optimal control inputs for the problems of maximizing the terminal response, minimizing the control effort, or minimizing the terminal time, and that these control inputs depend smoothly on the problem data. These predictions are validated using numerical analysis of problems of synchronization of coupled phase oscillators and spreading dynamics on time-varying networks.

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Optimization in Bifurcation Problems using a Continuation Method

Cited by 10 publications

References 1 publication

Optimization with equality and inequality constraints using parameter continuation

Optimization with equality and inequality constraints using parameter continuation

Optimization along families of periodic and quasiperiodic orbits in dynamical systems with delay

A Unified Analytical Framework for Optimal Control Problems on Networks With Input Homogeneity

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