Convex/concave relaxations of parametric ODEs using Taylor models

Sahlodin, Ali M.; Chachuat, Benoît

doi:10.1016/j.compchemeng.2011.01.031

Cited by 54 publications

(44 citation statements)

References 29 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…This gives Taylor models a clear advantage over traditional interval extensions or centered forms for sufficiently narrow domains, but conversely it may result in a large overestimation or may even be poorer than naive interval evaluation over wider domains. Nevertheless, this approach has proved successful in computing tight enclosures for the solutions of differential equations and implicit algebraic equations [24,33,42,49,50,53,60], and it has enabled complete search for a range of global optimization or constraint satisfaction problems that could not be tackled using interval techniques alone (see, e.g., [4,9,31,32,47,52]). Such higher-order inclusion techniques are indeed appealing in complete search applications based on branching or subdivision, where they can mitigate the clustering effect [15,62].…”

Section: Introductionmentioning

confidence: 99%

Chebyshev model arithmetic for factorable functions

et al. 2016

View full text Add to dashboard Cite

This article presents an arithmetic for the computation of Chebyshev models for factorable functions and an analysis of their convergence properties. Similar to Taylor models, Chebyshev models consist of a pair of a multivariate polynomial approximating the factorable function and an interval remainder term bounding the actual gap with this polynomial approximant. Propagation rules and local convergence bounds are established for the addition, multiplication and composition operations with Chebyshev models. The global convergence of this arithmetic as the polynomial expansion order increases is also discussed. A generic implementation of Chebyshev model arithmetic is available in the library MC++. It is shown through several numerical case studies that Chebyshev models provide tighter bounds than their Taylor model counterparts, but this comes at the price of extra computational burden.

show abstract

Section: Introductionmentioning

confidence: 99%

Chebyshev model arithmetic for factorable functions

et al. 2016

View full text Add to dashboard Cite

show abstract

“…In another approach, interval enclosures are derived from a Taylor model of the parametric ODE solutions [20,47,48]. This latter approach was recently extended to enable convex and concave bounds in [49] using so-called McCormick-Taylor models [50]. The computation of guaranteed state bounds has also been considered in different contexts, including reachability analysis and robust control [51][52][53].…”

Section: Output: Lower and Upper Boundsmentioning

confidence: 99%

“…Our implementation of Algorithm 3 is based on the ACADO Toolkit [26] as the local optimal control solversee Sect. 5.4.2-and uses the library MC++ [58] to compute the required nonlinearity bounds as well as the ODE enclosures based on Taylor models combined with rigorous remainder estimates [49]. Note that this is a prototype implementation and we do not report CPU times for this reason.…”

Section: Numerical Case Study: Optimal Control Of a Bioreactormentioning

confidence: 99%

Branch-and-Lift Algorithm for Deterministic Global Optimization in Nonlinear Optimal Control

Houska

Chachuat

2013

J Optim Theory Appl

View full text Add to dashboard Cite

This paper presents a branch-and-lift algorithm for solving optimal control problems with smooth nonlinear dynamics and potentially nonconvex objective and constraint functionals to guaranteed global optimality. This algorithm features a direct sequential method and builds upon a generic, spatial branch-and-bound algorithm. A new operation, called lifting, is introduced which refines the control parameterization via a Gram-Schmidt orthogonalization process, while simultaneously eliminating control subregions that are either infeasible or that provably cannot contain any global optima. Conditions are given under which the image of the control parameterization error in the state space contracts exponentially as the parameterization order is increased, thereby making the lifting operation efficient. A computational technique based on ellipsoidal calculus is also developed that satisfies these conditions. The practical applicability of branch-and-lift is illustrated in a numerical example.

show abstract

“…In this study, we employ verified ODE methods based on Taylor models to compute such over-approximations [Sahlodin and Chachuat, 2011]. Note that the resulting enclosuresŶ (t i ; P ) shrink as width(P ) → 0 and the set-inversion algorithm thus terminates finitely for any finite tolerances ǫ box > 0 and ǫ bnd > 0.…”

Section: Algorithmic Proceduresmentioning

confidence: 99%

“…Verified ODE integration based on Taylor models Stadtherr, 2007a, Sahlodin andChachuat, 2011] constructs a Taylor model T q x(ti),P of the state variables x(t i ; ·) on P at given times t i ∈ [t 0 , t N ]; that is, ∀p ∈ P :…”

Section: Taylor Model-based Boundsmentioning

confidence: 99%

Optimization-based Domain Reduction in Guaranteed Parameter Estimation of Nonlinear Dynamic Systems

Paulen

Villanueva

Chachuat

2013

IFAC Proceedings Volumes

View full text Add to dashboard Cite

This paper is concerned with guaranteed parameter estimation in nonlinear dynamic systems in a context of bounded measurement error. The problem consists of finding-or approximating as closely as possible-the set of all possible parameter values such that the predicted outputs match the corresponding measurements within prescribed error bounds. An exhaustive search procedure is applied, whereby the parameter set is successively partitioned into smaller boxes and exclusion tests are performed to eliminate some of these boxes, until a prespecified threshold on the approximation level is met. In order to enhance the convergence of this procedure, we investigate the use of optimization-based domain reduction techniques for tightening the parameter boxes before partitioning. We construct such bound-reduction problems as linear programs from the polyhedral relaxation of Taylor models of the predicted outputs. When applied to a simple case study, the proposed approach is found to reduce the computational burden significantly, both in terms of CPU time and number of iterations.

show abstract

Convex/concave relaxations of parametric ODEs using Taylor models

Cited by 54 publications

References 29 publications

Chebyshev model arithmetic for factorable functions

Chebyshev model arithmetic for factorable functions

Branch-and-Lift Algorithm for Deterministic Global Optimization in Nonlinear Optimal Control

Optimization-based Domain Reduction in Guaranteed Parameter Estimation of Nonlinear Dynamic Systems

Contact Info

Product

Resources

About