Numerical Algebraic Geometry for Optimal Control Applications

Rostalski, Philipp; Fotiou, Ioannis A.; Bates, Daniel J.; Beccuti, Marco; Morari, Manfred

doi:10.1137/090768308

Cited by 14 publications

(14 citation statements)

References 30 publications

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“…To find the pair (z 1 , z 2 ) in Step 3, one may solve the polynomial system (7)- (9). If there is a positivedimensional set of (complex) extremal points, then the procedures in (23) could be used to determine if the set contains a real point.…”

Section: Inputmentioning

confidence: 99%

“…Unless f and g are convex, solving (1) is a nonconvex problem, which can be challenging as standard local solvers run the risk of getting trapped in local minima (especially in high dimensions). This can be mitigated somewhat with techniques such as simulated annealing (6,7) or convex relaxation which has been successful for model invalidation (8)(9)(10), but there is generally no guarantee that a global minimum will be found. When f and g are polynomial, however, problem (1) can be solved globally by finding all roots of an associated polynomial system.…”

Section: Introductionmentioning

confidence: 99%

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Numerical algebraic geometry for model selection and its application to the life sciences

Gross

Davis

et al. 2016

J. R. Soc. Interface.

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Researchers working with mathematical models are often confronted by the related problems of parameter estimation, model validation and model selection. These are all optimization problems, well known to be challenging due to nonlinearity, non-convexity and multiple local optima. Furthermore, the challenges are compounded when only partial data are available. Here, we consider polynomial models (e.g. mass-action chemical reaction networks at steady state) and describe a framework for their analysis based on optimization using numerical algebraic geometry. Specifically, we use probability-one polynomial homotopy continuation methods to compute all critical points of the objective function, then filter to recover the global optima. Our approach exploits the geometrical structures relating models and data, and we demonstrate its utility on examples from cell signalling, synthetic biology and epidemiology.

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Section: Inputmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Numerical algebraic geometry for model selection and its application to the life sciences

Gross

Davis

et al. 2016

J. R. Soc. Interface.

Self Cite

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“…The survey paper discusses continuation methods and their application to OCPs. For the use of homotopy method in OCPs, see also .…”

Section: Homotopymentioning

confidence: 99%

Combined homotopy and neighboring extremal optimal control

Gupta

Bloch

Kolmanovsky

2016

Optim Control Appl Methods

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Summary This paper presents a new approach to trajectory optimization for nonlinear systems. The method exploits homotopy between a linear system and a nonlinear system and neighboring extremal optimal control, in combination with few iterations of a convergent optimizer at each step, to iteratively update the trajectory as the homotopy parameter changes. To illustrate the proposed method, a numerical example of a three‐dimensional orbit transfer problem for a spacecraft is presented. Copyright © 2016 John Wiley & Sons, Ltd.

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“…Parameter homotopies are not new and have been used in several areas of application [8,17,25,26] and implemented in at least two software packages for solving polynomial systems: Bertini [5] and PHCpack [29]. These implementations allow the user to run a single parameter homotopy from one parameter value p 0 with known solutions to the desired parameter value, p 1 , with the solutions at p 0 provided by the user.…”

Section: Introductionmentioning

confidence: 99%

Paramotopy: Parameter Homotopies in Parallel

Bates

Brake

Niemerg

2018

Mathematical Software – ICMS 2018

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Numerical algebraic geometry provides a number of efficient tools for approximating the solutions of polynomial systems. One such tool is the parameter homotopy, which can be an extremely efficient method to solve numerous polynomial systems that differ only in coefficients, not monomials. This technique is frequently used for solving a parameterized family of polynomial systems at multiple parameter values. Parameter homotopies have recently been useful in several areas of application and have been implemented in at least two software packages. This article describes Paramotopy, a new, parallel, optimized implementation of this technique, making use of the Bertini software package. The novel features of this implementation, not available elsewhere, include allowing for the simultaneous solutions of arbitrary polynomial systems in a parameterized family on an automatically generated (or manually provided) mesh in the parameter space of coefficients, front ends and back ends that are easily specialized to particular classes of problems, and adaptive techniques for solving polynomial systems near singular points in the parameter space. This last feature automates and simplifies a task that is important but often misunderstood by non-experts.1 In fact, this technique applies when the coefficients are holomorphic functions of the parameters [28], but we restrict to the case of polynomials as Bertini is restricted to polynomials.

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Numerical Algebraic Geometry for Optimal Control Applications

Cited by 14 publications

References 30 publications

Numerical algebraic geometry for model selection and its application to the life sciences

Numerical algebraic geometry for model selection and its application to the life sciences

Combined homotopy and neighboring extremal optimal control

Paramotopy: Parameter Homotopies in Parallel

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