Nonlinear parametric optimization using cylindrical algebraic decomposition

Fotiou, Ioannis A.; Parrilo, Pablo A.; Morari, Manfred

doi:10.1109/cdc.2005.1582743

Cited by 33 publications

(24 citation statements)

References 6 publications

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“…Symbolic methods based for instance on comprehensive Gröbner bases [21] or resultants, can deal with parametric problems arising in parametric optimal control, but tend to be quite expensive when it comes to computations [9], [10], [11]. Specifically, these techniques suffer from what is known as coefficient blowup and their precomputation phase can take a considerable amount of time.…”

Section: B Methods For Solving Polynomial Equationsmentioning

confidence: 99%

“…In this paper, we focus on the important subclass of polynomial systems with a general polynomial performance index, following [9], [10], [11] and show how insight from numerical algebraic geometry, based on homotopy continuation for polynomial systems over the complex numbers, can be used to exploit the structure of the problem and overcome certain shortcomings of general nonlinear optimal control methods.…”

Section: Introductionmentioning

confidence: 99%

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A numerical algebraic geometry approach to nonlinear constrained optimal control

Bates

Fotiou

Rostalski

2007

2007 46th IEEE Conference on Decision and Control

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A new method for nonlinear constrained optimal control based on numerical algebraic geometry is presented. First, the optimal control problem is formulated as a parametric optimization program. Then, certain structural information related to the optimization problem is computed off-line. Afterwards, given this information, numerical algebraic geometry techniques are used to efficiently obtain the optimal control input (i.e. optimal solution) of the original control problem in real time. By using homotopy continuation over the field of complex numbers, this approach has a probability-one guarantee of finding the global optimal solution to the problem at hand.

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Section: B Methods For Solving Polynomial Equationsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

A numerical algebraic geometry approach to nonlinear constrained optimal control

Bates

Fotiou

Rostalski

2007

2007 46th IEEE Conference on Decision and Control

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show abstract

“…We refer to [20] for a full account of the method. In the control community, CAD has been used for model predictive control [21]. The last reference also contains a lucid description of the principles of CAD, see also [25] for an introduction.…”

Section: Cylindrical Algebraic Decomposition and Experiments Designmentioning

confidence: 99%

“…This technique provides exact solution for constrained polynomial optimization problems. In the control literature, CAD has been applied to MPC [21], but it does not seem to have been used in an experiment design context.…”

Section: Introductionmentioning

confidence: 99%

Input design using cylindrical algebraic decomposition

Hjalmarsson

Egebrand

2011

IEEE Conference on Decision and Control and European Control Conference

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Abstract-Experiment design for system identification has seen significant progress in the last decade. One contribution has been to derive convex relaxations of such problems. Consider that only a scalar function of the system parameters is of interest. A standard step in such a case is to first linearize this function with respect to the estimated parameters. The objective of this contribution is twofold: firstly, to examine if there are cases where the linearized approximation is inadequate, and secondly to explore how to improve upon this approximation. By way of examples we show that it is not difficult to construct examples where linearization is insufficient. Furthermore, we introduce the use of higher order approximations and we formally show that this leads to polynomial optimization problems under Gaussian assumptions. We propose the use of cylindrical algebraic decomposition as a method to obtain exact solutions for this type of problems. Numerical examples are provided.

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“…The original CAD by Collins [1] was introduced as a tool for quantifier elimination over the reals. Since then CAD has also been applied to problems including epidemic modelling [9], parametric optimisation [18], theorem proving [22], motion planning [23] and reasoning with multi-valued functions and their branch cuts [14].…”

Section: Introductionmentioning

confidence: 99%

Choosing a Variable Ordering for Truth-Table Invariant Cylindrical Algebraic Decomposition by Incremental Triangular Decomposition

England

Bradford

Davenport

et al. 2014

Mathematical Software – ICMS 2014

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Cylindrical algebraic decomposition (CAD) is a key tool for solving problems in real algebraic geometry and beyond. In recent years a new approach has been developed, where regular chains technology is used to first build a decomposition in complex space. We consider the latest variant of this which builds the complex decomposition incrementally by polynomial and produces CADs on whose cells a sequence of formulae are truth-invariant. Like all CAD algorithms the user must provide a variable ordering which can have a profound impact on the tractability of a problem. We evaluate existing heuristics to help with the choice for this algorithm, suggest improvements and then derive a new heuristic more closely aligned with the mechanics of the new algorithm.

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Nonlinear parametric optimization using cylindrical algebraic decomposition

Cited by 33 publications

References 6 publications

A numerical algebraic geometry approach to nonlinear constrained optimal control

A numerical algebraic geometry approach to nonlinear constrained optimal control

Input design using cylindrical algebraic decomposition

Choosing a Variable Ordering for Truth-Table Invariant Cylindrical Algebraic Decomposition by Incremental Triangular Decomposition

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