Robust solutions to conic quadratic problems and their applications

In this study, a robust L 1 convex pose-graph optimisation solution for unmanned aerial vehicles (UAVs) monocular motion estimation with loop closing is presented. Most solutions proposed in literature formulate the pose-graph optimisation as a least-squares problem by minimising a cost function using iterative methods such as Gauss-Newton or Levenberg-Marquardt algorithms. However, with these algorithms, there is no guarantee to converge to a global minimum as they, with high probabilities, converge to a local minimum or even to an infeasible solution. The solution we propose in this work uses a new robust convex optimisation pose-graph technique, which efficiently corrects the UAV's pose after loop-closure detections. Uncertainty estimation using derivative method and its propagation through multiview geometry algorithms are included in the developed solution. The detection of the visual loop closures, in appearance-based navigation, is achieved with our innovative, fast and efficient method based on Bayes decision theory with Gaussian mixture model in combination with the KD-tree data structure. Our navigation solution has been validated using real-world data in both indoor and outdoor environments acquired by a UAV equipped with a monocular system.

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“…Similar to equation (5), this problem can efficiently be recast and solved using SOCP. 26 Overview of the proposed solution…”

Section: Robust Convex Optimisation Formulationmentioning

confidence: 99%

Robust L_∞ convex pose-graph optimisation for monocular localisation solution for unmanned aerial vehicles

Boulekchour

Aouf

Richardson

2014

Proceedings of the Institution of Mechanical Engineers, Part G:

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“…which has cardinality |H d | = d i=j N j , which exhibits exponential dependence 3 on d. 3 To see that, just consider the case when N j = N for each j = 1, . .…”

Section: Remark 2 (Complexity Of the Tensor Quadrature Formula)mentioning

confidence: 99%

“…Then, for a given risk level η ∈ (0, 1), the portfolio selection problem can be stated as follows (see e.g. [3], [14]):…”

Section: Application Example: the Portfolio Problemmentioning

confidence: 99%

Robust and chance-constrained optimization under polynomial uncertainty

Dabbene

Feng

Lagoa

2009

2009 American Control Conference

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A chance-constrained optimization problem, induced from a robust design problem with polynomial dependence on the uncertainties, is, in general, non-convex and difficult to solve. By introducing a novel concept -the kinship function -an easily computable convex relaxation of this problem is proposed. In particular, optimal polynomial kinship functions, which can be computed a priori and once for all, are introduced and used to bound the probability of constraint violation. Moreover, it is proven that the solution of the relaxed problem converges to that of the original robust optimization problem as the degree of the polynomial kinship function increases. Finally, by relying on quadrature formulae for computation of integrals of polynomials, it is shown that the computational complexity of the proposed approach is polynomial on the number of uncertainty parameters.

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“…The RGP (6) is a special type of robust convex optimization problem; see, e.g., Ben-Tal and Nemirovski (1998) for more on robust convex optimization. Unlike the various types of robust convex optimization problems that have been studied in the literature (e.g., Ben-Tal and Nemirovski 1999;Ben-Tal et al 2002;Lebret 1997, 1998;Goldfarb and Iyengar 2003;Boni et al 2007), the computational tractability of the RGP (6) is not clear; it is not yet known whether one can reformulate a general RGP as a tractable optimization problem that interior-point or other algorithms can efficiently solve.…”

Section: Robust Geometric Programmingmentioning

confidence: 99%

Tractable approximate robust geometric programming

2007

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The optimal solution of a geometric program (GP) can be sensitive to variations in the problem data. Robust geometric programming can systematically alleviate the sensitivity problem by explicitly incorporating a model of data uncertainty in a GP and optimizing for the worst-case scenario under this model. However, it is not known whether a general robust GP can be reformulated as a tractable optimization problem that interior-point or other algorithms can efficiently solve. In this paper we propose an approximation method that seeks a compromise between solution accuracy and computational efficiency.The method is based on approximating the robust GP as a robust linear program (LP), by replacing each nonlinear constraint function with a piecewise-linear (PWL) convex approximation. With a polyhedral or ellipsoidal description of the uncertain data, the resulting robust LP can be formulated as a standard convex optimization problem that interior-point methods can solve. The drawback of this basic method is that the number of terms in the PWL approximations required to obtain an acceptable approximation error can be very large. To overcome the "curse of dimensionality" that arises in directly approximating the nonlinear constraint functions in the original robust GP, we form a conservative approximation of the original robust GP, which contains only bivariate constraint functions. We show how to find globally optimal PWL approximations of these bivariate constraint functions.

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Robust solutions to conic quadratic problems and their applications

Cited by 15 publications

References 8 publications

Robust L_∞ convex pose-graph optimisation for monocular localisation solution for unmanned aerial vehicles

Robust L_∞ convex pose-graph optimisation for monocular localisation solution for unmanned aerial vehicles

Robust and chance-constrained optimization under polynomial uncertainty

Tractable approximate robust geometric programming

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Robust solutions to conic quadratic problems and their applications

Cited by 15 publications

References 8 publications

Robust L∞ convex pose-graph optimisation for monocular localisation solution for unmanned aerial vehicles

Robust L∞ convex pose-graph optimisation for monocular localisation solution for unmanned aerial vehicles

Robust and chance-constrained optimization under polynomial uncertainty

Tractable approximate robust geometric programming

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Product

Resources

About

Robust L_∞ convex pose-graph optimisation for monocular localisation solution for unmanned aerial vehicles

Robust L_∞ convex pose-graph optimisation for monocular localisation solution for unmanned aerial vehicles