Robust and chance-constrained optimization under polynomial uncertainty

Dabbene, Fabrizio; Feng, Chao; Lagoa, Constantino

doi:10.1109/acc.2009.5160248

Cited by 4 publications

(4 citation statements)

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“…To avoid this, in line with the results in [8], we adopt an algorithm to compute the so-called upper bound and lower bound polynomial approximations for I(y). Being different from the ordinary polynomial approximations constructed by interpolation, the upper/lower bound polynomial WeB03.5 approximations have to satisfy additional conditions so that the corresponding estimates are greater/less than the "true" probability for any probability measure f .…”

Section: Hard Bounds For Probabilistic Objective Functionsmentioning

confidence: 99%

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Distributional robustness analysis for polynomial uncertainty

Feng

Lagoa

2009

Proceedings of the 48h IEEE Conference on Decision and Control (CDC) Held Jointly With 2009 28th Chinese Control Conference

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This paper addresses the problem of computing the worst-case expected value of a polynomial function, over a class of admissible distributions. It is shown that this problem, for the class of distributions considered, is equivalent to a convex optimization problem for which efficient linear matrix inequality (LMI) relaxations are available. In case that the performance function is continuous (not necessarily polynomial), the worst-case expected value can be approximated by using its polynomial approximations. Moreover, the proposed approach is applied to compute hard bounds of the worst-case probability of a polynomial being negative. Numerical examples are presented which illustrate the application of the results in this paper.

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Section: Hard Bounds For Probabilistic Objective Functionsmentioning

confidence: 99%

“…According to [12], [8], the above two problems can be converted to standard semi-definite programs (SDPs). Hence, given interval [−α, β], weighting function W (y) and degree ̺, the "optimal" polynomial approximations can be easily computed by standard SDP solvers.…”

Section: Hard Bounds For Probabilistic Objective Functionsmentioning

confidence: 99%

Distributional robustness analysis for polynomial uncertainty

Feng

Lagoa

2009

Proceedings of the 48h IEEE Conference on Decision and Control (CDC) Held Jointly With 2009 28th Chinese Control Conference

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“…This method can only be applied to specific uncertainty structures. In ( [9], [10], [11]) convex relaxations of chance constrained problems are presented by introducing the concept of polynomial kinship function to estimate an upper bound on the probability of constraint. It is shown that as the degree of the polynomial kinship function increases, solutions to the relaxed problem converges to a solution of the original problem.…”

Section: A Previous Workmentioning

confidence: 99%

Convex Constrained Semialgebraic Volume Optimization: Application in Systems and Control

Jasour,

Lagoa

2017

Preprint

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In this paper, we generalize the chance optimization problems and introduce constrained volume optimization where enables us to obtain convex formulation for challenging problems in systems and control. We show that many different problems can be cast as a particular cases of this framework. In constrained volume optimization, we aim at maximizing the volume of a semialgebraic set under some semialgebraic constraints. Building on the theory of measures and moments, a sequence of semidefinite programs are provided, whose sequence of optimal values is shown to converge to the optimal value of the original problem. We show that different problems in the area of systems and control that are known to be nonconvex can be reformulated as special cases of this framework. Particularly, in this work, we address the problems of probabilistic control of uncertain systems as well as inner approximation of region of attraction and invariant sets of polynomial systems. Numerical examples are presented to illustrate the computational performance of the proposed approach.

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“…Moreover, components of q need to be independent and have computable finite generating functions. In [18,21,22] convex relaxations of chance constrained problems are presented. The concept of polynomial kinship function is used to estimate an upper bound on the probability of constraint violation.…”

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confidence: 99%

Semidefinite Programming For Chance Constrained Optimization Over Semialgebraic Sets

Jasour¹,

Aybat²,

Lagoa³

2015

SIAM J. Optim.

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In this paper, "chance optimization" problems are introduced, where one aims at maximizing the probability of a set defined by polynomial inequalities. These problems are, in general, nonconvex and computationally hard. With the objective of developing systematic numerical procedures to solve such problems, a sequence of convex relaxations based on the theory of measures and moments is provided, whose sequence of optimal values is shown to converge to the optimal value of the original problem. Indeed, we provide a sequence of semidefinite programs of increasing dimension which can arbitrarily approximate the solution of the original problem. To be able to efficiently solve the resulting large-scale semidefinite relaxations, a first-order augmented Lagrangian algorithm is implemented. Numerical examples are presented to illustrate the computational performance of the proposed approach.

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Robust and chance-constrained optimization under polynomial uncertainty

Cited by 4 publications

References 20 publications

Distributional robustness analysis for polynomial uncertainty

Distributional robustness analysis for polynomial uncertainty

Convex Constrained Semialgebraic Volume Optimization: Application in Systems and Control

Semidefinite Programming For Chance Constrained Optimization Over Semialgebraic Sets

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