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Inner and Outer Approximations of Star-Convex Semialgebraic Sets [article]

James Guthrie
2022 arXiv   pre-print
We also provide methods for establishing the star-convexity of a semialgebraic set by finding inner and outer approximations of its kernel.  ...  For the case of star-convex semialgebraic sets, we propose a novel objective which yields both an outer and an inner approximation while minimizing the ratio of their respective volumes.  ...  ACKNOWLEDGEMENTS The author thanks Enrique Mallada and the anonymous reviewers for their valuable feedback.  ... 
arXiv:2203.13071v2 fatcat:hjwgjdsnhbe4vafawjd6illn6a

Simple Approximations of Semialgebraic Sets and their Applications to Control [article]

Fabrizio Dabbene, Didier Henrion, Constantino Lagoa
2015 arXiv   pre-print
the hierarchy of outer approximations converges in volume (or, equivalently, almost everywhere and almost uniformly) to the original set.  ...  In this work, we move a step further, and propose possibly non-convex approximations , based on a small volume polynomial superlevel set of a single positive polynomial of given degree.  ...  In [7] a heuristic based on polynomial optimization and convex relaxations is proposed for computing small volume polytopic outer approximations of a compact semialgebraic set.  ... 
arXiv:1509.04200v1 fatcat:ucz5llidi5htjamyrdhgh2wsme

Simple approximations of semialgebraic sets and their applications to control

Fabrizio Dabbene, Didier Henrion, Constantino M. Lagoa
2017 Automatica  
the hierarchy of outer approximations converges in volume (or, equivalently, almost everywhere and almost uniformly) to the original set.  ...  In this work, we move a step further, and propose possibly non-convex approximations, based on a small volume polynomial superlevel set of a single positive polynomial of given degree.  ...  In [7] a heuristic based on polynomial optimization and convex relaxations is proposed for computing small volume polytopic outer approximations of a compact semialgebraic set.  ... 
doi:10.1016/j.automatica.2016.11.021 fatcat:qd3wg4hti5a5bg5mypii7kwmoi

Semidefinite Approximations of Reachable Sets for Discrete-time Polynomial Systems [article]

Victor Magron and Pierre-Loic Garoche and Didier Henrion and Xavier Thirioux
2019 arXiv   pre-print
We consider the problem of approximating the reachable set of a discrete-time polynomial system from a semialgebraic set of initial conditions under general semialgebraic set constraints.  ...  Then, one can outer approximate the reachable set as closely as desired with a hierarchy of super level sets of increasing degree polynomials.  ...  In some cases, it is possible to complement the hierarchy of convergent outer approximations of set of interest (ROA, MCI) by providing a sequence of inner approximations.  ... 
arXiv:1703.05085v3 fatcat:o57326vhgvgufa3doiybva5nyy

Semidefinite Approximations of Reachable Sets for Discrete-time Polynomial Systems

Victor Magron, Pierre-Loic Garoche, Didier Henrion, Xavier Thirioux
2019 SIAM Journal of Control and Optimization  
We consider the problem of approximating the reachable set of a discrete-time polynomial system from a semialgebraic set of initial conditions under general semialgebraic set constraints.  ...  Then, one can outer approximate the reachable set as closely as desired with a hierarchy of super level sets of increasing degree polynomials.  ...  In some cases, it is possible to complement the hierarchy of convergent outer approximations of set of interest (ROA, MCI) by providing a sequence of inner approximations.  ... 
doi:10.1137/17m1121044 fatcat:ccpjanjnaffldoiakgw7a7gwzm

Convex Algebraic Geometry of Curvature Operators [article]

Renato G. Bettiol, Mario Kummer, Ricardo A. E. Mendes
2021 arXiv   pre-print
For n≥5, algorithms using semidefinite programming are obtained from hierarchies of inner approximations by spectrahedral shadows and outer relaxations by spectrahedra.  ...  We study the structure of the set of algebraic curvature operators satisfying a sectional curvature bound under the light of the emerging field of Convex Algebraic Geometry.  ...  Theorem B relies on [BM17a] to produce the outer approximation by spectrahedra, and on an adaptation of the Lasserre hierarchy method for the inner approximation by spectrahedral shadows.  ... 
arXiv:1908.03713v2 fatcat:sgbigji7vfhh3g6hd2coatztxu

Randomized Approximations of the Image Set of Nonlinear Mappings with Applications to Filtering∗∗This research was partly funded by CNR-CNRS bilateral project No. 134562

Fabrizio Dabbene, Didier Henrion, Constantino Lagoa, Pavel Shcherbakov
2015 IFAC-PapersOnLine  
These filters represent a natural extension and rapprochement of Gaussian and set-valued filters, and bear similarities with modern tools such as particle filters.  ...  The aim of this paper is twofold: In the first part, we leverage recent results on scenario design to develop randomized algorithms for approximating the image set of a nonlinear mapping, that is, a (possibly  ...  Contrary to classical approximating sets, which are generally either outer or inner, the ensuing approximating set "optimally describes" the set without neither containing nor being contained in it.  ... 
doi:10.1016/j.ifacol.2015.09.430 fatcat:aqtivmfijnb67in6rcu76rpdya

Randomized Approximations of the Image Set of Nonlinear Mappings with Applications to Filtering [article]

Fabrizio Dabbene, Didier Henrion, Constantino Lagoa, Pavel Shcherbakov
2015 arXiv   pre-print
idea is then exploited to develop a new family of randomized prediction-corrector filters.These filters represent a natural extension and rapprochement of Gaussian and set-valued filters,and bear similarities  ...  noisy) mapping of a set via a nonlinear function.We introduce minimum-volume approximations which have the characteristic of guaranteeing a low probability of violation, i.e.  ...  Contrary to classical approximating sets, which are generally either outer or inner, the ensuing approximating set "optimally describes" the set without neither containing nor being contained in it.  ... 
arXiv:1507.08032v1 fatcat:ob6puumodfd6vp5dkar7vnlhnq

Decomposed Structured Subsets for Semidefinite and Sum-of-Squares Optimization [article]

Jared Miller, Yang Zheng, Mario Sznaier, Antonis Papachristodoulou
2021 arXiv   pre-print
Semidefinite programs (SDPs) are standard convex problems that are frequently found in control and optimization applications.  ...  To improve scalability, SDPs can be approximated with lower and upper bounds through the use of structured subsets (e.g., diagonally-dominant and scaled-diagonally dominant matrices).  ...  Structured subset methods restrict (1) to simple subsets K inner ⊂ S n + ⊂ K outer to develop inner and outer approximations resulting in optima p * outer ≤ p * SDP ≤ p * inner .  ... 
arXiv:1911.12859v4 fatcat:nortxjtqbnfizjcof7rdyyrtxe

Semidefinite Approximations of the Polynomial Abscissa

Roxana Heß, Didier Henrion, Jean-Bernard Lasserre, Tien Son Pham
2016 SIAM Journal of Control and Optimization  
Keywords Linear systems control, non-convex non-smooth optimization, polynomial approximations, semialgebraic optimization, semidefinite programming.  ...  Our approximations are computed by a hierarchy of finite-dimensional convex semidefinite programming problems.  ...  The set of feasible solutions for the dual LP (2) is a bounded subset of M + (Z) with Z compact and therefore it is weak-star compact.  ... 
doi:10.1137/15m1033198 fatcat:lj3yfty2cbdexbin4t4btsrlwa

An SDP method for Fractional Semi-infinite Programming Problems with SOS-convex polynomials [article]

Feng Guo, Meijun Zhang
2021 arXiv   pre-print
The convergence rate of the lower bounds and outer approximations are also discussed.  ...  Moreover, for a set defined by infinitely many s.o.s-convex polynomial inequalities, we obtain a procedure to construct a convergent sequence of outer approximations which have semidefinite representations  ...  This is a rather mild assumption and satisfied by, for instance, convex bodies, sets that are star-shaped with respect to a ball.  ... 
arXiv:2110.04737v1 fatcat:ywvch5tggfetxadqro7rwdfgtq

Robust Sensing of Low-Rank Matrices with Non-Orthogonal Sparse Decomposition [article]

Johannes Maly
2022 arXiv   pre-print
We propose a variational formulation that lends itself to alternating minimization and whose global minimizers provably approximate X up to noise level.  ...  Working with a variant of robust injectivity, we derive reconstruction guarantees for various choices of A including sub-gaussian, Gaussian rank-1, and heavy-tailed measurements.  ...  The author furthermore thanks the anonymous reviewers for their detailed and helpful comments.  ... 
arXiv:2103.05523v2 fatcat:4mwsldbuv5egxgz5smcgzhezgu

Semidefinite Outer Approximation of the Backward Reachable Set of Discrete-time Autonomous Polynomial Systems [article]

Weiqiao Han, Russ Tedrake
2018 arXiv   pre-print
Finally we solve the SDP's and obtain a sequence of outer approximations of the backward reachable set. We demonstrate our approach on three dynamical systems.  ...  As a special case, we also show how to approximate the preimage of a compact semi-algebraic set under a polynomial map.  ...  The benefit of this framework is that we solve convex problems instead of generally non-convex problems, and theoretically the approximations to the real set are closer and closer with higher and higher  ... 
arXiv:1803.07725v2 fatcat:iyl2tnezdfdmzdasu4hhyismuu

Safe Trajectory Synthesis for Autonomous Driving in Unforeseen Environments [article]

Shreyas Kousik, Sean Vaskov, Matthew Johnson-Roberson, Ramanarayan Vasudevan
2017 arXiv   pre-print
By bounding the time for this set intersection and subsequent path selection, this paper proves a lower bound for the FRS time horizon and sensing horizon to guarantee safety.  ...  The presented method begins by computing a conservative Forward Reachable Set (FRS) of a high-fidelity model's trajectories produced when tracking trajectories of a low-fidelity model over a finite time  ...  superlevel set of h l is an inner approximation to K safe .  ... 
arXiv:1705.00091v1 fatcat:by4irsndzjcnvhjse4kmspwh2y

Fitting Tractable Convex Sets to Support Function Evaluations [article]

Yong Sheng Soh, Venkat Chandrasekaran
2019 arXiv   pre-print
The geometric problem of estimating an unknown compact convex set from evaluations of its support function arises in a range of scientific and engineering applications.  ...  We provide a geometric characterization of the asymptotic behavior of our estimators, and our analysis relies on the property that certain sets which admit semialgebraic descriptions are Vapnik-Chervonenkis  ...  A*STAR (Agency for Science, Technology and Research, Singapore) fellowship.  ... 
arXiv:1903.04194v1 fatcat:bvsikvucfzanreuxbitaw6qms4
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