74 research outputs found
Minimally Constrained Stable Switched Systems and Application to Co-simulation
We propose an algorithm to restrict the switching signals of a constrained
switched system in order to guarantee its stability, while at the same time
attempting to keep the largest possible set of allowed switching signals. Our
work is motivated by applications to (co-)simulation, where numerical stability
is a hard constraint, but should be attained by restricting as little as
possible the allowed behaviours of the simulators. We apply our results to
certify the stability of an adaptive co-simulation orchestration algorithm,
which selects the optimal switching signal at run-time, as a function of
(varying) performance and accuracy requirements.Comment: Technical report complementing the following conference publication:
Gomes, Cl\'audio, Beno\^it Legat, Rapha\"el Jungers, and Hans Vangheluwe.
"Minimally Constrained Stable Switched Systems and Application to
Co-Simulation." In IEEE Conference on Decision and Control. Miami Beach, FL,
USA, 201
Low-Rank Univariate Sum of Squares Has No Spurious Local Minima
We study the problem of decomposing a polynomial into a sum of
squares by minimizing a quadratically penalized objective . This objective is nonconvex
and is equivalent to the rank- Burer-Monteiro factorization of a
semidefinite program (SDP) encoding the sum of squares decomposition. We show
that for all univariate polynomials , if then
has no spurious second-order critical points, showing that all local optima are
also global optima. This is in contrast to previous work showing that for
general SDPs, in addition to genericity conditions, has to be roughly the
square root of the number of constraints (the degree of ) for there to be no
spurious second-order critical points. Our proof uses tools from computational
algebraic geometry and can be interpreted as constructing a certificate using
the first- and second-order necessary conditions. We also show that by choosing
a norm based on sampling equally-spaced points on the circle, the gradient
can be computed in nearly linear time using fast Fourier
transforms. Experimentally we demonstrate that this method has very fast
convergence using first-order optimization algorithms such as L-BFGS, with
near-linear scaling to million-degree polynomials.Comment: 18 pages, to appear in SIAM Journal on Optimizatio
Piecewise semi-ellipsoidal control invariant sets
Computing control invariant sets is paramount in many applications. The
families of sets commonly used for computations are ellipsoids and polyhedra.
However, searching for a control invariant set over the family of ellipsoids is
conservative for systems more complex than unconstrained linear time invariant
systems. Moreover, even if the control invariant set may be approximated
arbitrarily closely by polyhedra, the complexity of the polyhedra may grow
rapidly in certain directions. An attractive generalization of these two
families are piecewise semi-ellipsoids. We provide in this paper a convex
programming approach for computing control invariant sets of this family.Comment: 7 pages, 3 figures, to be published in IEEE Control Systems Letter
Flexible Differentiable Optimization via Model Transformations
We introduce DiffOpt.jl, a Julia library to differentiate through the
solution of optimization problems with respect to arbitrary parameters present
in the objective and/or constraints. The library builds upon MathOptInterface,
thus leveraging the rich ecosystem of solvers and composing well with modeling
languages like JuMP. DiffOpt offers both forward and reverse differentiation
modes, enabling multiple use cases from hyperparameter optimization to
backpropagation and sensitivity analysis, bridging constrained optimization
with end-to-end differentiable programming. DiffOpt is built on two known rules
for differentiating quadratic programming and conic programming standard forms.
However, thanks ability to differentiate through model transformation, the user
is not limited to these forms and can differentiate with respect to the
parameters of any model that can be reformulated into these standard forms.
This notably includes programs mixing affine conic constraints and convex
quadratic constraints or objective function
MathOptInterface: a data structure for mathematical optimization problems
We introduce MathOptInterface, an abstract data structure for representing
mathematical optimization problems based on combining pre-defined functions and
sets. MathOptInterface is significantly more general than existing data
structures in the literature, encompassing, for example, a spectrum of problems
classes from integer programming with indicator constraints to bilinear
semidefinite programming. We also outline an automated rewriting system between
equivalent formulations of a constraint. MathOptInterface has been implemented
in practice, forming the foundation of a recent rewrite of JuMP, an open-source
algebraic modeling language in the Julia language. The regularity of the
MathOptInterface representation leads naturally to a general file format for
mathematical optimization we call MathOptFormat. In addition, the automated
rewriting system provides modeling power to users while making it easy to
connect new solvers to JuMP
Stability of planar switched systems under delayed event detection
This is an accepted manuscript of an article published by IEEE in 2020 59th IEEE Conference on Decision and Control (CDC), available online: https://ieeexplore.ieee.org/document/9304152
The accepted version of the publication may differ from the final published version.In this paper, we analyse the impact of delayed event detection on the stability of a 2-mode planar hybrid automata. We consider hybrid automata with a unique equilibrium point for all the modes, and we find the maximum delay that preserves stability of that equilibrium point. We also show for the class of hybrid automata treated that the instability of the equilibrium point for the equivalent hybrid automaton with delay in the transitions is equivalent to the existence of a closed orbit in the hybrid state space, a result that is inspired by the Joint Spectral Radius theorem. This leads to an algorithm for computing the maximum stable delay exactly. Other potential applications of our technique include co-simulation, networked control systems and delayed controlled switching with a state feedback control.Published versio
Set programming : theory and computation
The complexity of systems that are relevant to engineering today has grown tremendously. The control techniques based on frequency analysis that were perfectly adequate for simple systems tend to be difficult to use for more complex systems. An important challenge arising for these complex systems is the need to obtain sets satisfying given properties. Once the intended end-use of the sets as well as the properties it should satisfy are clarified, a specific family of sets, called template, is chosen to formulate the search of the appropriate set as a problem that is numerically tractable. In this thesis, we introduce Set Programming as an interface between the requirements that sets should satisfy and the numerical algorithms used to compute sets satisfying given requirements. Several templates are studied in this thesis including the classical templates (polyhedra, zonotopes and ellipsoids) but also more elaborate ones that provide a richer family of sets but are more complicated to implement both theoretically and algorithmically. These includes polysets, piecewise semi-ellipsoids, and piecewise polysets. We study two questions in detail about set programs that both examine different aspects of duality: 1) Conic duality: we analyse what the infeasibility of a set program for a specific template means for the set programs for other templates, e.g., is it infeasible only for this template or for all of them ? 2) Geometric duality: we discuss the choices of represention of convex sets, either in the primal or dual space, depending on the class of set programs that should be solved. Interestingly, this choice seems to be mostly template-independent as it mostly relies on geometric arguments. Finally, we apply our results to several applications ranging from cruise control to energy and information theory.(FSA - Sciences de l'ingénieur) -- UCL, 202
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