Herve Lebret
@inria.fr
Inria Startup Studio
Inria
RESEARCH, TEACHING, or OTHER INTERESTS
Computer Science
Scopus Publications
Scholar Citations
Scholar h-index
Scholar i10-index
Scopus Publications
S. D. Cleyn and G. Festel
Edward Elgar Publishing
Miguel Sousa Lobo, Lieven Vandenberghe, Stephen Boyd, and Hervé Lebret
Elsevier BV
Laurent El Ghaoui, Francois Oustry, and Hervé Lebret
Society for Industrial & Applied Mathematics (SIAM)
In this paper we consider semidefinite programs (SDPs) whose data depend on some unknown but bounded perturbation parameters. We seek "robust" solutions to such programs, that is, solutions which minimize the (worst-case) objective while satisfying the constraints for every possible value of parameters within the given bounds. Assuming the data matrices are rational functions of the perturbation parameters, we show how to formulate sufficient conditions for a robust solution to exist as SDPs. When the perturbation is "full," our conditions are necessary and sufficient. In this case, we provide sufficient conditions which guarantee that the robust solution is unique and continuous (Holder-stable) with respect to the unperturbed problem's data. The approach can thus be used to regularize ill-conditioned SDPs. We illustrate our results with examples taken from linear programming, maximum norm minimization, polynomial interpolation, and integer programming.
H. Lebret and S. Boyd
Institute of Electrical and Electronics Engineers (IEEE)
We show that a variety of antenna array pattern synthesis problems can be expressed as convex optimization problems, which can be (numerically) solved with great efficiency by recently developed interior-point methods. The synthesis problems involve arrays with arbitrary geometry and element directivity, constraints on far- and near-field patterns over narrow or broad frequency bandwidth, and some important robustness constraints. We show several numerical simulations for the particular problem of constraining the beampattern level of a simple array for adaptive and broadband arrays.
Laurent El Ghaoui and Hervé Lebret
Society for Industrial & Applied Mathematics (SIAM)
We consider least-squares problems where the coefficient matrices A,b are unknown but bounded. We minimize the worst-case residual error using (convex) second-order cone programming, yielding an algorithm with complexity similar to one singular value decomposition of A. The method can be interpreted as a Tikhonov regularization procedure, with the advantage that it provides an exact bound on the robustness of solution and a rigorous way to compute the regularization parameter. When the perturbation has a known (e.g., Toeplitz) structure, the same problem can be solved in polynomial-time using semidefinite programming (SDP). We also consider the case when A,b are rational functions of an unknown-but-bounded perturbation vector. We show how to minimize (via SDP) upper bounds on the optimal worst-case residual. We provide numerical examples, including one from robust identification and one from robust interpolation.
Hervé Lebret
Springer Science and Business Media LLC
Herve Lebret
SPIE
Antenna pattern synthesis deals with choosing control parameters (the complex weights) of an array of antennas, in order to achieve a set of given specifications. It appears that these problems can often be formulated as convex optimization problems, which can be numerically solved with algorithms such as the ellipsoid algorithm of interior point methods. After a brief presentation of antenna pattern synthesis and of convex optimization, we illustrate then with simulations results. We first minimize the sidelobe levels of a cosecant diagram. Then we show an example of interference cancellation and compare it to adaptive techniques. We will also introduce the important problem of robust antenna arrays.