Indico Feed [Laboratoire de Mathématiques de Limoges]https://indico.math.cnrs.fr/category/33/events.atom2019-01-28T10:00:00ZPyAtomBlock-sparse FGLMhttps://indico.math.cnrs.fr/event/3901/2018-09-27T08:30:00Z<div>Consider a zero-dimensional ideal $I$ in $K[X_1,\dots,X_n]$. Inspired by Faug\`ere and Mou's Sparse FGLM algorithm, we use Krylov sequences based on multiplication matrices of $I$ in order to compute a description of its zero set by means of univariate polynomials. Using the block-Wiedemann techniques, we present an algorithm that is easily parallelizable.</div>
<div> </div>A convex optimization approach to the broadband impedance matching problem in finite dimensionhttps://indico.math.cnrs.fr/event/3916/2018-10-03T12:00:00Z<p>The synthesis of filters and antennas present in microwave reception or emission chains are usually done by assuming a fix resistive reference (usually 50 Ohm) at their connecting ports. The antenna's mismatch is then either ignored or compensated by an additional matching network, resulting either in a degradation of the energy efficiency or in an increased footprint of the overall hardware. The aim this talk is to develop a synthesis technique for matching filters, allowing to handle the matching and filtering requirements in a single filter and rendering superfluous the use of an extra matching network. We will begin the talk by a review of the classical approach due to D.C Youla R.M. Fano, as well as the more mathematical framework developed by J.W. Helton for his theory of broad-band matching in infinite dimension. We will then present a convex optimisation approach, based on the use of the notion of Pick matrix in order to characterise convex admissible sets of passive electrical responses related to the matching problem. We will show, how the latter yields upper and lower bounds, as well as near optimal responses, for the original broad-band matching problem in finite dimension. The resulting non-linear semi-definite optimization problem (NL-SDP) will be discussed as well as practical cases considered.</p>Local convergence analysis of a primal-dual method for nonlinear programming without constraint qualificationhttps://indico.math.cnrs.fr/event/3908/2018-10-05T09:30:00Z<div class="page">
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<p><span>In nonlinear optimization, the lack of the Mangasarian-Fromovitz constraint qualification (MFCQ) may lead to numerical difficulties and in particular to slow down the convergence of an optimization algorithm. In this talk we analyse the local behavior of an algorithm based on a mixed logarithmic barrier-augmented Lagrangian method for solving a nonlinear optimization problem. This work has been motivated by the good efficiency and robustness of this algorithm, even in the degenerate case in which MFCQ does not hold. Furthermore, we detail different updating rules of the parameters of the algorithm to obtain a rapid (superlinear or quadratic) rate of convergence of the sequence of iterates. The local convergence analysis is done by using a stability theorem of Hager and Gowda, as well as a property of uniform boundedness of the inverse of the regularized Jacobian matrix used in the primal-dual method. Numerical results on degenerate problems are also presented. </span></p>
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</div>On relaxation Methods for Mathematical Programs with Complementarity Constraintshttps://indico.math.cnrs.fr/event/3915/2018-10-26T09:30:00Z<p>We consider the Mathematical Program with Complementarity Constraints of minimizing a function f under :<br />
- the inequality constraints $g(x) \leq 0$,<br />
- the equality constraints $h(x)=0$,<br />
- and the complementary constraints $0 \leq G(x) \perp H(x) \geq 0$.<br />
All functions are assumed to be continuously differentiable.</p>
<div>We propose a new family of relaxation schemes for mathematical programs with complementarity constraints that extends the relaxations converging to an M-stationary point [1, 2, 3]. We discuss the properties of the sequence of relaxed non-linear programs as well as stationarity properties of limiting points. We prove under a new and weak constraint qualification, that our relaxation schemes have the desired property of converging to an M-stationary point.</div>
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<div>Unfortunately, in practice, relaxed problems are only solved up to approximate stationary points and the guarantee of convergence to an M-stationary point is lost [4].</div>
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<div>We define a new strong approximate stationarity condition and prove that we can maintain our guarantee of convergence and attain the desired goal of computing an M-stationary point.</div>
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<div>A comprehensive numerical comparison between existing relaxations methods is performed and shows promising results for our new methods. We also propose different extensions to tackle MPVC (vanishing constraints) and MOCC (cardinality constraints) problems.</div>
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<div>[1] Flegel, Michael L and Kanzow, Christian, <em>Abadie-type constraint qualification for mathematical programs with equilibrium constraints</em>, 2005.<br />
[2] Kadrani, Abdeslam and Dussault, Jean-Pierre and Benchakroun, Abdelhamid, <em>A new regularization scheme for mathematical programs with complementarity constraints</em>, 2009.<br />
[3] Schwartz, Alexandra, <em>Mathematical programs with complementarity constraints: Theory, methods, and applications</em>, 2011.<br />
[4] Kanzow, Christian and Schwartz, Alexandra, <em>The Price of Inexactness: Convergence Properties of Relaxation Methods for Mathematical Programs with Complementarity Constraints Revisited</em>, 2015.</div>à définirhttps://indico.math.cnrs.fr/event/3868/2019-01-28T10:00:00Z<p> </p>
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