Papers by François Oustry
Journal of Asset Management, 2006
One of the current challenges of risk modelling consists in building global risk models from loca... more One of the current challenges of risk modelling consists in building global risk models from local ones: from a set of local market risk forecasts (local covariance matrices) and cross-markets correlations, a global covariance matrix preserving local market estimations and restoring a positive semidefinite matrix must be computed. Convex optimization, taking advantages of convex properties of dual functions, is an original and very performing approach for such an issue. In this paper, a particular semidefinite program is posed and solved with dual convex algorithms for correlation matrices in order to build a global risk model starting from a set local market covariance, and cross-correlation. Some numerical illustrations are given.
We show that it is fruitful to dualize the integrality constraints in a combinatorial optimizatio... more We show that it is fruitful to dualize the integrality constraints in a combinatorial optimization problem. First, this reproduces the known SDP relaxations of the max-cut and max-stable problems. Then we apply the approach to general combinatorial problems. We show that the resulting duality gap is smaller than with the classical Lagrangian relaxation; we also show that linear constraints need a special treatment.
Journal of Convex Analysis, 1998
. In this paper we compare two dierent approaches to analyse the second-order behaviour of a conv... more . In this paper we compare two dierent approaches to analyse the second-order behaviour of a convex function. The rst one is classical, we call it the horizontal approach; the second one is more recent, it is the vertical approach. We prove equivalences between horizontal and vertical growth conditions. Then we derive well-known directional results. Finally we show that the vertical approach is particularly interesting to get more than a rst-order (and more than directional) analysis of the maximum eigenvalue function. Key words. Convex analysis, second-order derivative, approximate subdierential, semidenite programming. 1. Introduction. Let f be a nite-valued convex function from the m-dimens- ional Euclidean subspace R m to R. The behaviour of the "-subdierential of f at a point x 2 R m , as a multifunction of the real nonnegative parameter ", is commonly known \to give information" on the second-order behaviour of f at x. In order to examine to what extent this idea is...
This paper describes a linear matrix inequality (LMI)-based algorithm for the static and reduced-... more This paper describes a linear matrix inequality (LMI)-based algorithm for the static and reduced-order output-feedback synthesis problems of nth-order linear time-invariant (LTI) systems with nu (respectively, n y) independent inputs (respectively, outputs). The algorithm is based on a "cone complementarity" formulation of the problem and is guaranteed to produce a stabilizing controller of order m n 0 max(n u ; n y), matching a generic stabilizability result of Davison and Chatterjee [7]. Extensive numerical experiments indicate that the algorithm finds a controller with order less than or equal to that predicted by Kimura's generic stabilizability result (m n0nu 0ny+1). A similar algorithm can be applied to a variety of control problems, including robust control synthesis.
Transactions of the American Mathematical Society, Sep 21, 1999
At a given point p, a convex function f is differentiable in a certain subspace U (the subspace a... more At a given point p, a convex function f is differentiable in a certain subspace U (the subspace along which ∂f (p) has 0-breadth). This property opens the way to defining a suitably restricted second derivative of f at p. We do this via an intermediate function, convex on U. We call this function the U-Lagrangian; it coincides with the ordinary Lagrangian in composite cases: exact penalty, semidefinite programming. Also, we use this new theory to design a conceptual pattern for superlinearly convergent minimization algorithms. Finally, we establish a connection with the Moreau-Yosida regularization.
APS, Nov 1, 1999
ABSTRACT The ground state variational problem for a many-electron system may be formulated in ter... more ABSTRACT The ground state variational problem for a many-electron system may be formulated in terms of reduced density matrices instead of the complete wavefunction [A. J. Coleman, ``Structure of fermion density matrices", Rev. Mod. Phys., 35:668--689, (1963)]. The calculation of ground-state properties then reduces to a linear optimization problem subject to the representability constraints, which are incompletely known but of which the simplest ones are a finite set of linear equalities and bounds on eigenvalues, as in semidefinite programming. We have found new representability conditions through numerical solution of certain dual semidefinite programs. We are exploring numerically the strength of the known representability conditions by calculations on model systems, in which we compare the ground state energy found by the optimization approach with the ground state energy found using full configuration interaction.
Society for Industrial and Applied Mathematics eBooks, 2000
International series in management science/operations research, 2000
Mathematical Programming, Nov 1, 2000
IFAC Proceedings Volumes, Oct 1, 1997
We present a nonsmooth algorithm with a superlinear rate of convergence for minimizing generalize... more We present a nonsmooth algorithm with a superlinear rate of convergence for minimizing generalized eigenvalues of symmetric pencils depending affinely on a parameter x E Rm. The key idea of the algorithm consists in solving, with a Newton method, an equation v(>.) = 0, where vC) is the optimal value function of a perturbed semidefinite program. We indicate how to implement numerically the algorithm and we give an application in robust control (computing the largest lower bound on the decay rate in Popov stability analysis).
Papiers d'Economie Mathématique et Applications, 1997
Siam Journal on Optimization, 1999
Nonconvex Optimization and Its Applications, 2001
A successful technique for some problems in combinatorial optimization is the so-called SDP relax... more A successful technique for some problems in combinatorial optimization is the so-called SDP relaxation, essentially due to L. Lovasz, and much developed by M. Goemans and D.P. Williamson. As observed by S. Poljak, F. Rendl and H. Wolkowicz, this technique can be interpreted from the point of view of Lagrangian duality. A central tool for this is dualization of quadratic constraints, an operation pioneered by N.Z. Shor. We synthesize these various operations, in a language close to that of nonlinear programming. Then we show how the approach can be applied to general combinatorial problems.
At a given point p, a convex function f is differentiable in a certain subspace U (the subspace a... more At a given point p, a convex function f is differentiable in a certain subspace U (the subspace along which @f(p) has 0-breadth). This property opens the way to defining a second derivative of f at p, along U . We do this via an intermediate function, convex on U . We call this function the U-Lagrangian; it coincides with the ordinary Lagrangian in composite cases: exact penalty, semidefinite programming. Also, we use this new theory to design a conceptual pattern for superlinearly convergent minimization algorithms. Finally, we establish a connection with the Moreau-Yosida regularization.
IFAC Proceedings Volumes
Abstract We present a nonsmooth algorithm with a superlinear rate of convergence for minimizing g... more Abstract We present a nonsmooth algorithm with a superlinear rate of convergence for minimizing generalized eigenvalues of symmetric pencils depending affinely on a parameter x ∈ R m . The key idea of the algorithm consists in solving, with a Newton method, an equation v (λ) = 0, where v(·) is the optimal value function of a perturbed semidefinite program. We indicate how to implement numerically the algorithm and we give an application in robust control (computing the largest lower bound on the decay rate in Popov stability analysis).
Optimization, 2009
We consider robust formulations of the mid-term optimal power management problem. For this type o... more We consider robust formulations of the mid-term optimal power management problem. For this type of problems, classical approaches minimize the expected generation cost over a horizon of one year, and model the uncertain future by means of scenario trees. In this setting, extreme scenarios-with low probability in the scenario tree-may fail to be well represented. More precisely, when extreme events occur, strategies devised with the classical approach can result in significant financial losses. By contrast, robust techniques can handle well extreme cases. We consider two robust formulations that preserve the separable structure of the original problem, a fundamental issue when solving real-life problems. Numerical results assess the validity and practicality of the approaches.
Advances in Linear Matrix Inequality Methods in Control, 2000
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Papers by François Oustry