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This paper presents an overview of some recent, and significant, progress in the theory of optimization problems with perturbations. We put the emphasis on methods based on upper and lower estimates of the objective function of the perturbed problems. These methods allow one to compute expansions of the optimal value function and approximate optimal… (More)

In this paper we discuss second order optimality conditions in optimization problems subject to abstract constraints. Our analysis is based on various concepts of second order tangent sets and parametric duality. We introduce a condition, called second order regularity, under which there is no gap between the corresponding second order necessary and second… (More)

We present a perturbation theory for nite dimensional optimization problems subject to abstract constraints satisfying a second order regularity condition. We derive Lipschitz and HH older expansions of approximate optimal solutions, under a directional constraint qualiication hypothesis and various second order suucient conditions that take into account… (More)

In the framework of Galichon, Henry-Labordère and Touzi [9], we consider the model-free no-arbitrage bound of variance option given the marginal distributions of the underlying asset. We first make some approximations which restrict the computation on a bounded domain. Then we propose a gradient projection algorithm together with a finite difference scheme… (More)

This paper presents a second-order analysis for a simple model optimal control problem of a partial diierential equation, namely a well-posed semilinear elliptic system with constraints on the control variable only. The cost to be minimized is a standard quadratic functional. Assuming the feasible set to be polyhedric, we state necessary and suucient second… (More)

The paper deals with an optimal control problem with a scalar first-order state constraint and a scalar control. In presence of (nonessen-tial) touch points, the arc structure of the trajectory is not stable. We show how to perform a sensitivity analysis that predicts which touch points will, under a small perturbation, become inactive, remain touch points… (More)

We present an extension, for nonlinear optimization under linear constraints, of an algorithm for quadratic programming using a trust region idea introduced by Ye and Tse [Math. Due to the nonlinearity of the cost, we use a linesearch in order to reduce the step if necessary. We prove that, under suitable hypotheses, the algorithm converges to a point… (More)

The literature on interior point algorithms shows impressive results related to the speed of convergence of the objective values, but very little is known about the convergence of the iterate sequences. This paper studies the horizontal linear complementarity problem, and derives general convergence properties for algorithms based on Newton iterations. This… (More)