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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)

The paper deals with optimal control problems with only one control variable and one state constraint, of arbitrary order. We consider the case of finitely many boundary arcs and touch times. We obtain a no-gap theory of second-order conditions, allowing to characterize second-order quadratic growth.

We present new models, numerical simulations and rigorous analysis for the optimization of the velocity in a race. In a seminal paper, Keller [11, 12] explained how a runner should determine his speed in order to run a given distance in the shortest time. We extend this analysis, based on the equation of motion and aerobic energy, to include a balance of… (More)