J. Frédéric Bonnans

Optimization Problems with Perturbations: A Guided Tour

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 solutions in situations where the set of Lagrange multipliers is not a singleton, may be unbounded, or is even empty. We give rather complete results for nonlinear programming problems and describe some extensions of the method to more general problems. We illustrate the results by computing the equilibrium position of a chain that is almost vertical or horizontal.

Local analysis of Newton-type methods for variational inequalities and nonlinear programming

This paper presents some new results in the theory of Newton-type methods for variational inequalities, and their application to nonlinear programming. A condition of semistability is shown to ensure the quadratic convergence of Newtons method and the superlinear convergence of some quasi-Newton algorithms, provided the sequence defined by the algorithm exists and converges. A partial extension of these results to nonsmooth functions is given. The second part of the paper considers some particular variational inequalities with unknowns (x, λ), generalizing optimality systems. Here only the question of superlinear convergence of {xk} is considered. Some necessary or sufficient conditions are given. Applied to some quasi-Newton algorithms they allow us to obtain the superlinear convergence of {xk}. Application of the previous results to nonlinear programming allows us to strengthen the known results, the main point being a characterization of the superlinear convergence of {xk} assuming a weak second-order condition without strict complementarity.

Second Order Optimality Conditions Based on Parabolic Second Order Tangent Sets

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 order sufficient conditions. We show that the second order regularity condition always holds in the case of semidefinite programming.

Avoiding the Maratos effect by means of a nonmonotone line search II. Inequality constrained problems—feasible iterates

When solving inequality constrained optimization problems via Sequential Quadratic Programming (SQP), it is potentially advantageous to generate iterates that all satisfy the constraints: all quadratic programs encountered are then feasible and there is no need for a surrogate merit function. (Feasibility of the successive iterates is in fact required in many contexts such as in real-time applications or when the objective function is not defined outside the feasible set.) It has recently been shown that this is, indeed, possible, by means of a suitable perturbation of the original SQP iteration, without losing superlinear convergence. In this context, the well-known Maratos effect is compounded by the possible infeasibility of the full step of one even close to a solution. These difficulties have been accommodated by making use of a suitable modification of a “bending” technique proposed by Mayne and Polak, requiring evaluation of the constraints function at an auxiliary point at each iteration.In Part I...

Perturbation analysis of second-order cone programming problems

We discuss first and second order optimality conditions for nonlinear second-order cone programming problems, and their relation with semidefinite programming problems. For doing this we extend in an abstract setting the notion of optimal partition. Then we state a characterization of strong regularity in terms of second order optimality conditions. This is the first time such a characterization is given for a nonpolyhedral conic problem.

Consistency of Generalized Finite Difference Schemes for the Stochastic HJB Equation

We analyze a class of numerical schemes for solving the HJB equation for stochastic control problems, which enters the framework of Markov chain approximations and generalizes the usual finite difference method. The latter is known to be monotonic, and hence valid, only if the scaled covariance matrix is dominant diagonal. We generalize this result by, given the set of neighboring points allowed to enter the scheme, showing how to compute effectively the class of covariance matrices that is consistent with this set of points. We perform this computation for several cases in dimensions 2, 3, and 4.

An Extension of Pontryagin's Principle for State-Constrained Optimal Control of Semilinear Elliptic Equations and Variational Inequalities

This paper deals with state-constrained optimal control problems governed by semilinear elliptic equations or variational inequalities. By using Ekelands principle, a minimum principle of Pontryagins type under some stability conditions of the optimal cost with respect to the state constraints is derived.

Convergence of interior point algorithms for the monotone linear complementarity problem

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 problem provides a simple and general framework for most existing primal-dual interior point methods. The conclusion is that most of the published algorithms of this kind generate convergent sequences. In many cases whenever the convergence is not too fast in a certain sense, the sequences converge to the analytic center of the optimal face.

Computation of order conditions for symplectic partitioned Runge-Kutta schemes with application to optimal control

We derive order conditions for the discretization of (unconstrained) optimal control problems, when the scheme for the state equation is of Runge-Kutta type. This problem appears to be essentially the one of checking order conditions for symplectic partitioned Runge-Kutta schemes. We show that the computations using bi-coloured trees are naturally expressed in this case in terms of oriented free tree. This gives a way to compute them by an appropriate computer program.

Sensitivity Analysis of Optimization Problems Under Second Order Regular Constraints

We present a perturbation theory for finite dimensional optimization problems subject to abstract constraints satisfying a second order regularity condition. This is a technical condition that is always satisfied in the case of semi-definite optimization. We derive Lipschitz and Holder expansions of approximate optimal solutions, under a directional constraint qualification hypothesis and various second order sufficient conditions that take into account the curvature of the set defining the constraints of the problem. We show how the theory applies to semi-infinite programs in which the contact set is a smooth manifold and the quadratic growth condition in the constraint space holds, and discuss the differentiability of metric projections as well as the Moreau-Yosida regularization. Finally we show how the theory applies to semi-definite optimization.