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Papers by Frédéric Bonnans

Research paper thumbnail of Perturbed Optimization in Banach Spaces III: Semi-Infinite Optimization

SIAM Journal on Control and Optimization, 1996

This paper is devoted to the study of perturbed semi-infinite optimization problems, i.e., minimi... more This paper is devoted to the study of perturbed semi-infinite optimization problems, i.e., minimization over R n with an infinite number of inequality constraints. We obtain the second-order expansion of the optimal value function and the first-order expansion of approximate optimal solutions in two cases: (i) when the number of binding constraints is finite and (ii) when the inequality constraints are parametrized by a real scalar.

Research paper thumbnail of A Trust Region Interior Point Algorithm for Linearly Constrained Optimization

Siam Journal on Optimization - SIAM J OPTIMIZATION, 1997

Abstract. We present an extension, for nonlinear optimization under linear constraints, of an alg... more Abstract. 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. Programming, 44 (1989), pp. 157{179] and extended by Bonnans and Bouhtou [RAIRO Rech. Oper., 29 (1995), pp. 195{217]. 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 satisfying the rst-order optimality system, and we analyze under which conditions the unit stepsize will be asymptotically accepted.

Research paper thumbnail of Optimization Problems with Perturbations: A Guided Tour

SIAM Review, 1998

This paper presents an overview of some recent, and significant, progress in the theory of optimi... more 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.

Research paper thumbnail of Avoiding the Maratos Effect by Means of a Nonmonotone Line Search. II. Inequality Constrained Problems—Feasible Iterates

SIAM Journal on Numerical Analysis, 1992

Research paper thumbnail of Optimal Control Problems with Partially Polyhedric Constraints

SIAM Journal on Control and Optimization, 1999

Research paper thumbnail of An Extension of Pontryagin’s Principle for State-Constrained Optimal Control of Semilinear Elliptic Equations and Variational Inequalities

SIAM Journal on Control and Optimization, 1995

Research paper thumbnail of Perturbed Optimization in Banach Spaces I: A General Theory Based on a Weak Directional Constraint Qualification

SIAM Journal on Control and Optimization, 1996

Using a directional form of constraint qualification weaker than Robinson's, we derive an implici... more Using a directional form of constraint qualification weaker than Robinson's, we derive an implicit function theorem for inclusions and use it for firstand second-order sensitivity analyses of the value function in perturbed constrained optimization. We obtain H61der and Lipschitz properties and, under a no-gap condition, first-order expansions for exact and approximate solutions. As an application, differentiability properties of metric projections in Hilbert spaces are obtained, using a condition generalizing polyhedricity. We also present in the appendix a short proof of a generalization of the convex duality theorem in Banach spaces.

Research paper thumbnail of Fast convergence of the simplified largest step path following algorithm

Mathematical Programming, 1997

Each master iteration of a simpli ed Newton algorithm for solving a system of equations starts by... more Each master iteration of a simpli ed Newton algorithm for solving a system of equations starts by computing the Jacobian matrix and then uses this matrix in the computation of p Newton steps: the rst of these steps is exact, and the other are called \simpli ed".

Research paper thumbnail of Perturbed Optimization in Banach Spaces III: Semi-Infinite Optimization

SIAM Journal on Control and Optimization, 1996

This paper is devoted to the study of perturbed semi-infinite optimization problems, i.e., minimi... more This paper is devoted to the study of perturbed semi-infinite optimization problems, i.e., minimization over R n with an infinite number of inequality constraints. We obtain the second-order expansion of the optimal value function and the first-order expansion of approximate optimal solutions in two cases: (i) when the number of binding constraints is finite and (ii) when the inequality constraints are parametrized by a real scalar.

Research paper thumbnail of A Trust Region Interior Point Algorithm for Linearly Constrained Optimization

Siam Journal on Optimization - SIAM J OPTIMIZATION, 1997

Abstract. We present an extension, for nonlinear optimization under linear constraints, of an alg... more Abstract. 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. Programming, 44 (1989), pp. 157{179] and extended by Bonnans and Bouhtou [RAIRO Rech. Oper., 29 (1995), pp. 195{217]. 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 satisfying the rst-order optimality system, and we analyze under which conditions the unit stepsize will be asymptotically accepted.

Research paper thumbnail of Optimization Problems with Perturbations: A Guided Tour

SIAM Review, 1998

This paper presents an overview of some recent, and significant, progress in the theory of optimi... more 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.

Research paper thumbnail of Avoiding the Maratos Effect by Means of a Nonmonotone Line Search. II. Inequality Constrained Problems—Feasible Iterates

SIAM Journal on Numerical Analysis, 1992

Research paper thumbnail of Optimal Control Problems with Partially Polyhedric Constraints

SIAM Journal on Control and Optimization, 1999

Research paper thumbnail of An Extension of Pontryagin’s Principle for State-Constrained Optimal Control of Semilinear Elliptic Equations and Variational Inequalities

SIAM Journal on Control and Optimization, 1995

Research paper thumbnail of Perturbed Optimization in Banach Spaces I: A General Theory Based on a Weak Directional Constraint Qualification

SIAM Journal on Control and Optimization, 1996

Using a directional form of constraint qualification weaker than Robinson's, we derive an implici... more Using a directional form of constraint qualification weaker than Robinson's, we derive an implicit function theorem for inclusions and use it for firstand second-order sensitivity analyses of the value function in perturbed constrained optimization. We obtain H61der and Lipschitz properties and, under a no-gap condition, first-order expansions for exact and approximate solutions. As an application, differentiability properties of metric projections in Hilbert spaces are obtained, using a condition generalizing polyhedricity. We also present in the appendix a short proof of a generalization of the convex duality theorem in Banach spaces.

Research paper thumbnail of Fast convergence of the simplified largest step path following algorithm

Mathematical Programming, 1997

Each master iteration of a simpli ed Newton algorithm for solving a system of equations starts by... more Each master iteration of a simpli ed Newton algorithm for solving a system of equations starts by computing the Jacobian matrix and then uses this matrix in the computation of p Newton steps: the rst of these steps is exact, and the other are called \simpli ed".