Ivar Ekeland

On the variational principle

The variational principle states that if a differentiable functional F attains its minimum at some point zi, then F’(C) = 0; it has proved a valuable tool for studying partial differential equations. This paper shows that if a differentiable function F has a finite lower bound (although it need not attain it), then, for every E > 0, there exists some point u( where 11 F’(uJj* < l , i.e., its derivative can be made arbitrarily small. Applications are given to Plateau’s problem, to partial differential equations, to nonlinear eigenvalues, to geodesics on infinite-dimensional manifolds, and to control theory.

Nonconvex minimization problems

I. The central result. The grandfather of it all is the celebrated 1961 theorem of Bishop and Phelps (see [7], [8]) that the set of continuous linear functionals on a Banach space E which attain their maximum on a prescribed closed convex bounded subset X c E is norm-dense in £*. The crux of the proof lies in introducing a certain convex cone in E, associating with it a partial ordering, and applying to the latter a transfinite induction argument (Zorns lemma). This argument was later used in different settings by Brondsted and Rockafellar (see [9]) and by F. Browder (see [11]). The various situations can be adequately summarized in a diagram:

Selected new aspects of the calculus of variations in the large

We discuss some of the recent developments in variational methods while emphasizing new applications to nonlinear problems. We touch on several issues: (i) the formulation of variational set-ups which provide more information on the location of critical points and therefore on the qualitative properties of the solutions of corresponding Euler-Lagrange equations; (ii) the relationships between the energy of variationally generated solutions, their Morse indices, and the Hausdorff measure of their nodal sets; (iii) the gluing of several topological obstructions; (iv) the preservation of critical levels after deformation of functionals; (v) and the various ways to recover compactness in certain borderline variational problems.

The micro economics of group behavior: General characterization

Abstract We study the demand function of a group of S members facing a global budget constraint. Any vector belonging to the budget set can be consumed within the group, with no restriction on the form of individual preferences, the nature of individual consumptions or the form of the decision process beyond efficiency. Moreover, only the group aggregate behavior, summarized by its demand function, is observable. We provide necessary and (locally) sufficient restrictions that fully characterize the groups demand function, with and without distribution factors. We show that the private or public nature of consumption within the group is not testable from aggregate data on group behavior.

Convex Hamiltonian energy surfaces and their periodic trajectories

In this paper we introduce symplectic invariants for convex Hamiltonian energy surfaces and their periodic trajectories and show that these quentities satisfy several nontrivial relations. In particular we show that they can be used to prove multiplicity results for the number of periodic trajectories.

On the Malliavin approach to Monte Carlo approximation of conditional expectations

Abstract.Given a multi-dimensional Markov diffusion X, the Malliavin integration by parts formula provides a family of representations of the conditional expectation E[g(X2)|X1]. The different representations are determined by some localizing functions. We discuss the problem of variance reduction within this family. We characterize an exponential function as the unique integrated mean-square-error minimizer among the class of separable localizing functions. For general localizing functions, we prove existence and uniqueness of the optimal localizing function in a suitable Sobolev space. We also provide a PDE characterization of the optimal solution which allows to draw the following observation : the separable exponential function does not minimize the integrated mean square error, except for the trivial one-dimensional case. We provide an application to a portfolio allocation problem, by use of the dynamic programming principle.

Transversality conditions for some infinite horizon discrete time optimization problems

This paper considers a class of discrete-time, infinite-horizon optimization problems arising in economics. Necessary optimality conditions for such problems consist of an Euler equation and a transversality condition at infinity. Our concern here is with the latter condition. Previous results for this type of problem have required such restrictions as concavity and boundedness of the utility functions whose sum is being maximized. Here we derive a transversality condition that remains valid under considerably more general hypotheses. In doing so, we introduce methods that may also be applicable to other situations, such as optimization in continuous time.

Estimates of the Duality Gap in Nonconvex Optimization

We associate with every real-valued function a number which measures its lack of convexity. This number is used to estimate the duality gap in optimization problems where the criterion and/or the constraints are nonconvex. It is shown that when the number of variables is very great with respect to the number of constraints, this duality gap is small in relative value. Approximating in this way problems where the criterion and constraints are given as integrals, we show that the duality gap vanishes.

Une théorie de Morse pour les systèmes hamiltoniens convexes

Resume On s’interesse a des systemes hamiltoniens convexes. On demontre que, sur une surface d’energie donnee, ou bien les trajectoires fermees sont en nombre infini, ou bien elles verifient une relation de resonance. On en deduit que generiquement, les trajectoires fermees sont en nombre infini. La demonstration repose sur une forme duale du principe de moindre action, sur la theorie de Morse, sur une formule d’iteration pour l’index, et sur le theoreme de transversalite de Thom.

A theory of bond portfolios

We introduce a bond portfolio management theory based on foundations similar to those of stock portfolio management. A general continuous-time zero-coupon market is considered. The problem of optimal portfolios of zero-coupon bonds is solved for general utility functions, under a condition of no-arbitrage in the zero-coupon market. A mutual fund theorem is proved, in the case of deterministic volatilities. Explicit expressions are given for the optimal solutions for several utility functions.