Georgia Institute of Technology College of Sciences

Stochastic programming is a powerful approach for decision-making under uncertainty. Unfortunately, the solution may be misleading if the underlying distribution of the involved random parameters is not known exactly. In this talk, we study distributionally robust stochastic programming (DRSP), in which the decision hedges against the worst possible distribution that belongs to an ambiguity set. More specifically, we consider the DRSP with the ambiguity set comprising all distributions that are close to some reference distribution in terms of Wasserstein distance. We derive a tractable reformulation of the DRSP problem by constructing the worst-case distribution explicitly via the first-order optimality condition of the dual problem. Our approach has several theoretical and computational implications. First, using the precise characterization of the worst-case distribution, we show that the DRSP can be approximated by robust programs to arbitrary accuracy, and thus many DRSP problems become tractable with tools from robust optimization. Second, when the objective is concave in the uncertainty, the robust-program approximation is exact and equivalent to a saddle-point problem, which can be solved by a Mirror-Prox algorithm. Third, our framework can also be applied to problems other than stochastic programming, such as a class of distributionally robust transportation problems. Furthermore, we perform sensitivity analysis with respect to the radius of the Wasserstein ball, and apply our results to the newsvendor problem, two-stage linear program with uncertainty-affected recourse, and worst-case Value-at-risk analysis.

Decomposition techniques such as atomic, molecular, wavelet and wave-packet expansions provide a multi-scale refinement of Fourier analysis and exploit a rather simple concept: “waves with very different frequencies are almost invisible to each other”. Starting with the classical Calderon-Zygmund and Littlewood-Paley decompositions, many of these useful techniques have been developed around the study of singular integral operators. By breaking an operator or splitting the functions on which it acts into non-interacting almost orthogonal pieces, these tools capture subtle cancelations and quantify properties of an operator in terms of norm estimates in function spaces. This type of analysis has been used to study linear operators with tremendous success. More recently, similar decomposition techniques have been pushed to the analysis of new multilinear operators that arise in the study of (para) product-like operations, commutators, null-forms and other nonlinear functional expressions. In this talk we will present some of our contributions in the study of multilinear singular integrals, function spaces, and the analysis of nanostructure in biological tissues, not all immediately connected topics, yet all centered on some notion of almost orthogonality.

For $\mathbb B^n$ the unit ball of $\mathbb C^n$, we consider Bergman-Orlicz spaces of holomorphic functions in $L_\alpha^\Phi(\mathbb B^n)$, which are generalizations of classical Bergman spaces. Weobtain their atomic decomposition and then prove weak factorization theorems involving the Bloch space and Bergman-Orlicz space and also weak factorization involving two Bergman-Orlicz spaces. This talk is based on joint work with D. Bekolle and A. Bonami.

The closed geodesic problem is a classical topic of dynamical systems, differential geometry and variational analysis, which can be chased back at least to Poincar\'e. A famous conjecture claims the existence of infinitely many distinct closed geodesics on every compact Riemaniann manifold. But so far this is only proved for the 2-dimentional case. On the other hand, Riemannian metrics are quadratic reversible Finsler metrics, and the existence of at least one closed geodesic on every compact Finsler manifold is well-known because of the famous work of Lyusternik and Fet in 1951. In 1973 A. Katok constructed a family of remarkable Finsler metrics on every sphere $S^d$ which possesses precisely $2[(d+1)/2]$ distinct closed geodesics. In 2004, V. Bangert and the author proved the existence of at least $2$ distinct closed geodesics for every Finsler metric on $S^2$, and this multiplicity estimate on $S^2$ is sharp by Katok's example. Since this work, many new results on the multiplicity and stability of closed geodesics have been established. In this lecture, I shall give a survey on the study of closed geodesics on compact Finsler manifolds, including a brief history and results obtained in the last 10 years. Then I shall try to explain the most recent results we obtained for the multiplicity and stability of closed geodesics on compact simply connected Finsler manifolds, sketch the ideas of their proofs, and then propose some further open problems in this field.