In this paper we propose a robust formulation for discrete time dynamic programming (DP). The objective of the robust formulation is to systematically mitigate the sensitivity of the DP optimal policy to ambiguity in the underlying transition probabilities. The ambiguity is modeled by associating a set of conditional measures with each state-action pair. Consequently, in the robust formulation each policy has a set of measures associated with it. We prove that when this set of measures has a certain "Rectangularity" property all the main results for finite and infinite horizon DP extend to natural robust counterparts. We identify families of sets of conditional measures for which the computational complexity of solving the robust DP is only modestly larger than solving the DP, typically logarithmic in the size of the state space. These families of sets are constructed from the confidence regions associated with density estimation, and therefore, can be chosen to guarantee any desired level of confidence in the robust optimal policy. Moreover, the sets can be easily parameterized from historical data. We contrast the performance of robust and non-robust DP on small numerical examples.

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Abstract. This paper gives a brief overview of the nonparametric techniques that are useful for financial econometric problems. The problems include estimation and inference for instantaneous returns and volatility functions of time-homogeneous and time-dependent diffusion processes, and estimation of transition densities and state price densities. We first briefly describe the problems and then outline the main techniques and main results. Some useful probabilistic aspects of diffusion processes are also briefly summarized to facilitate our presentation and applications.

We describe a two-stage robust optimization approach for solving network flow and design problems with uncertain demand. In two-stage network optimization one defers a subset of the flow decisions until after the realization of the uncertain demand. Availability of such a recourse action allows one to come up with less conservative solutions compared to single-stage optimization. However, this advantage often comes at a price: two-stage optimization is, in general, significantly harder than singe-stage optimization. For network flow and design under demand uncertainty we give a characterization of the first-stage robust decisions with an exponential number of constraints and prove that the corresponding separation problem is N P-hard even for a network flow problem on a bipartite graph. We show, however, that if the second-stage network topology is totally ordered or an arborescence, then the separation problem is tractable. Unlike single-stage robust optimization under demand uncertainty, two-stage robust optimization allows one to control conservatism of the solutions by means of an allowed “budget for demand uncertainty.” Using a budget of uncertainty we provide an upper

In this paper we study ambiguous chance constrained problems where the distributions of the random parameters in the problem are themselves uncertain. We primarily focus on the special case where the uncertainty set Q of the distributions is of the form Q = {Q : # p (Q, Q 0 ) # #}, where # p denotes the Prohorov metric. The ambiguous chance constrained problem is approximated by a robust sampled problem where each constraint is a robust constraint centered at a sample drawn according to the central measure Q 0 . The main contribution of this paper is to show that the robust sampled problem is a good approximation for the ambiguous chance constrained problem with high probability. This result is established using the Strassen-Dudley Representation Theorem that states that when the distributions of two random variables are close in the Prohorov metric one can construct a coupling of the random variables such that the samples are close with high probability. We also show that the robust sampled problem can be solved e#ciently both in theory and in practice. 1

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informs ® doi 10.1287/opre.1070.0385 © 2007 INFORMS In the area of dynamic revenue management, optimal pricing policies are typically computed on the basis of an underlying demand rate model. From the perspective of applications, this approach implicitly assumes that the model is an accurate representation of the real-world demand process and that the parameters characterizing this model can be accurately calibrated using data. In many situations, neither of these conditions are satisfied. Indeed, models are usually simplified for the purpose of tractability and may be difficult to calibrate because of a lack of data. Moreover, pricing policies that are computed under the assumption that the model is correct may perform badly when this is not the case. This paper presents an approach to single-product dynamic revenue management that accounts for errors in the underlying model at the optimization stage. Uncertainty in the demand rate model is represented using the notion of relative entropy, and a tractable reformulation of the “robust pricing problem ” is obtained using results concerning the change of probability measure for point processes. The optimal pricing policy is obtained through a version of the so-called Isaacs ’ equation for stochastic differential games, and the structural properties of the optimal solution are obtained through an analysis of this equation. In particular, (i) closed-form solutions for the special case of an exponential nominal demand rate model,

In this paper we study robust convex quadratically constrained programs, a subset of the class of robust convex programs introduced by Ben-Tal and Nemirovski [4]. Unlike [4], our focus in this paper is to identify uncertainty structures that allow the corresponding robust quadratically constrained programs to be reformulated as second-order cone programs. We propose three classes of uncertainty sets that satisfy this criterion and present examples where these classes of uncertainty sets are natural. 1 Problem formulation A generic quadratically constrained program (QCP) is defined as follows.

In this we paper we study techniques for generating valid convex constraints for mixed 0-1 conic programs. We show that many of the techniques developed for generating linear cuts for mixed 0-1 linear programs, such as the Gomory cuts, the lift-and-project cuts, and cuts from other hierarchies of tighter relaxations, extend in a straightforward manner to mixed 0-1 conic programs. We also show that simple extensions of these techniques lead to methods for generating convex quadratic cuts. Gomory cuts for mixed 0-1 conic programs have interesting implications for comparing the semidefinite programming and the linear programming relaxations of combinatorial optimization problems, e.g. we show that all the subtour elimination inequalities for the traveling salesman problem are rank-1 Gomory cuts with respect to a single semidefinite constraint. We also include results from our preliminary computational experiments with these cuts.

Abstract. In this survey we review the many faces of the S-lemma, a result about the correctness of the S-procedure. The basic idea of this widely used method came from control theory but it has important consequences in quadratic and semidefinite optimization, convex geometry, and linear algebra as well. These were all active research areas, but as there was little interaction between researchers in these different areas, their results remained mainly isolated. Here we give a unified analysis of the theory by providing three different proofs for the S-lemma and revealing hidden connections with various areas of mathematics. We prove some new duality results and present applications from control theory, error estimation, and computational geometry. Key words. S-lemma, S-procedure, control theory, nonconvex theorem of alternatives, numerical range, relaxation theory, semidefinite optimization, generalized convexities