We present different robust frameworks using probabilistic ambiguity descriptions of the input data in the least squares problems. The three probability ambiguity descriptions are given by: (1) confidence interval over the first two moments; (2) bounds on the probability measure with moments constraints; (3) confidence interval over the probability measure by using the Kantorovich probability … Read more
Affinely-Adjustable Robust Counterparts provide tractable alternatives to (two-stage) robust programs with arbitrary recourse. We apply them to robust network design with polyhedral demand uncertainty, introducing the affine routing principle. We compare the affine routing to the well-studied static and dynamic routing schemes for robust network design. All three schemes are embedded into the general framework … Read more
The Kantorovich function $(x^TAx)( x^T A^{-1} x)$, where $A$ is a positive definite matrix, is not convex in general. From a matrix or convex analysis point of view, it is interesting to address the question: When is this function convex? In this paper, we prove that the 2-dimensional Kantorovich function is convex if and only … Read more
The wide applicability of chance-constrained programming, together with advances in convex optimization and probability theory, has created a surge of interest in finding efficient methods for processing chance constraints in recent years. One of the successes is the development of so-called safe tractable approximations of chance-constrained programs, where a chance constraint is replaced by a … Read more
This paper introduces a new method for proving global stability of fluid flows through the construction of Lyapunov functionals. For finite dimensional approximations of fluid systems, we show how one can exploit recently developed optimization methods based on sum-of-squares decomposition to construct a polynomial Lyapunov function. We then show how these methods can be extended … Read more
In this paper we study relations between the minimax, risk averse and nested formulations of multistage stochastic programming problems. In particular, we discuss conditions for time consistency of such formulations of stochastic problems. We also describe a connection between law invariant coherent risk measures and the corresponding sets of probability measures in their dual representation. … Read more
In this paper we consider the adjustable robust approach to multistage optimization, for which we derive dynamic programming equations. We also discuss this from a point of view of risk averse stochastic programming. As an example we consider a robust formulation of the classical inventory model and show that, similar to the risk neutral case, … Read more
The structural properties of graphs are usually characterized in terms of invariants, which are functions of graphs that do not depend on the labeling of the nodes. In this paper we study convex graph invariants, which are graph invariants that are convex functions of the adjacency matrix of a graph. Some examples include functions of … Read more
We propose a tractable approximation scheme for convex (not necessarily linear) multi-stage robust optimization problems. We approximate the adaptive decisions by finite linear combinations of prescribed basis functions and demonstrate how one can optimize over these decision rules at low computational cost through constraint randomization. We obtain a-priori probabilistic guarantees on the feasibility properties of … Read more
In this exposition the robust counterpart approach by Ben-Tal, El Ghaoui and Nemirovski is investigated with respect to its costs and benefits, with the focus on the costs of robustification. Although robust optimization has gained more and more interest among both academics and practitioners and although this certainly represents a well-established theory, it is to … Read more