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 survey the primary research, both theoretical and applied, in the field of Robust Optimization (RO). Our focus will be on the computational attractiveness of RO approaches, as well as the modeling power and broad applicability of the methodology. In addition to surveying the most prominent theoretical results of RO over the past decade, we will also present some recent results linking RO to adaptable models for multi-stage decision-making problems. Finally, we will highlight successful applications of RO across a wide spectrum of domains, including, but not limited to, finance, statistics, learning, and engineering.

Abstract Robust Optimization is a rapidly developing methodology for handling optimization problems affected by non-stochastic “uncertain-butbounded” data perturbations. In this paper, we overview several selected topics in this popular area, specifically, (1) recent extensions of the basic concept of robust counterpart of an optimization problem with uncertain data, (2) tractability of robust counterparts, (3) links between RO and traditional chance constrained settings of problems with stochastic data, and (4) a novel generic application of the RO methodology in Robust Linear Control. Keywords optimization under uncertainty · robust optimization · convex programming · chance constraints · robust linear control

Abstract—Despite the celebrated success of dynamic programming for optimizing quadratic cost functions over linear systems, such an approach is limited by its inability to tractably deal with even simple constraints. In this paper, we present an alternative approach based on results from robust optimization to solve the stochastic linear-quadratic control (SLQC) problem. In the unconstrained case, the problem may be formulated as a semidefinite optimization problem (SDP). We show that we can reduce this SDP to optimization of a convex function over a scalar variable followed by matrix multiplication in the current state, thus yielding an approach that is amenable to closed-loop control and analogous to the Riccati equation in our framework. We also consider a tight, second-order cone (SOCP) approximation to the SDP that can be solved much more efficiently when the problem has additional constraints. Both the SDP and SOCP are tractable in the presence of control and state space constraints; moreover, compared to the Riccati approach, they provide much greater control over the stochastic behavior of the cost function when the noise in the system is distributed normally. Index Terms—Control with constraints, linear-quadratic control, robust optimization, semidefinite optimization. I.

doi 10.1287/opre.1080.0646

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We consider the design of linear precoders (beamformers) for broadcast channels with Quality of Service (QoS) constraints for each user, in scenarios with uncertain channel state information (CSI) at the transmitter. We consider a deterministically-bounded model for the channel uncertainty of each user, and our goal is to design a robust precoder that minimizes the total transmission power required to satisfy the users ’ QoS constraints for all channels within a specified uncertainty region around the transmitter’s estimate of each user’s channel. Since this problem is not known to be computationally tractable, we will derive three conservative design approaches that yield convex and computationally-efficient restrictions of the original design problem. The three approaches yield semidefinite program (SDP) formulations that offer different trade-offs between the degree of conservatism and the size of the SDP. We will also show how these conservative approaches can be used to derive efficiently-solvable quasi-convex restrictions of some related design problems, including the robust counterpart to the problem of maximizing the minimum signal-to-interference-plus-noise-ratio (SINR) subject to a given power constraint. Our simulation results indicate that in the presence of uncertain CSI the proposed approaches can satisfy the users ’ QoS requirements for a significantly larger set of uncertainties than existing methods, and require less transmission power to do so.

I hereby declare that I am the sole author of this thesis. I authorize the University of Waterloo to lend this thesis to other institutions or individuals for the purpose of scholarly research.

This survey paper is intended for the graduate students and researchers who are interested in Operations Research, have solid understanding of linear optimization but are not familiar with Semidefinite Programming (SDP). Here, I provide a very gentle introduction to SDP, some entry points for further look into the SDP literature, and brief introductions to some selected well-known applications which may be attractive to such audience and in turn motivate them to learn more about semidefinite optimization.

In engineering design, an optimized solution often turns out to be suboptimal, when errors are encountered. While the theory of robust convex optimization has taken significant strides over the past decade, all approaches fail if the underlying cost function is not explicitly given; it is even worse if the cost function is nonconvex. In this work, we present a robust optimization method, which is suited for unconstrained problems with a nonconvex cost function as well as for problems based on simulations such as large PDE solvers, response surface, and kriging metamodels. Moreover, this technique can be employed for most real-world problems, because it operates directly on the response surface and does not assume any specific structure of the problem. We present this algorithm along with the application to an actual engineering problem in electromagnetic multiple-scattering of aperiodically arranged dielectrics, relevant to nano-photonic design. The corresponding objective function is highly nonconvex and resides in a 100-dimensional design space. Starting from an “optimized ” design, we report a robust solution with a significantly lower worst case cost, while maintaining optimality. We further generalize this algorithm to address a nonconvex optimization problem under both implementation errors and parameter uncertainties.