In this paper we consider uncertain second-order cone (SOC) and semidefinite programming (SDP) constraints with polyhedral uncertainty, which are in general computationally intractable. We propose to reformulate an uncertain SOC or SDP constraint as a set of adjustable robust linear optimization constraints withan ellipsoidal or semidefinite representable uncertainty set, respectively. The resulting adjustable problem can then (approximately) be solved by using adjustable robust linear optimization techniques. For example, we show that if linear decision rules are used, then the final robust counterpart consists of SOC or SDP constraints, respectively, which have the same computational complexity as the nominal version of the original constraints. We propose an efficient method to obtain good lower bounds. Moreover, we extend our approach to other classes of robust optimization problems, such as nonlinear problems that contain waitand-see variables or linear problems that contain bilinear uncertainty. Numerically, we apply our approach to reformulate the problem on finding the minimum volume circumscribing ellipsoid of a polytope, and solvethe resulting reformulation with linear and quadratic decision rules as well as Fourier-Motzkin elimination. We demonstrate the effectiveness and efficiency of the proposed approach by comparing it with the state-ofthe-art copositive approach. Moreover, we apply the proposed approach to a robust regression problem and a robust sensor network problem, and use linear decision rules to solve the resulting adjustable robust linear optimization problems, which solves the problem to (near) optimality.

In this paper we derive and exploit duality in general two-stage adaptive linear optimization models. The equivalent dualized formulation we derive is again a two-stage adaptive linear optimization model. Therefore, all existing solution approaches for two-stage adaptive models can be used to solve or approximate the dual formulation. The new dualized model differs from the primal formulation in its dimension and uses a different description of the uncertainty set. We show that the optimal primal affine policy can be directly obtained from the optimal affine policy in the dual formulation. We provide empirical evidence that the dualized model in the context of two-stage lot-sizing on a network and two-stage facility location problems solves an order of magnitude faster than the primal formulation with affine policies. We also provide an explanation and associated empirical evidence that offer insight on which characteristics of the dualized formulation make computations faster. Furthermore, the affine policy of the dual formulations can be used to provide stronger lower bounds on the optimality of affine policies.