ROmodel: modeling robust optimization problems in Pyomo

This paper introduces ROmodel, an open source Python package extending the modeling capabilities of the algebraic modeling language Pyomo to robust optimization problems. ROmodel helps practitioners transition from deterministic to robust optimization through modeling objects which allow formulating robust models in close analogy to their mathematical formulation. ROmodel contains a library of commonly used uncertainty sets which can be generated using their matrix representations, but it also allows users to define custom uncertainty sets using Pyomo constraints. ROmodel supports adjustable variables via linear decision rules. The resulting models can be solved using ROmodels solvers which implement both the robust reformulation and cutting plane approach. ROmodel is a platform to implement and compare custom uncertainty sets and reformulations. We demonstrate ROmodel’s capabilities by applying it to six case studies. We implement custom uncertainty sets based on (warped) Gaussian processes to show how ROmodel can integrate data-driven models with optimization.

A New Affinely Adjustable Robust Model for Security Constrained Unit Commitment under Uncertainty

Currently, optimization models for the safe and reliable operation of power systems deal with two major challenges: the first one is the reduction of the computational load when considering N−1 contingencies; the second one is the adequate modeling of the uncertainty of intermittent generation and demand. This paper proposes a new affinely adjustable robust model to solve the security constrained unit commitment problem considering these sources of uncertainty. Linear decision rules, which take into account the forecasts and forecast errors of the different sources of uncertainty, are used for the affine formulation of the dispatch variables, thus allowing the tractability of the model. Another major novelty is that the evaluation of the N−1 security constraints is performed by incorporating a novel method, proposed in the literature, based on the user-cuts concept. This method efficiently and dynamically adds only the binding N−1 security constraints, increasing the computational efficiency of the model when transmission line contingencies are considered. Finally, Monte Carlo simulations on the post-optimization results were run to demonstrate the effectiveness, feasibility and robustness of the solutions provided by the proposed model.

Robust Optimization for Models with Uncertain Second-Order Cone and Semidefinite Programming Constraints

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 with an 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 wait-and-see variables, linear problems that contain bilinear uncertainty, and general conic constraints. Numerically, we apply our approach to reformulate the problem on finding the minimum volume circumscribing ellipsoid of a polytope and solve the 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-of-the-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 solve the problem to (near) optimality. Summary of Contribution: Computing robust solutions for nonlinear optimization problems with uncertain second-order cone and semidefinite programming constraints are of much interest in real-life applications, yet they are in general computationally intractable. This paper proposes a computationally tractable approximation for such problems. Extensive computational experiments on (i) computing the minimum volume circumscribing ellipsoid of a polytope, (ii) robust regressions, and (iii) robust sensor networks are conducted to demonstrate the effectiveness and efficiency of the proposed approach.