A tractable approach for designing piecewise affine policies in two-stage adjustable robust optimization

Quadratically adjustable robust linear optimization with inexact data via generalized S-lemma: Exact second-order cone program reformulations

In this paper, we consider adjustable robust versions of convex optimization problems with uncertain constraints and objectives and show that under fairly general assumptions, a static robust solution provides a good approximation for these adjustable robust problems. An adjustable robust optimization problem is usually intractable since it requires to compute a solution for all possible realizations of uncertain parameters, while an optimal static solution can be computed efficiently in most cases if the corresponding deterministic problem is tractable. The performance of the optimal static robust solution is related to a fundamental geometric property, namely, the symmetry of the uncertainty set. Our work allows for the constraint and objective function coefficients to be uncertain and for the constraints and objective functions to be convex, thereby providing significant extensions of the results in Bertsimas and Goyal (Math Oper Res 35:284–305, 2010) and Bertsimas et al. (Math Oper Res 36: 24–54, 2011b) where only linear objective and linear constraints were considered. The models in this paper encompass a wide variety of problems in revenue management, resource allocation under uncertainty, scheduling problems with uncertain processing times, semidefinite optimization among many others. To the best of our knowledge, these are the first approximation bounds for adjustable robust convex optimization problems in such generality.

In this paper we propose a methodology for constructing decision rules for integer and continuous decision variables in multiperiod robust linear optimization problems. This type of problem finds application in, for example, inventory management, lot sizing, and manpower management. We show that by iteratively splitting the uncertainty set into subsets, one can differentiate the later-period decisions based on the revealed uncertain parameters. At the same time, the problem’s computational complexity stays at the same level, as for the static robust problem. This also holds in the nonfixed recourse situation. In the fixed recourse situation our approach can be combined with linear decision rules for the continuous decision variables. We provide theoretical results on splitting the uncertainty set by identifying sets of uncertain parameter scenarios to be divided for an improvement in the worst-case objective value. Based on this theory, we propose several splitting heuristics. Numerical examples entailing a capital budgeting and a lot sizing problem illustrate the advantages of the proposed approach.

To enhance the resilience of transmission networks with offshore wind farms (OWFs) and pump hydro energy storage systems (PHESSs) towards typhoons, a resilient unit commitment scheme is proposed. This problem is formulated as a two-stage robust optimization problem, where the uncertain impacts of typhoons on OWF output, transmission line failures, and demands are captured by a robust uncertain set with chance constraints. This uncertainty set is embedded into a two-stage robust unit commitment model, where the commitment of thermal units is optimized in the first-stage and PHESSs are scheduled in the second-stage optimization as the emergency resources. The two-stage robust optimization problem is solved using the column and constraint generation (C&CG) algorithm. Simulations are performed on the modified IEEE-RTS test systems with real OWF output during Typhoon Higos, and the effectiveness of proposed scheme is verified regarding the reduction of worst case load shedding.

In active distribution networks, topology change is an important scheme to cope with the rapid fluctuations of distributed renewable generations. This paper studies the real-time distribution network reconfiguration (DNR) problem while ensuring a reliable power supply under set-based uncertainty. Considering the intertemporal constraint on the maximum number of action times for each switchable line, real-time DNR is formulated as a multi-stage robust optimization (MRO) problem. To solve this problem, a customized implicit policy method is proposed. First, the non-anticipative constraints of each switchable line are specifically devised to generate the upper and lower bounds for line status, ensuring that the intertemporal constraint is always satisfied as long as the bounds are met. Second, by augmenting the stage to optimize the above bounds, the original MRO problem is reformulated as a two-stage robust optimization problem with mixed-integer recource by relaxing the intertemporal constraints and aggregating stages, thereby suggesting an OPF-based real-time DNR policy that is robustly feasible and parameterized in the bounds of line status. Finally, these bounds are obtained by solving the two-stage robust optimization problem through the nested column-and-constraint generation algorithm. Tests on IEEE 33-, 69-, and 123-bus distribution systems comprehensively verify the effectiveness of the proposed method, as well as its advantages in scalability and optimality over state-of-the-art methods.

The present paper addresses the class of two-stage robust optimization problems which can be formulated as mathematical programs with uncertainty on the right-hand side coefficients (RHS uncertainty). The wide variety of applications and the fact that many problems in the class have been shown to be NP-hard, motivates the search for efficiently solvable special cases. Accordingly, the first objective of the paper is to provide an overview of the most important applications and of various polynomial or pseudo-polynomial special cases identified so far. The second objective is to introduce a new subclass of polynomially solvable robust optimization problems with RHS uncertainty based on the concept of state-space representable uncertainty sets . A typical application to a multi period energy production problem under uncertain customer load requirements is described into details, and computational results including a comparison between optimal two-stage solutions and exact optimal multistage strategies are discussed.

This paper presents a new approach aims to solve robust multidisciplinary design optimization MDO problem called Improved Multi-objective Robust Collaborative Optimization . This method combines the Multi-objective Robust Collaborative Optimization method, the Worst Possible Point constraint cuts and the Genetic algorithm NSGA-II type as an optimizer to solve the robust optimization problem of complex structure named Y-stiffened panel under interval uncertainty . The proposed approach hierarchically decomposes the optimization problem into a structure level considered as an upper level in the Y-stiffened panel and a second level considered as a lower level of the studied panel. A robust multi-objective optimization problem intended to optimize the eigenfrequency, the global mass and the displacement at a fixed point of the Y-stiffened panel at the first level and each structure’s robust optimization problem allows optimizing its eigenfrequency and mass limited by their local constraint functions at the second one. Tor demonstrate our method, an engineering example of Y-stiffened panel is treated. A good performance of proposed method is proved by a comparison between obtained results and Non-Distributed Multi-objective Robust Optimization .

In this paper, we consider chance-constrained Stochastic Model Predictive Control problems for uncertain linear systems subject to additive disturbance. A popular method for solving the associated chance-constrained optimization problem is by means of randomization, in which the chance constraints are replaced by a finite number of sampled constraints, each corresponding to a disturbance realization. Earlier approaches in this direction lead to computationally expensive problems, whose solutions are typically very conservative both in terms of cost and violation probabilities. One way of overcoming this conservatism is to use piecewise affine (PWA) policies, which offer more flexibility than conventional open-loop and affine policies. Unfortunately, the straight-forward application of randomized methods towards PWA policies will lead to computationally demanding problems, that can only be solved for problems of small sizes. To address this issue, we propose an alternative method based on a combination of randomized and robust optimization. We show that the resulting approximation can greatly reduce conservatism of the solution while exhibiting favorable scaling properties with respect to the prediction horizon.

In “Dual Approach for Two-Stage Robust Nonlinear Optimization,” de Ruiter, Zhen, and den Hertog study adjustable robust minimization problems where the objective or constraints depend in a convex way on the adjustable variables. They reformulate the original adjustable robust nonlinear problem with a polyhedral uncertainty set into an equivalent adjustable robust linear problem, for which all existing approaches for adjustable robust linear problems can be used. The reformulation is obtained by first dualizing over the adjustable variables and then over the uncertain parameters. The polyhedral structure of the uncertainty set then appears in the linear constraints of the dualized problem, and the nonlinear functions of the adjustable variables in the original problem appear in the uncertainty set of the dualized problem. The authors show how to recover linear decision rules to the original primal problem and how to generate bounds on its optimal objective value.

Surrogate duality for robust optimization

Fast robust optimization of ORC based on an artificial neural network for waste heat recovery

We consider distributionally robust two-stage stochastic linear optimization problems with higher-order (say \(p\ge 3\) and even possibly irrational) moment constraints in their ambiguity sets. We suggest to solve the dual form of the problem by a semi-infinite programming approach, which deals with a much simpler reformulation than the conic optimization approach. Some preliminary numerical results are reported.

In modern taxi networks, large amounts of taxi occupancy status and location data are collected from networked in-vehicle sensors in real- time. They provide knowledge of system models on passenger demand and mobility patterns for efficient taxi dispatch and coordination strategies. Such approaches face a new challenge: how to deal with uncertainties of predicted customer demand while fulfilling the system's performance requirements, including minimizing taxis' total idle mileage and maintaining service fairness across the whole city. To address this problem, we develop a data- driven robust taxi dispatch framework to consider spatial-temporally correlated demand uncertainties. The robust optimization problem has an objective function that is concave in the uncertain demand and convex in the decision variables, and constraints that are convex in the decision variables according to the practical requirements. We design a data-driven algorithm for constructing uncertainty sets of random demand vectors that provide a probabilistic guarantee for the performance of robust taxi dispatch solutions. It guarantees that the probability that the actual dispatch cost under the true demand vector being smaller than the optimal cost of the robust dispatch solutions with the uncertainty sets is greater than a desired probability value. We prove equivalent computationally tractable forms of the robust dispatch problem with the constructed closed convex polytope and second order cone types of uncertainty sets, using the minimax theorem and strong duality. Evaluations on four years of taxi trip data for New York City show that the probabilistic guarantee is satisfied. By selecting a probabilistic guarantee level that does not sacrifice the average cost much, the average demand- supply ratio error is reduced by 31.7%, and the average total idle driving distance is reduced by 10.13% compared with non-robust dispatch solutions.

Robust optimization is proving to be a fruitful tool to study problems with uncertain data. In this paper we deal with the minmax aproach to robust multiobjective optimization. We survey the main features of this problem with particular reference to results concerning linear scalarization and sensitivity of optimal values with respect to changes in the uncertainty set. Furthermore we prove results concerning sensitivity of optimal solutions with respect to changes in the uncertainty set. Finally we apply the presented results to mean-variance portfolio optimization.

Designing networks with resiliency to edge failures using two-stage robust optimization