scholarly journals Robust optimization of uncertain multistage inventory systems with inexact data in decision rules

2016 ◽  
Vol 14 (1) ◽  
pp. 45-66 ◽  
Author(s):  
Frans J. C. T. de Ruiter ◽  
Aharon Ben-Tal ◽  
Ruud C. M. Brekelmans ◽  
Dick den Hertog
Keyword(s):  
Robust Optimization ◽  
Decision Rules ◽  
Inventory Systems ◽  
Inexact Data

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

2021 ◽  
Author(s):  
Jianzhe Zhen ◽  
Frans J. C. T. de Ruiter ◽  
Ernst Roos ◽  
Dick den Hertog
Keyword(s):  
Robust Optimization ◽  
Semidefinite Programming ◽  
Optimization Problems ◽  
Linear Optimization ◽  
Decision Rules ◽  
Second Order ◽  
Minimum Volume ◽  
Second Order Cone ◽  
Linear Decision Rules ◽  
Effectiveness And Efficiency

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.

scholarly journals Optimal Transport-Based Distributionally Robust Optimization: Structural Properties and Iterative Schemes

2021 ◽  
Author(s):  
Jose Blanchet ◽  
Karthyek Murthy ◽  
Fan Zhang
Keyword(s):  
Robust Optimization ◽  
Optimal Transport ◽  
Decision Rules ◽  
Stochastic Gradient Descent ◽  
Transport Cost ◽  
Distributionally Robust Optimization ◽  
Worst Case ◽  
Strongly Convex ◽  
Distributionally Robust ◽  
Convexity Assumptions

We consider optimal transport-based distributionally robust optimization (DRO) problems with locally strongly convex transport cost functions and affine decision rules. Under conventional convexity assumptions on the underlying loss function, we obtain structural results about the value function, the optimal policy, and the worst-case optimal transport adversarial model. These results expose a rich structure embedded in the DRO problem (e.g., strong convexity even if the non-DRO problem is not strongly convex, a suitable scaling of the Lagrangian for the DRO constraint, etc., which are crucial for the design of efficient algorithms). As a consequence of these results, one can develop efficient optimization procedures that have the same sample and iteration complexity as a natural non-DRO benchmark algorithm, such as stochastic gradient descent.

Technical Note—Two-Stage Sample Robust Optimization

2021 ◽  
Author(s):  
Dimitris Bertsimas ◽  
Shimrit Shtern ◽  
Bradley Sturt
Keyword(s):  
Robust Optimization ◽  
Approximation Scheme ◽  
Optimization Problems ◽  
Linear Optimization ◽  
Decision Rules ◽  
Technical Note ◽  
Data Driven ◽  
Two Stage ◽  
Linear Optimization Problems ◽  
Stochastic Linear Optimization

In “Two-Stage Sample Robust Optimization,” Bertsimas, Shtern, and Sturt investigate a simple approximation scheme, based on overlapping linear decision rules, for solving data-driven two-stage distributionally robust optimization problems with the type-infinity Wasserstein ambiguity set. Their main result establishes that this approximation scheme is asymptotically optimal for two-stage stochastic linear optimization problems; that is, under mild assumptions, the optimal cost and optimal first-stage decisions obtained by approximating the robust optimization problem converge to those of the underlying stochastic problem as the number of data points grows to infinity. These guarantees notably apply to two-stage stochastic problems that do not have relatively complete recourse, which arise frequently in applications. In this context, the authors show through numerical experiments that the approximation scheme is practically tractable and produces decisions that significantly outperform those obtained from state-of-the-art data-driven alternatives.

Primal and dual linear decision rules in stochastic and robust optimization

2009 ◽  
Vol 130 (1) ◽  
pp. 177-209 ◽  
Author(s):  
Daniel Kuhn ◽  
Wolfram Wiesemann ◽  
Angelos Georghiou
Keyword(s):  
Robust Optimization ◽  
Decision Rules ◽  
Linear Decision Rules ◽  
Primal And Dual

scholarly journals Piecewise Constant Decision Rules via Branch-and-Bound Based Scenario Detection for Integer Adjustable Robust Optimization

2020 ◽  
Author(s):  
Ward Romeijnders ◽  
Krzysztof Postek
Keyword(s):  
Robust Optimization ◽  
Branch And Bound ◽  
Decision Rules ◽  
Route Planning ◽  
Static Problem ◽  
General Purpose ◽  
Planning Problem ◽  
Uncertain Parameters ◽  
Piecewise Constant ◽  
Uncertainty Set

Multistage problems with uncertain parameters and integer decisions variables are among the most difficult applications of robust optimization (RO). The challenge in these problems is to find optimal here-and-now decisions, taking into account that the wait-and-see decisions have to adapt to the revealed values of the uncertain parameters. An existing approach to solve these problems is to construct piecewise constant decision rules by adaptively partitioning the uncertainty set. The partitions of this set are iteratively updated by separating so-called criticial scenarios, and methods for identifying these critical scenarios are available. However, these methods are most suitable for problems with continuous decision variables and many uncertain constraints, providing no mathematically rigorous methodology for partitioning in case of integer decisions. In particular, they are not able to identify sets of critical scenarios for integer problems with uncertainty in the objective function only. In this paper, we address this shortcoming by introducing a general critical scenario detection method. The new method leverages the information embedded in the dual vectors of the LP relaxations at the nodes of the branch-and-bound tree used to solve the corresponding static problem. Numerical experiments on a route planning problem show that our general-purpose method outperforms a problem-specific approach from the literature.

scholarly journals Staying Ahead of the Game: Adaptive Robust Optimization for Dynamic Allocation of Threat Screening Resources

2017 ◽  
Author(s):  
Sara Marie Mc Carthy ◽  
Phebe Vayanos ◽  
Milind Tambe
Keyword(s):  
Robust Optimization ◽  
Decision Rules ◽  
Dynamic Allocation ◽  
Arrival Times ◽  
X Ray ◽  
Solution Approach ◽  
X Ray Imaging ◽  
Solution Methods ◽  
Linear Decision Rules ◽  
And Performance

We consider the problem of dynamically allocating screening resources of different efficacies (e.g., magnetic or X-ray imaging) at checkpoints (e.g., at airports or ports) to successfully avert an attack by one of the screenees. Previously, the Threat Screening Game model was introduced to address this problem under the assumption that screenee arrival times are perfectly known. In reality, arrival times are uncertain, which severely impedes the implementability and performance of this approach. We thus propose a novel framework for dynamic allocation of threat screening resources that explicitly accounts for uncertainty in the screenee arrival times. We model the problem as a multistage robust optimization problem and propose a tractable solution approach using compact linear decision rules combined with robust reformulation and constraint randomization. We perform extensive numerical experiments which showcase that our approach outperforms (a) exact solution methods in terms of tractability, while incurring only a very minor loss in optimality, and (b) methods that ignore uncertainty in terms of both feasibility and optimality.

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