Extending Scope of Robust Optimization: Comprehensive Robust Counterparts of Uncertain Problems (original) (raw)

Foreword: special issue on robust optimization

Mathematical Programming, 2006

In this paper, we propose a new methodology for handling optimization problems with uncertain data. With the usual Robust Optimization paradigm, one looks for the decisions ensuring a required performance for all realizations of the data from a given bounded uncertainty set, whereas with the proposed approach, we require also a controlled deterioration in performance when the data is outside the uncertainty set.

Two-Stage Robust Optimization Under Decision Dependent Uncertainty

IEEE/CAA Journal of Automatica Sinica, 2022

In the conventional robust optimization (RO) context, the uncertainty is regarded as residing in a predetermined and fixed uncertainty set. In many applications, however, uncertainties are affected by decisions, making the current RO framework inapplicable. This paper investigates a class of twostage RO problems that involve decision-dependent uncertainties. We introduce a class of polyhedral uncertainty sets whose righthand-side vector has a dependency on the here-and-now decisions and seek to derive the exact optimal wait-and-see decisions for the second-stage problem. A novel iterative algorithm based on the Benders dual decomposition is proposed where advanced optimality cuts and feasibility cuts are designed to incorporate the uncertainty-decision coupling. The computational tractability, robust feasibility and optimality, and convergence performance of the proposed algorithm are guaranteed with theoretical proof. Four motivating application examples that feature the decisiondependent uncertainties are provided. Finally, the proposed solution methodology is verified by conducting case studies on the pre-disaster highway investment problem.

Chapter One. Uncertain Linear Optimization Problems and their Robust Counterparts

2009

In this chapter, we introduce the concept of the uncertain Linear Optimization problem and its Robust Counterpart, and study the computational issues associated with the emerging optimization problems. 1.1 DATA UNCERTAINTY IN LINEAR OPTIMIZATION Recall that the Linear Optimization (LO) problem is of the form min x c T x + d : Ax ≤ b , (1.1.1) where x ∈ R n is the vector of decision variables, c ∈ R n and d ∈ R form the objective, A is an m × n constraint matrix, and b ∈ R m is the right hand side vector. Clearly, the constant term d in the objective, while affecting the optimal value, does not affect the optimal solution, this is why it is traditionally skipped. As we shall see, when treating the LO problems with uncertain data there are good reasons not to neglect this constant term. The structure of problem (1.1.1) is given by the number m of constraints and the number n of variables, while the data of the problem are the collection (c, d, A, b), which we will arrange into an (m + 1) × (n + 1) data matrix D = c T d A b. Usually not all constraints of an LO program, as it arises in applications, are of the form a T x ≤ const; there can be linear "≥" inequalities and linear equalities as well. Clearly, the constraints of the latter two types can be represented equivalently by linear "≤" inequalities, and we will assume henceforth that these are the only constraints in the problem. 1.1.1 Introductory Example Consider the following very simple linear optimization problem: Example 1.1.1. A company produces two kinds of drugs, DrugI and DrugII, containing a specific active agent A, which is extracted from raw materials purchased on the market. There are two kinds of raw materials, RawI and RawII, which can be used as sources of the active agent. The related production, cost, and resource data are given in table 1.1. The goal is to find the production plan that maximizes the profit of the company.

Robust optimal control with adjustable uncertainty sets

Automatica, 2017

In this paper, we develop a unified framework for studying constrained robust optimal control problems with adjustable uncertainty sets. In contrast to standard constrained robust optimal control problems with known uncertainty sets, we treat the uncertainty sets in our problems as additional decision variables. In particular, given a finite prediction horizon and a metric for adjusting the uncertainty sets, we address the question of determining the optimal size and shape of the uncertainty sets, while simultaneously ensuring the existence of a control policy that will keep the system within its constraints for all possible disturbance realizations inside the adjusted uncertainty set. Since our problem subsumes the classical constrained robust optimal control design problem, it is computationally intractable in general. We demonstrate that by restricting the families of admissible uncertainty sets and control policies, the problem can be formulated as a tractable convex optimization problem. We show that our framework captures several families of (convex) uncertainty sets of practical interest, and illustrate our approach on a demand response problem of providing control reserves for a power system.

On the design of robust controllers for arbitrary uncertainty structures

IEEE Transactions on Automatic Control, 2003

The focal point of this note is the design of robust controllers for linear time-invariant uncertain systems. Given bounds on performance (defined by a convex performance evaluator) the algorithm converges to a controller that robustly satisfies the specifications. The procedure introduced has its basis on stochastic gradient algorithms and it is proven that the probability of performance violation tends to zero with probability one. Moreover, this algorithm can be applied to any uncertain plant, independently of the uncertainty structure. As an example of application of this new approach, we demonstrate its usefulness in the design of robust controllers.

Two-Stage Robust Optimization Problems with Two-Stage Uncertainty

2021

We consider robust two-stage optimization problems, which can be considered as a game between the decision maker and an adversary. After the decision maker fixes part of the solution, the adversary chooses a scenario from a specified uncertainty set. Afterwards, the decision maker can react to this scenario by completing the partial first-stage solution to a full solution. We extend this classic setting by adding another adversary stage after the second decision-maker stage, which results in min-max-min-max problems, thus pushing twostage settings further towards more general multi-stage problems. We focus on budgeted uncertainty sets and consider both the continuous and discrete case. In the former, we show that a wide range of robust combinatorial problems can be decomposed into polynomially many subproblems, which in turn can often be solved in polynomial time. For the latter, we prove NP-hardness for a wide range of problems, but note that the special case where firstand second-...

Optimization under Decision-Dependent Uncertainty

SIAM Journal on Optimization, 2018

The efficacy of robust optimization spans a variety of settings with uncertainties bounded in predetermined sets. In many applications, uncertainties are affected by decisions and cannot be modeled with current frameworks. This paper takes a step towards generalizing robust linear optimization to problems with decisiondependent uncertainties. In general settings, we show these problems to be NP-complete. To alleviate the computational inefficiencies, we introduce a class of uncertainty sets whose size depends on binary decisions. We propose reformulations that improve upon alternative standard linearization techniques. To illustrate the advantages of this framework, a shortest path problem is discussed, where the uncertain arc lengths are affected by decisions. Beyond the modeling and performance advantages, the proposed notion of proactive uncertainty control also mitigates over conservatism of current robust optimization approaches.

Data-Driven Robust Optimization Using Scenario-Induced Uncertainty Sets

2021

Uncertainty sets are at the heart of robust optimization (RO) because they play a key role in determining the RO models’ tractability, robustness, and conservativeness. Different types of uncertainty sets have been proposed that model uncertainty from various perspectives. Among them, polyhedral uncertainty sets are widely used due to their simplicity and flexible structure to model the underlying uncertainty. However, the conventional polyhedral uncertainty sets present certain disadvantages; some are too conservative while others lead to computationally expensive RO models. This paper proposes a systematic approach to develop data-driven polyhedral uncertainty sets that mitigate these drawbacks. The proposed uncertainty sets are polytopes induced by a given set of scenarios, capture correlation information between uncertain parameters, and allow for direct trade-offs between tractability and conservativeness issue of conventional polyhedral uncertainty sets. To develop these uncer...

Robust control design for systems with probabilistic uncertainty

NASA report, TP-2005-213531, 2005

This paper presents a reliability-and robustness-based formulation for robust control synthesis for systems with probabilistic uncertainty. In a reliability-based formulation, the probability of violating design requirements prescribed by inequality constraints is minimized. In a robustness-based formulation, a metric which measures the tendency of a random variable/process to cluster close to a target scalar/function is minimized. A multi-objective optimization procedure, which combines stability and performance requirements in time and frequency domains, is used to search for robustly optimal compensators. Some of the fundamental differences between the proposed strategy and conventional robust control methods are: (i) unnecessary conservatism is eliminated since there is not need for convex supports, (ii) the most likely plants are favored during synthesis allowing for probabilistic robust optimality, (iii) the tradeoff between robust stability and robust performance can be explored numerically, (iv) the uncertainty set is closely related to parameters with clear physical meaning, and (v) compensators with improved robust characteristics for a given control structure can be synthesized. Several numerical methods for estimation, including the Hammersley sequence sampling method, the First Order Reliability method, and the First-and Second-Moment-Second-Order-Methods, are compared. Examples using output-feedback and full-state feedback with state estimation are used to demonstrate and validate the methodology.

A New Robust Optimization Approach to Deal with Dependent Uncertain Parameters

2020

In the optimization problems with uncertain parameters, a solution is said to be robust if it is feasible with high probability regarding the realization of uncertain parameters. In this paper, a new robust approach is developed for the linear problems in which the model parameters are dependent on each other. The proposed approach converts the linear model to an equivalent integer linear programming one using the primal and dual theorem. The results of the paper indicate the ability of the new approach in fixing some inconsistency of the common robust optimization approach for the mentioned problem.