The Price of Robustness

TLDR: In this paper, the authors propose an approach that attempts to make this trade-off more attractive by flexibly adjusting the level of conservatism of the robust solutions in terms of probabilistic bounds of constraint violations.

Abstract: A robust approach to solving linear optimization problems with uncertain data was proposed in the early 1970s and has recently been extensively studied and extended. Under this approach, we are willing to accept a suboptimal solution for the nominal values of the data in order to ensure that the solution remains feasible and near optimal when the data changes. A concern with such an approach is that it might be too conservative. In this paper, we propose an approach that attempts to make this trade-off more attractive; that is, we investigate ways to decrease what we call the price of robustness. In particular, we flexibly adjust the level of conservatism of the robust solutions in terms of probabilistic bounds of constraint violations. An attractive aspect of our method is that the new robust formulation is also a linear optimization problem. Thus we naturally extend our methods to discrete optimization problems in a tractable way. We report numerical results for a portfolio optimization problem, a knapsack problem, and a problem from the Net Lib library.

Full text

The price of robustness
IA meeting 14/12/2020
Bertsimas, Dimitris, and Melvyn Sim. "The price of
robustness." Operations research 52.1 (2004): 35-53.

The price of robustness
Context
Quote from the case study by Ben-Tal and Nemirovski (2000):!
«!In real-world applications of Linear Programming, one cannot
ignore the possibility that a small uncertainty in the data can
make the usual optimal solution completely meaningless from a
practical viewpoint.!»!
This observation raises the natural question of designing solution
approaches that are immune to data uncertainty; that is, they
are «"robust"».#
#
This paper designs a new robust approach that adresses the
issue of over-conservatism.

The price of robustness
Data uncertainty in linear optimization
Data uncertainty is in the matrix A. !
The coefficients a_ij that are subjected to parameter uncertainty takes
values according to a symmetric distribution with a mean equal to the
nominal value a_ij in the interval [a_ij- â_ij, a_ij + â_ij].!
Row i -> J_i coefficients subject to uncertainty !
Gamma_i = parameter to adjust the robustness of the proposed method
against the level of conservatism of the solution.!
0 <= Gamma_i <= J_i -> only a subset of the coefficients will change in
order to adversely affect the solution.#
The higher Gamma_i, the more robust the solution is. With Gamma_i
= J_i -> maximum protection.
Linear optimization problem:

The price of robustness
Zero-one knap sack problem (MILP)
MILP:
An application of this problem is to maximize the
total value of goods to be loaded on a cargo that
has strict weight restrictions. The weight of the
individual item is assumed to be uncertain,
independent of other weights, and follows a
symmetric distribution.

The price of robustness
Zero-one knap sack problem (MILP)
The zero-one knapsack problem is the following discrete
optimization problem:
Let J the set of uncertain parameters ωj, with 0 ≤ |J| ≤ N. The weights ωj
with j ∈ J are subjected to parameter uncertainty takes values according to
a symmetric distribution with a mean equal to the nominal value ωj in the
interval [ωj − ωˆj, ωj + ωˆj]. The parameter to adjust the robustness of the
approach is Γ, with 0 ≤ Γ ≤ |J| ≤ N.