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.