The price of robustness

The price of robustness

2004 | Bertsimas, Dimitris, and Melvyn Sim
The paper "The Price of Robustness" by Dimitris Bertsimas and Melvyn Sim addresses the issue of data uncertainty in linear optimization problems, particularly in the context of linear programming and mixed-integer linear programming (MILP). The authors highlight the importance of designing robust solutions that are resilient to small uncertainties in data, as these uncertainties can render optimal solutions meaningless in practical applications. In the context of linear optimization, the paper introduces a robust approach to handle parameter uncertainty in the matrix \( \mathbf{A} \). The coefficients \( \mathbf{a}_{ij} \) are assumed to follow a symmetric distribution around their nominal values, with a parameter \( \mathbf{\Gamma}_i \) controlling the level of robustness. Higher values of \( \mathbf{\Gamma}_i \) result in more robust solutions, with \( \mathbf{\Gamma}_i = \mathbf{J}_i \) providing maximum protection. The paper also applies this robust approach to the zero-one knapsack problem (MILP), which involves maximizing the total value of goods subject to weight constraints. The weights of the items are uncertain and follow a symmetric distribution. The robust formulation of the knapsack problem includes a protection function \( \beta(\mathbf{x}^*, \Gamma) \) that ensures a worst-case scenario within a specified probability bound. The robust formulation is formulated as a mixed-integer linear program (MILP) and is shown to reduce the objective function value only slightly compared to non-robust solutions, effectively reducing the "price of robustness." A use case is provided where the goal is to maximize the total value of goods while allowing a maximum 1% chance of constraint violation. The optimal value of the robust formulation is compared to the non-robust formulation, showing a reduction in the objective function value from 5,992 to 5,283, representing a 5.5% reduction. Additionally, the paper demonstrates that for a 0.57% chance of constraint violation, the objective function value is reduced by 1.54% when \( \mathbf{\Gamma}_i = 37 \). Overall, the paper shows that the robust approach successfully reduces the penalty on the objective function value while maintaining robustness against constraint violations.The paper "The Price of Robustness" by Dimitris Bertsimas and Melvyn Sim addresses the issue of data uncertainty in linear optimization problems, particularly in the context of linear programming and mixed-integer linear programming (MILP). The authors highlight the importance of designing robust solutions that are resilient to small uncertainties in data, as these uncertainties can render optimal solutions meaningless in practical applications. In the context of linear optimization, the paper introduces a robust approach to handle parameter uncertainty in the matrix \( \mathbf{A} \). The coefficients \( \mathbf{a}_{ij} \) are assumed to follow a symmetric distribution around their nominal values, with a parameter \( \mathbf{\Gamma}_i \) controlling the level of robustness. Higher values of \( \mathbf{\Gamma}_i \) result in more robust solutions, with \( \mathbf{\Gamma}_i = \mathbf{J}_i \) providing maximum protection. The paper also applies this robust approach to the zero-one knapsack problem (MILP), which involves maximizing the total value of goods subject to weight constraints. The weights of the items are uncertain and follow a symmetric distribution. The robust formulation of the knapsack problem includes a protection function \( \beta(\mathbf{x}^*, \Gamma) \) that ensures a worst-case scenario within a specified probability bound. The robust formulation is formulated as a mixed-integer linear program (MILP) and is shown to reduce the objective function value only slightly compared to non-robust solutions, effectively reducing the "price of robustness." A use case is provided where the goal is to maximize the total value of goods while allowing a maximum 1% chance of constraint violation. The optimal value of the robust formulation is compared to the non-robust formulation, showing a reduction in the objective function value from 5,992 to 5,283, representing a 5.5% reduction. Additionally, the paper demonstrates that for a 0.57% chance of constraint violation, the objective function value is reduced by 1.54% when \( \mathbf{\Gamma}_i = 37 \). Overall, the paper shows that the robust approach successfully reduces the penalty on the objective function value while maintaining robustness against constraint violations.
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