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ARO (adjustable robust optimization)

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Introduction:

Adjustable Robust Optimization (ARO) is a mathematical programming framework that aims to minimize the cost of a decision-making problem under uncertainty. ARO provides a method for determining a robust solution that can be adjusted based on the level of conservatism desired by the decision-maker. In this article, we will explain what ARO is, its key components, and how it can be applied in practice.

What is ARO? ARO is a mathematical programming framework that provides a decision-making tool for optimization under uncertainty. It is based on the concept of robust optimization, which aims to identify a decision that performs well under a wide range of uncertain scenarios. ARO extends robust optimization by allowing the decision-maker to adjust the level of conservatism of the solution. This flexibility makes ARO a powerful tool for decision-making under uncertainty.

Key Components of ARO:

The key components of ARO are the objective function, the decision variables, the uncertainty set, and the adjustable parameter.

  1. Objective function: The objective function is a mathematical expression that describes the goal of the decision-making problem. In ARO, the objective function is typically a cost function that the decision-maker wants to minimize.
  2. Decision Variables: The decision variables are the parameters that the decision-maker can control. In ARO, the decision variables are typically continuous or discrete, and they represent the decision-maker's choices.
  3. Uncertainty Set: The uncertainty set is a set of scenarios that the decision-maker believes may occur. In ARO, the uncertainty set can be defined in a variety of ways, depending on the nature of the problem. It can be a set of possible values for a parameter, a range of values for a random variable, or a set of probability distributions.
  4. Adjustable Parameter: The adjustable parameter is a parameter that determines the level of conservatism of the solution. In ARO, the adjustable parameter can be used to control the size of the uncertainty set, the confidence level of the solution, or the trade-off between the objective function and the robustness of the solution.

ARO Formulation: The ARO problem can be formulated as follows:

min f(x) subject to g(x, w) <= 0, for all w in W, x in X

where f(x) is the objective function, g(x,w) is a constraint function that depends on the decision variables x and the uncertain parameters w, and W is the uncertainty set. The ARO problem aims to find a solution x that minimizes the objective function f(x) while satisfying the constraint function g(x,w) for all w in W.

The ARO problem can be extended to include an adjustable parameter, alpha, as follows:

min f(x) subject to g(x, w) <= alpha, for all w in W, x in X

where alpha is an adjustable parameter that determines the level of conservatism of the solution. If alpha is set to a small value, the solution will be more conservative, i.e., it will perform well under a wider range of uncertain scenarios. If alpha is set to a large value, the solution will be less conservative, i.e., it will perform well under a narrower range of uncertain scenarios.

Solving ARO Problems:

Solving ARO problems can be challenging because of the complexity of the uncertainty set and the non-convexity of the problem. However, several methods have been proposed to solve ARO problems efficiently. These methods can be classified into three categories: robust optimization, adjustable robust optimization, and distributionally robust optimization.

  1. Robust Optimization: Robust optimization is a method that seeks a solution that is optimal under the worst-case scenario in the uncertainty set. The worst-case scenario is typically defined as the scenario that leads to the highest cost or the lowest profit. Robust optimization methods are based on the idea of minimizing the maximum of the objective function over the uncertainty set. This approach ensures that the solution is robust to all scenarios in the uncertainty set, but it can be too conservative for some applications.
  2. Adjustable Robust Optimization: Adjustable Robust Optimization, as previously mentioned, extends robust optimization by allowing the decision-maker to adjust the level of conservatism of the solution. The adjustable parameter alpha can be used to control the size of the uncertainty set or the trade-off between the objective function and the robustness of the solution. The solution can be found using optimization algorithms such as column generation or branch-and-bound.
  3. Distributionally Robust Optimization: Distributionally Robust Optimization is a method that seeks a solution that performs well under a range of probability distributions rather than a fixed set of scenarios. The method aims to minimize the expected cost over a set of probability distributions that are close to a nominal distribution. This approach ensures that the solution is robust to model uncertainty, but it can be computationally expensive.

Applications of ARO:

ARO has a wide range of applications in different fields, including finance, engineering, and logistics. Here are some examples of how ARO can be applied in practice:

  1. Portfolio Optimization: ARO can be used to optimize investment portfolios under uncertainty. The decision-maker can adjust the level of conservatism of the solution based on their risk appetite. The uncertainty set can include different sources of risk, such as market risk, credit risk, and liquidity risk.
  2. Supply Chain Management: ARO can be used to optimize supply chain decisions under uncertainty. The decision-maker can adjust the level of conservatism of the solution based on the reliability of the supply chain. The uncertainty set can include different sources of uncertainty, such as demand variability, lead time variability, and transportation disruptions.
  3. Power System Optimization: ARO can be used to optimize power system operations under uncertainty. The decision-maker can adjust the level of conservatism of the solution based on the reliability of the power system. The uncertainty set can include different sources of uncertainty, such as renewable energy generation variability, demand variability, and transmission line failures.

Conclusion:

ARO is a mathematical programming framework that provides a decision-making tool for optimization under uncertainty. ARO extends robust optimization by allowing the decision-maker to adjust the level of conservatism of the solution. The adjustable parameter can be used to control the size of the uncertainty set or the trade-off between the objective function and the robustness of the solution. ARO has a wide range of applications in different fields, including finance, engineering, and logistics.