Scenario optimization

The scenario approach or scenario optimization approach is a technique for obtaining solutions to robust optimization and chance-constrained optimization problems based on a sample of the constraints. It also relates to inductive reasoning in modeling and decision-making. The technique has existed for decades as a heuristic approach and has more recently been given a systematic theoretical foundation. In optimization, robustness features translate into constraints that are parameterized by the uncertain elements of the problem. In the scenario method, a solution is obtained by only looking at a random sample of constraints (heuristic approach) called scenarios and a deeply-grounded theory tells the user how “robust” the corresponding solution is related to other constraints. This theory justifies the use of randomization in robust and chance-constrained optimization.

The scenario approach or scenario optimization approach is a technique for obtaining solutions to robust optimization and chance-constrained optimization problems based on a sample of the constraints. It also relates to inductive reasoning in modeling and decision-making. The technique has existed for decades as a heuristic approach and has more recently been given a systematic theoretical foundation. In optimization, robustness features translate into constraints that are parameterized by the uncertain elements of the problem. In the scenario method, a solution is obtained by only looking at a random sample of constraints (heuristic approach) called scenarios and a deeply-grounded theory tells the user how “robust” the corresponding solution is related to other constraints. This theory justifies the use of randomization in robust and chance-constrained optimization. \max_x \min_{i=1,\dots,N} R_{\delta_i}(x). \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (1) This corresponds to choosing a portfolio vector x so as to obtain the best possible return in the worst-case scenario. After solving (1), an optimal investment strategy x^\ast is achieved along with the corresponding optimal return R^\ast . While R^\ast has been obtained by looking at N possible market states only, the scenario theory tells us that the solution is robust up to a level \varepsilon , that is, the return R^\ast will be achieved with probability 1 - \varepsilon for other market states. In quantitative finance, the worst-case approach can be overconservative. One alternative is to discard some odd situations to reduce pessimism; moreover, scenario optimization can be applied to other risk-measures including CVaR – Conditional Value at Risk – so adding to the flexibility of its use.

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