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,[1][2][3] 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.
More often, however, scenarios are instances of the uncertain constraints that are obtained as observations (data-driven science).
For constraints that are convex (e.g. in semidefinite problems, involving LMIs (Linear Matrix Inequalities)), a deep theoretical analysis has been established which shows that the probability that a new constraint is not satisfied follows a distribution that is dominated by a Beta distribution.
regularization has also been considered,[5] and handy algorithms with reduced computational complexity are available.
[6] Extensions to more complex, non-convex, set-ups are still objects of active investigation.
As this procedure moves on, the user constructs an empirical “curve of values”, i.e. the curve representing the value achieved after the removing of an increasing number of constraints.
The scenario theory provides precise evaluations of how robust the various solutions are.
A remarkable advance in the theory has been established by the recent wait-and-judge approach:[10] one assesses the complexity of the solution (as precisely defined in the referenced article) and from its value formulates precise evaluations on the robustness of the solution.
These results shed light on deeply-grounded links between the concepts of complexity and risk.
A related approach, named "Repetitive Scenario Design" aims at reducing the sample complexity of the solution by repeatedly alternating a scenario design phase (with reduced number of samples) with a randomized check of the feasibility of the ensuing solution.
possible market states only, the scenario theory tells us that the solution is robust up to a level
One alternative is to discard some odd situations to reduce pessimism;[7] moreover, scenario optimization can be applied to other risk-measures including CVaR – Conditional Value at Risk – so adding to the flexibility of its use.
[14] Fields of application include: prediction, systems theory, regression analysis (Interval Predictor Models in particular), Actuarial science, optimal control, financial mathematics, machine learning, decision making, supply chain, and management.