# Robust optimization

### From Glossary

A term given to an approach to deal with uncertainty, similar to the recourse model of stochastic programming, except that feasibility for all possible realizations (called *scenarios*) is replaced by a penalty function in the objective. As such, the approach integrates goal programming with a *scenario-based* description of problem data. To illustrate, consider the LP:

where and are random variables with possible realizations
( number of scenarios). The robust optimization model for this LP is:

where f is a function that measures the cost of the policy, is a penalty function, and (a parameter to trade off the cost of infeasibility). One example of is the expected value:
where probability of scenario s. In a *worst-case model*,
The penalty function is defined to be zero if is feasible (for all scenarios) -- i.e., In addition, satisfies a form of monotonicity: worse violations incur greater penalty. This often has the form
-- i.e., the "up" and "down" penalties, where and are strictly increasing functions.

The above makes robust optimization similar (at least in the model) to a goal program. Recently, the robust optimization community defines it differently - it optimizes for the worst-case scenario. Let the uncertain MP be given by

where is some set of scenarios (like parameter values). The * robust optimization model* (according to this more recent definition) is:

The policy is required to be feasible no matter what parameter value (scenario) occurs; hence, it is requied to be in the intersection of all possible The inner maximization yields the worst possible objective value among all scenarios. There are variations, such as "adjustability" (i.e., recourse).