# Classical robust optimization

Author Names: Andre Ramirez-Cedeno
Steward: Fengqi You, Dajun Yue

# Classical Robust Optimization

## Indroduction and History

Robust optimization is a subset of optimization theory that deals with a certain measure of robustness vs uncertainty. This balance of robustness and uncertainty is represented as variability in the parameters of the problem at hand and or its solution. Robust optimization dates back to the beginning of modern decision theory in the 1950’s. It became a discipline of its own in the 1970’s with paralleled development in other technological fields. Robust optimization has many applications in statistics, chemical engineering, finance, pharmaceuticals and computer science. In engineering, this theory often is referred to as “Robust Design Optimization’ of “Reliability Based Design”.

## What is Robustness?

Robustness refers to the ability of a system to cope with errors during an execution. It can also be defined as the ability of an algorithm to continue operating despite abnormalities in calculations. Most algorithms try to find a balance between robustness and efficiency/execution time.

## Simple Mathematical Example

$Max 5X + 2Y$
s.t. $cX + dY <= 15$
$X <= 0$
$Y <= 0$
For All $(c,d) \in P$

This problem is a simple example of a robust optimization problem. The last clause; “for all (c,d) ϵ P” makes it a robust optimization problem because it implies that for a pair (X,Y) to be acceptable, the constraint cX + dY <= 15 must be satisfied for all values of (c,d) including the worst (c,d) pair that maximized the value of cX + dY for the given values of (x,y). For this example, P is simplified to a finite set meaning that for each (c,d) within the set, there is a constraint cX + dY <= 15.

## Engineering Design Example

Rolls-Royce. Robust design allows variation in the design process and the consideration of the appropriate selection of the nominal design point.

Step 1: Define – Understand what is important to the client and formulate problem in engineering language. Choose design concepts with variation in mind.

Step 2: Characterize – Generate measurable “critical to quality” (CTQ’s) criteria. For each CTQ, understand the possible sources of variation and measure the effects of variation.

Step 3: Optimize – For each CTQ, choose a strategy to reduce the effects of variation.

Step 4: Verify – Use knowledge of variation from previous steps to determine its effects in construction and design plan.

${\sigma_Y}^2 = {\sigma_{X1}}^2$ $\left( \frac{dY}{dX} \right)$