Input variables are system parameters that can be changed to improve the system performance such as beam dimensions, material properties, diameter, number of bolts, and so on. Input variables can be of different types with respect to their physical and/or numerical nature. Physically they can be dimensions, material properties, locations, etc. Numerically they can be continuous numbers, discrete, integer, real, strings. Optimization formulations are grouped as continuous, discrete or mixed continuous-discrete with respect to the numerical properties of the input variables.

Selection of input variables may not be unique as seen in the example below. In this example, one can either pick ri and ro or ri and t as input variables.

Figure 1: Example for Input Variables Definition

Input variables should be independent of each other. Also, for a larger search space and increased flexibility in the solution, we need to use as many as possible independent input variables but for computational affordability, we need to identify and only include the most important ones.

Objective functions are output responses that are required to be minimized (maximized). These output responses are functions of the input variables. Mass, stress, displacement, frequency and pressure drop are some examples of objective functions.

Objective functions numerically can be grouped as linear and non-linear functions.

Optimization problems can be grouped as single or multi-objective optimization problems depending on how many objectives are considered. Multi-objective optimization (MOO) objective functions are formulated as:

such that

When dealing with multiple objectives it is unlikely that one design will have minimum objective function values for all objectives.  As a result, in MOO applications an optimal Pareto front is searched for instead of an optimal design.  Optimal Pareto front is a collection of non-dominated designs. A non-dominated design has a lower objective function value than others with respect to at least one objective.

Usually the computational effort required to solve MOO problems are significantly more compared to single objective optimization problems. In cases where solving MOO problems are prohibitive, these problems have been converted to a single objective problem by summing all the objectives (Weighted Sum Method).

When dealing with probabilistic variables, the objective function also has an associated distribution. When doing robust optimization, instead of the deterministic objective, the objective function is the value of the objective distribution at a specified value of the cumulative distribution function (CDF). A minimization problem might use the 95% value of the CDF (default value), and a maximization problem might use the 5% value of the CDF. For more information on probabilistic variables, refer to Uncertainty in Design.

Constraint functions are system requirements that need to be satisfied for the design to be acceptable such as requirements on displacement, frequency, pressure drop, cost, etc. These functions are also functions of the input variables.

Objective

min f(x)

min cost ($)

Constraints

g(x) ≤ 0.0

h(x) = 0.0

σ < σallowable

Design Space

lower xi ≤ xi ≤ upper xi

2.5 mm < thickness < 5.0 mm

number of bolts ∈ (20, 22, 24, 26, 28, 30)

The point or design that minimized (maximized) the objective function and at the same time satisfy all the constraints.

Constraint that is not satisfied

One that is satisfied exactly; equality constraints are active for feasible designs.

One that satisfied but not on the bound.

A point or a design that satisfies all the constraints.

A design that violates one or more constraints.

System Identification helps to move a set of output responses towards their target values. Examples for typical applications are experimental curve fitting or parameter fitting. The objective function is formulated as a least squares formula, as described below, where is the target value of the ith output response.

Optimization problem formulation used to solve problems where the maximum (or minimum) of an output response is minimized (or maximized). The MinMax problem is formulated as follows:

These problems are solved using the Beta-method. In this method the problem is transformed into a regular optimization problem by introducing an additional input variable such that: