Criteria for Formulating an Optimization Problem

In order to formulate a design problem as an optimization problem you must identify input variables, objective functions, and constraint functions.

When you put input variables, objectives and constraints together, you get the optimization formulation for a design problem as shown in Table 1.

Refer to Objectives and Constraints for more information.

Table 1.
Type Formula(s) Example
Objectives min f ( x ) min cost ( $ )
Constraints

g ( x ) 0.0

h ( x ) = 0.0

σ < σ a l l o w a b l e
Design Space

l o w e r   x i x i u p p e r   x i

2.5 m m < t h i c k n e s s < 5.0 m m

number of bolts  ( 20 , 22 , 24 , 26 , 28 , 30 )
Where:
  • f(x) MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzaiaacI cacaWG4bGaaiykaaaa@3934@ is the vector of system output responses that are used as objectives.
  • g ( x ) MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4zaiaacI cacaWG4bGaaiykaaaa@3935@ and h(x) MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiAaiaacI cacaWG4bGaaiykaaaa@3936@ is the vector or system output responses that are used as inequality and equality constraints.
  • x MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaaaa@36F0@ is the vector of input variables.
Some of the results that would be reported after an optimization run include but not limited to:
Optimum Design
The point or design that minimized (maximized) the objective function and at the same time satisfy all the constraints.
Violated Constraint
Constraint that is not satisfied.
Active Constraint
Constraint that is satisfied exactly; equality constraints are active for feasible designs.
Inactive Constraint
Constraint that satisfied but not on the bound.
Feasible Design
A point or a design that satisfies all the constraints.
Infeasible Design
A design that violates one or more constraints.
Figure 1. Design Space Definitions