Multi-objective optimization

4. Multi-objective optimization#

In this chapter, we’ll cover how to solve multi-objective optimization problem using scipy. As a reminder, nonlinear constrained optimization considers:

(4.1)#minx⁡f1(x),f2(x)

with:

  • f1(x) and f2(x), the linear or nonlinear objective functions.

  • x, the n design variables

  • Constraints and bounds as for single-objective optimization problems.

Model#

Three different ways of solving multi-objective optimization problems were introduced, which all effectively convert the problem to a single-objective optimization problem. All of this is assuming minimization problems:

  1. Weighted objective function: setting pre-determined weight on the two objectives. In general this requires the two objectives to have a comparable unit:

(4.2)#minx⁡(δ1,predefined⋅f1(x)+δ2predefined⋅f2(x))

  1. Goal attainment, minimizing the maximum difference with respect to two goal values for the objectives. Again, this requires the two objectives to have a comparable unit:

(4.3)#minx⁡(max(f1(x)−f1,goal,f2(x)−f2,goal))

  1. Pareto front: finding many possible optimal solution for a large set weights. It’s good practise to normalize the objective functions.

(4.4)#minx(δi⋅f1,normalized(x)+δj⋅f2,normalized(x))with 0<δi<1 and δj=1−δi

All of these methods could also be applied to problems which include more than two goals.

Test yourself

Normalize objective functions#

Normalizing the objectives functions can be done by setting the domain of every goal f between 0 and 1 by finding (or estimating) the lower and upper bounds for these objective functions within the domain:

(4.5)#fnormalized(x)=f(x)−minx(f(x))maxx(f(x))−minx(f(x))with minx(f(x)),maxx(f(x)) for xil≤xi≤xiu with i=1,n
Test yourself

Method#

Because the models are all single-objective, we can use our earlier methods to solve these problems.

Questions, discussions and comments#