1. EMGT 6412/MATH 6665 Mathematical Programming Spring 2016
Introduction to Multi-objective Optimization
Dincer Konur
Engineering Management and Systems Engineering

2. Outline Introduction Solution Concepts Formulation Pareto Efficiency
Basic Properties
Solution Concepts
Reduction to single-objective
Pareto Front Approximation
Weighted approach
Epsilon Constraint

3. Outline Introduction Solution Concepts Formulation Pareto Efficiency
Basic Properties
Solution Concepts
Reduction to single-objective
Pareto Front Approximation
Weighted approach
Epsilon Constraint

4. Introduction Optimization problems that involve multiple objectives
Sometimes, the problems do not have single objective
In many decision making problems, you have two or more objectives
Minimize cost? Well, minimize time as well, maximize service quality as well, minimize risk as well, etc.
Formulation is just the same as regular models

5. Formulation A generic formulation: 𝒙: decision variables vector
𝑓 𝑖 (𝒙): ith objective function, i=1,2,…,n
𝑚𝑖𝑛𝑖𝑚𝑖𝑧𝑒
𝑓 1 (𝒙)
𝑓 2 (𝒙) …
𝑓 𝑛 (𝒙)
𝑠𝑢𝑏𝑗𝑒𝑐𝑡 𝑡𝑜
𝒙∈𝑋

6. Pareto Efficiency
For single-objective optimization models, we seek an optimum solution
Optimum solution: A solution such that there does not exist another solution which is better in terms of the single objective!
For multi-objective optimization models, the optimum cannot be defined because of multiple objectives
Based on each objective, we have an optimum solution
Let 𝒙 𝑖 be the optimum solution with ith objective function only, i.e., 𝒙 𝑖 =𝑎𝑟𝑔𝑚𝑖𝑛{ 𝑓 𝑖 𝒙 :𝒙∈𝑋}

7. Pareto Efficiency
For multi-objective optimization models, we define Pareto efficient solutions
Pareto efficient solution: A solution is Pareto efficient (Pareto optimum) if there does not exist another solution that is better in terms of all objectives
𝒙 𝑎 is Pareto efficient if and only if ∄ 𝒙 𝑏 ∈𝑋 such that 𝑓 𝑖 𝒙 𝑏 ≤ 𝑓 𝑖 𝒙 𝑎 for all i=1,2,…,n and 𝑓 𝑖 𝒙 𝑏 < 𝑓 𝑖 𝒙 𝑎 for at least one i.
This is also referred to as weak-Pareto efficiency
If all <= is replace with < then we have strong-Pareto efficiency

8. Basic Properties Recall:
Let 𝒙 𝑖 be the optimum solution with ith objective function only, i.e., 𝒙 𝑖 =𝑎𝑟𝑔𝑚𝑖𝑛{ 𝑓 𝑖 𝒙 :𝒙∈𝑋}
Property: 𝒙 𝑖 is Pareto efficient
Proof: ?

9. Basic Properties More definitions:
Pareto Front: Set of Pareto efficient (non-dominated) solutions

10. Basic Properties The properties of dominance relation:
Not reflexive: x does not dominate itself
Not symmetric: if x dominates y, y does not dominate x
Not asymmetric: if x dominates y, y cannot dominate x
Transitive: if x dominates y, y dominates t, then x dominates t
Proof?
If x does not dominate y, it does not mean that y dominates x
They can be both non-dominant

11. Basic Properties
Given a discrete finite set of solutions, one can determine all non-dominated solutions within the set by pair-wise comparison
C(.) and E(.) are the objective functions to be minimized
See more approaches and their complexity:

12. Basic Properties More definitions: Pareto Point vs. Pareto solution
A Pareto point is the point defined by the objective function values of a Pareto efficient solution

13. Basic Properties More definitions:
Ideal Point: A point on the objective space such that 𝑓 𝑖 = 𝑓 𝑖 𝒙 𝑖 ∀𝑖 where 𝒙 𝑖 =𝑎𝑟𝑔𝑚𝑖𝑛{ 𝑓 𝑖 𝒙 :𝒙∈𝑋}
Generally, there is no solution corresponding to the ideal point! 
Because, most of the times, objective functions are conflicting!
There are lots of versions of multi-objective optimization problems
Continuous, integer/binary, mixed-integer
Bi-objective, tri-objective, many-objective

14. Basic Properties Consider a bi-objective optimization problem
Let 𝒙 𝟏 =𝑎𝑟𝑔𝑚𝑖𝑛{ 𝑓 1 𝒙 :𝒙∈𝑋}, 𝒙 2 =𝑎𝑟𝑔𝑚𝑖𝑛{ 𝑓 2 𝒙 :𝒙∈𝑋}
Property (box): If 𝒙 is Pareto efficient then
𝑓 1 𝒙 𝟏 ≤ 𝑓 1 𝒙 ≤ 𝑓 1 𝒙 2
𝑓 2 𝒙 𝟐 ≤ 𝑓 2 𝒙 ≤ 𝑓 2 𝒙 1
Proof?
This can be generalized to more than two objectives!
This defines the boundaries of the objective space of interest
𝑚𝑖𝑛𝑖𝑚𝑖𝑧𝑒
𝑓 1 (𝒙)
𝑓 2 (𝒙)
𝑠𝑢𝑏𝑗𝑒𝑐𝑡 𝑡𝑜
𝒙∈𝑋

15. Basic Properties Consider a bi-objective optimization problem
Let 𝒙 ∗ =𝑎𝑟𝑔𝑚𝑖𝑛{ 𝑓 1 𝒙 : 𝑓 2 𝒙 ≤A,𝒙∈𝑋}, where 𝑓 2 𝒙 𝟐 ≤𝐴≤ 𝑓 2 𝒙 1
Property: Then 𝒙 ∗ is Pareto efficient
Proof?
This can also be generalized to more than two objectives
𝑚𝑖𝑛𝑖𝑚𝑖𝑧𝑒
𝑓 1 (𝒙)
𝑓 2 (𝒙)
𝑠𝑢𝑏𝑗𝑒𝑐𝑡 𝑡𝑜
𝒙∈𝑋

16. Basic Properties Continuous multi-objective optimization problems:
Here, we consider that the decision variables are continuous
Linear multi-objective optimization
Convex multi-objective optimization
Non-convex multi-objective optimization

17. Basic Properties Linear multi-objective optimization (continuous)
The Pareto front in the objective space is convex!
Proof?
For bi-objective case, the Pareto front consists of connected line segments
Convex multi-objective optimization (continuous)
Given objective functions are convex
The feasible region is convex
Then Pareto front is convex

18. Basic Properties
When we have integer or mixed-integer programming models with multiple objectives, the analyses get more complicated
Surprised? No!

19. Outline Introduction Solution Concepts Formulation Pareto Efficiency
Basic Properties
Solution Concepts
Reduction to single-objective
Pareto Front Approximation
Weighted approach
Epsilon Constraint

20. Solution Concepts There are two main approaches:
Reduction to single objective optimization
Weighted (or normalized weighted) approach: Assign weights to the objective functions (when the objective functions have different weights one can use normalized weights)
Use upper and lower bounds!
Ideal point defines the lower bounds
Nadir point defines the upper bounds
How to select the weights?
Min-max deviation approach: Minimize the maximum deviation from the individual optimum!
These return a single solution to the decision maker

21. Solution Concepts Pareto Front generation (approximation)
Try to generate all or (some) of the Pareto efficient solutions
Returns multiple solutions
The decision maker can select one of them to implement
Demonstrates the trade-offs, i.e., the Pareto front
The issues:
How to determine the Pareto efficient solutions effectively
How to generate (or approximate) the Pareto front
How good an approximation is compared to the exact Pareto front

22. Epsilon constraint method
It is the most well-known and most straight-forward method
It is an approximation method for continuous models
For continuous models, the Pareto front can be a continuous set (or combination of continuous sets), so, it is not possible to generate all of them
It can be an exact method for integer models
Just might need to make all parameters of the objective function integer

23. Epsilon constraint method
Recall our properties:
Property (box): If 𝒙 is Pareto efficient then
𝑓 1 𝒙 𝟏 ≤ 𝑓 1 𝒙 ≤ 𝑓 1 𝒙 2
𝑓 2 𝒙 𝟐 ≤ 𝑓 2 𝒙 ≤ 𝑓 2 𝒙 1
Property: Let 𝒙 ∗ =𝑎𝑟𝑔𝑚𝑖𝑛{ 𝑓 1 𝒙 : 𝑓 2 𝒙 ≤A,𝒙∈𝑋}, where 𝑓 2 𝒙 𝟐 ≤𝐴≤ 𝑓 2 𝒙 1 . Then 𝒙 ∗ is Pareto efficient
The epsilon constraint method uses these two properties and iteratively generates Pareto efficient solutions

24. Epsilon constraint method
Here is the outline of the algorithm:
Let A= 𝑓 2 𝒙 1
Let e be a small value and PF={}
While A>= 𝑓 2 𝒙 𝟐
Solve 𝒙 ∗ =𝑎𝑟𝑔𝑚𝑖𝑛{ 𝑓 1 𝒙 : 𝑓 2 𝒙 ≤A,𝒙∈𝑋},
Let PF:=PF U { 𝒙 ∗ }
Set A:=A-e
End
Diversity? Which function to use in the constraint? How small e should be?

25. Epsilon constraint method
Extension to integer and mixed-integer models
Still the main properties hold true
You can use the classical epsilon constraint method still
There are two main issues
If e is too small you might have the same 𝒙 ∗ repetitively
If e is too larger, you might miss some solutions!
There are adaptive version of the epsilon constraint method
See: Laumanns, M., Thiele, L., Zitzler, E., An ecient, adaptive parameter variation scheme for metaheuristics based on the epsilon-constraint method. European Journal of Operational Research 169 (3),