EMGT 6412/MATH 6665
Mathematical Programming
Spring 2016
Introduction to Multi-objective
Optimization
Dincer Konur
Engineering Management and Systems
Engineering
1
Outline
• Introduction
– Formulation
– Pareto Efficiency
– Basic Properties
• Solution Concepts
– Reduction to single-objective
– Pareto Front Approximation
• Weighted approach
• Epsilon Constraint
2
Outline
• Introduction
– Formulation
– Pareto Efficiency
– Basic Properties
• Solution Concepts
– Reduction to single-objective
– Pareto Front Approximation
• Weighted approach
• Epsilon Constraint
3
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
4
Formulation
• A generic formulation:
– 𝒙: decision variables vector
– 𝑓𝑖 (𝒙): ith objective function, i=1,2,…,n
𝑚𝑖𝑛𝑖𝑚𝑖𝑧𝑒
𝑓1 (𝒙)
𝑚𝑖𝑛𝑖𝑚𝑖𝑧𝑒
𝑓2 (𝒙)
…
…
𝑚𝑖𝑛𝑖𝑚𝑖𝑧𝑒
𝑓𝑛 (𝒙)
𝑠𝑢𝑏𝑗𝑒𝑐𝑡 𝑡𝑜
𝒙∈𝑋
5
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 𝒙𝑖 be the optimum solution with ith objective
function only, i.e., 𝒙𝑖 = 𝑎𝑟𝑔𝑚𝑖𝑛{𝑓𝑖 𝒙 : 𝒙 ∈ 𝑋}
6
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
– 𝒙𝑎 is Pareto efficient if and only if ∄𝒙𝑏 ∈ 𝑋 such that
𝑓𝑖 𝒙𝑏 ≤ 𝑓𝑖 𝒙𝑎 for all i=1,2,…,n and 𝑓𝑖 𝒙𝑏 < 𝑓𝑖 𝒙𝑎 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
7
Basic Properties
• Recall:
– Let 𝒙𝑖 be the optimum solution with ith objective
function only, i.e., 𝒙𝑖 = 𝑎𝑟𝑔𝑚𝑖𝑛{𝑓𝑖 𝒙 : 𝒙 ∈ 𝑋}
– Property: 𝒙𝑖 is Pareto efficient
– Proof: ?
8
Basic Properties
• More definitions:
– Pareto Front: Set of Pareto efficient (non-dominated) solutions
9
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
10
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:
http://cims.nyu.edu/~gn387/glp
/lec1.pdf
11
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
12
Basic Properties
• More definitions:
– Ideal Point: A point on the objective space such that
𝑓𝑖 = 𝑓𝑖 𝒙𝑖 ∀𝑖 where 𝒙𝑖 = 𝑎𝑟𝑔𝑚𝑖𝑛{𝑓𝑖 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
13
Basic Properties
• Consider a bi-objective optimization problem
𝑚𝑖𝑛𝑖𝑚𝑖𝑧𝑒
𝑓1 (𝒙)
𝑚𝑖𝑛𝑖𝑚𝑖𝑧𝑒
𝑓2 (𝒙)
𝑠𝑢𝑏𝑗𝑒𝑐𝑡 𝑡𝑜
𝒙∈𝑋
• Let 𝒙𝟏 = 𝑎𝑟𝑔𝑚𝑖𝑛{𝑓1 𝒙 : 𝒙 ∈ 𝑋}, 𝒙2 = 𝑎𝑟𝑔𝑚𝑖𝑛{𝑓2 𝒙 : 𝒙 ∈ 𝑋}
– Property (box): If 𝒙 is Pareto efficient then
• 𝑓1 𝒙𝟏 ≤ 𝑓1 𝒙 ≤ 𝑓1 𝒙2
• 𝑓2 𝒙𝟐 ≤ 𝑓2 𝒙 ≤ 𝑓2 𝒙1
– Proof?
– This can be generalized to more than two objectives!
– This defines the boundaries of the objective space of
14
interest
Basic Properties
• Consider a bi-objective optimization problem
𝑚𝑖𝑛𝑖𝑚𝑖𝑧𝑒
𝑓1 (𝒙)
𝑚𝑖𝑛𝑖𝑚𝑖𝑧𝑒
𝑓2 (𝒙)
𝑠𝑢𝑏𝑗𝑒𝑐𝑡 𝑡𝑜
𝒙∈𝑋
• Let 𝒙∗ = 𝑎𝑟𝑔𝑚𝑖𝑛{𝑓1 𝒙 : 𝑓2 𝒙 ≤ A, 𝒙 ∈ 𝑋}, where
𝑓2 𝒙𝟐 ≤ 𝐴 ≤ 𝑓2 𝒙1
– Property: Then 𝒙∗ is Pareto efficient
– Proof?
– This can also be generalized to more than two objectives
15
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
16
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
• Proof?
17
Basic Properties
• When we have integer or mixed-integer
programming models with multiple objectives, the
analyses get more complicated
– Surprised? No!
18
Outline
• Introduction
– Formulation
– Pareto Efficiency
– Basic Properties
• Solution Concepts
– Reduction to single-objective
– Pareto Front Approximation
• Weighted approach
• Epsilon Constraint
19
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
20
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
21
Epsilon constraint method
• It is the most well-known and most straightforward 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
22
Epsilon constraint method
• Recall our properties:
– Property (box): If 𝒙 is Pareto efficient then
• 𝑓1 𝒙𝟏 ≤ 𝑓1 𝒙 ≤ 𝑓1 𝒙2
• 𝑓2 𝒙𝟐 ≤ 𝑓2 𝒙 ≤ 𝑓2 𝒙1
– Property: Let 𝒙∗ = 𝑎𝑟𝑔𝑚𝑖𝑛{𝑓1 𝒙 : 𝑓2 𝒙 ≤ A, 𝒙 ∈ 𝑋},
where 𝑓2 𝒙𝟐 ≤ 𝐴 ≤ 𝑓2 𝒙1 . Then 𝒙∗ is Pareto efficient
• The epsilon constraint method uses these two
properties and iteratively generates Pareto
efficient solutions
23
Epsilon constraint method
• Here is the outline of the algorithm:
– Let A=𝑓2 𝒙1
– Let e be a small value and PF={}
1. While A>=𝑓2 𝒙𝟐
2.
Solve 𝒙∗ = 𝑎𝑟𝑔𝑚𝑖𝑛{𝑓1 𝒙 : 𝑓2 𝒙 ≤ A, 𝒙 ∈ 𝑋},
3.
Let PF:=PF U {𝒙∗ }
4.
Set A:=A-e
5. End
• Diversity? Which function to use in the
constraint? How small e should be?
24
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., 2006. An ecient, adaptive
parameter variation scheme for metaheuristics based on the epsilonconstraint method. European Journal of Operational Research 169 (3), 932942.
25
http://www.sciencedirect.com/science/article/pii/S0377221704005715