Multi optimization lectures for the the understanding of the multi variable decision making

Lecture 2 – Intro to optimization, MOOP
DSc PhD Eng. Joanna Szłapczyńska, GUT Assoc. Prof.
PG WETI, KASK, Gdańsk 2025, e-mail: joanna.szlapczynska@pg.edu.pl
Multi-objective Optimization
Algorithms (MOOA)
algorytmy optymalizacji
wielokryterialnej (AOW)

Lecture plan
2
 Introduction to optimization and decision making
 Multi-Criteria Decision Making (MCDM)
 Optimization problems in general
 Multi-Objective Optimization (MOO) model
 Simplifying the MOO
 AI meta-heuristics used in MOO, selected meta-heuristic MOO algorithms
 Programming frameworks and libraries for MOO
 Methods of taking into account the decision-maker's preferences in MOO
 Performance and efficiency testing of MOO algorithms
Multi-objective Optimization Algorithms (MOOA/AOW)
Lecture 2

Classification of optimization methods
3 Multi-objective Optimization Algorithms (MOOA/AOW) http://dx.doi.org/10.1155/2021/5521951

Single-Objective Optimization (SOO)
4
 SOO is when we have just one optimization goal (objective) function
 In real problems there is usually no pure SOO
 Different objectives can be reduced to a single one by means of
• aggregated objective e.g. weighted average of different objectives
• handling some of the objectives as constraints
 Problems can be constrained
 The result is (in most cases) a single solution

Multi-Objective Optimization (MOO)
5
 MOO is when there are more than 2 objectives
• for 5 and more objectives it is called many-objective instead
 Every objective is taken into account separately
 MOO can also be constrained
 It is hardly possible that a single solution would optimize all the objective functions at once
• so-called Pareto dominance is utilized to find solutions to MOO

Classic Pareto dominance
6
 Concept of dominance was originally created by Vilfredo Pareto (1848–1923),
Italian civil engineer and economist
 In Pareto Multi-Objective Optimization Problem (MOOP)
• let’s say we have two MOOP solutions x and y
• we state that solution x dominates solution y if and only if
x is better than y for at least one objective
and x is not worse than y for all the other objectives
 Minimization of every goal is assumed by default
 The result is a set of solutions that are not Pareto dominated (non-dominated)

Pareto 80/20 rule (Pareto principle)
7
 Another concept – the „80/20 rule” aka „Pareto principle” - was named after
Vilfredo Pareto
• proposed in 1941 by J.M. Juran
• for many outcomes, roughly 80% of consequences come from 20% of causes
• V. Pareto published in one of his last works that approximately 80% of the land
in the Kingdom of Italy was owned by 20% of the population
 This principle applies to many economic, political and engineering issues
 Pareto 80/20 rule ≠ Pareto dominance !!!

Classic Pareto dominance (formally)
8
 Assumptions of classic Pareto dominance
• minimization for every goal function
• all the solutions are feasible (no direct constraint handling!)
 In Pareto MOOP (with n objectives) solution x dominates y if and only if
 The result is a set of Pareto non-dominated (optimal) solutions, for which a Pareto Front (PF)
is created
 Pareto non-dominated solutions are often called efficient
∀𝑖:1,..,𝑛 𝑓𝑖 𝑥 ≤ 𝑓𝑖 𝑦 and ∃𝑗:1,..,𝑛 𝑓𝑗 𝑥 < 𝑓𝑗 𝑦

Strong and weak Pareto dominance
9
 Strong Pareto dominance is when we get the improvement for all of the objectives
 Weak Pareto dominance is when Pareto dominance does occur, but it is not strong
 Classic Pareto dominance is either weak or strong
∀𝑖:1,..,𝑛 𝑓𝑖 𝑥 < 𝑓𝑖 𝑦 ⟺ 𝑓(𝑥) ≺ 𝑓(𝑦) ⟺ 𝑥 ≺ 𝑦
∀𝑖:1,..,𝑛 𝑓𝑖 𝑥 ≤ 𝑓𝑖 𝑦 and ∃𝑗:1,..,𝑛 𝑓𝑗 𝑥 < 𝑓𝑗 𝑦 and ∃𝑘:1,..,𝑛;𝑘≠𝑗 𝑓𝑘 𝑥 = 𝑓𝑘 𝑦 ⇔
⇔ 𝑓(𝑥) ≼ 𝑓(𝑦) ⟺ 𝑥 ≼ 𝑦

Pareto dominance – example 1 (strong dominance)
10
https://commons.wikimedia.org/w/index.php?curid=143545224

Pareto dominance – example 2 (weak dominance)
11 Multi-objective Optimization Algorithms (MOOA/AOW)

Pareto dominance – example 3 (indifference)
12
https://commons.wikimedia.org/w/index.php?curid=143545224

Decision space vs. objective space
13
 k: number of decision variables
 n: number of objective functions
 Decision space
• k-dimensional
• comprises of potential solutions to the problem (values of decision variables)
 Objective space (aka solution space)
• n-dimensional
• space in which the objective function vectors are represented
Definitions

Decision space vs. objective space
14
http://dx.doi.org/10.1109/IPIN.2017.8115908 https://doi.org/10.1016/B978-0-323-91781-0.00002-8

Pareto optimal set vs. Pareto front
15
https://doi.org/10.1016/j.asoc.2019.105631

Constraints in MOO
16
 Constraints can be considered for
• decision space
• objective space
• combined or elements in-between the decision and objective spaces
 All constraints should be satisfied for a feasible solution
 Feasible region represents the set of all solutions (in decision space) that satisfy the
constraints
 Feasible Pareto Front is the part of PF (in objective space) that is constructed by the feasible
region
 Basic Pareto-dominance definition does not have constraint handling built in

Basic constraint handling techniques in MOO
17
 Penalty functions
• additional elements added to the objective function(s) to degrade their values in
cases when a constraint is violated
 Domination of constraints
• basic Pareto dominance can be extended to handle constraints by differentiating
the impact and level of violation for particular constraints
 Brute force
• infeasible solutions/PF elements are removed from the final set of solution/PF

Can we solve MOO problems in a deterministic way?
18
 What does it mean to solve MOO problem?
• to find the feasible Pareto optimal set of solutions (determine feasible Pareto Front)
 Can we solve MOO problem by a deterministic method?
• in general – no, we cannot 
• in some particular cases – yes, we can 
only if strong assumptions are imposed on the model e.g. as in
Multi-Objective Linear Programming (MOLP) multi-objective gradient method

Multi-Objective Linear Programming (MOLP)
19
 MOLP model
• objective (goal) functions, linearly dependent on decision variables
• constraints linearly dependent on decision variables
• MOLP model example
• Decision variables: x1, x2, x3
• Objective functions: f1: -x1 -2x2  min
f2:-x1 +2x3  min
f3: x1 -x3  min
• Constraints: x1 +x2  1
x2  2
x1 -x2 +x3  4

MOLP - deterministic solvers
20
 Benson’s algorithm
• finds the efficient extreme points in the outcome set
• free Benson’s solver (C, Matlab) for MOLP http://bensolve.org/
 Variations of SIMPLEX algorithm for MOLP
• pivot base solution until no further improvements are possible

MOLP – Benson’s algorithm (1)
21 Multi-objective Optimization Algorithms (MOOA/AOW)
 The main tasks of a typical iteration of the Benson’s algorithm are as follows:
• solve a single linear program that is parameterized by a vertex of the current outer
approximation and
• apply vertex enumeration to obtain the vertices of the next outer approximation.
 In certain situations, the execution time of the algorithm is dominated by the time for
solving the sequence of linear programs

MOLP – Benson’s algorithm (2)
22
https://doi.org/10.1007/978-3-319-95165-2_46

Original LP SIMPLEX algorithm
23
1. Find a base LP solution
2. Improve the base LP solution
• pivot the base in one dimension – select a new base!
• recalculate the solution and current value of goal function
3. Check if current solution is optimal
• yes -> THE END
• no -> go to 2.

MOLP – extended SIMPLEX algorithm
24
 Similar to the original LP SIMPLEX, but
• how to select an initial base?
• how to pivot the base?
• how to check if the solution is optimal?

MOLP – extended SIMPLEX algorithm (M. Ehrgot)
25
https://www.lamsade.dauphine.fr/~projet_cost/ALGORITHMIC_DECISION_THEORY/pdf/Ehrgott/HanLecture2_ME.pdf
!

Can we simplify MOO problems to solve it more easily?
26
 Solving MOO in deterministic fashion is not that easy
• we can try simplifying MOO in order to utilize some more straight-forward solvers
 Possible techniques of simplifying MOO
• Weighted Objectives Method
• Lexicographic or Hierarchical Optimization Method
• Epsilon-Constrained (aka Trade-Off) Method
 Each simplifying technique converts MOO into SOO

Simplifying MOO - Weighted Objectives Method (1)
27
 In Weighted Objectives Method (WOM), instead of n objective functions, we introduce
and optimize a new single objective function G(x) using
• values of the original n objective functions fi, i=1..n (x)
• weights assigned to the objectives wi, i=1..n
𝐺 𝑥 = ෍
𝑖=1
𝑛
𝑤𝑖𝑓𝑖(𝑥)
where
𝑤𝑖 ∈ 0.0; 1.0 , ෍
𝑖=1
𝑛
𝑤𝑖 = 1.0

Simplifying MOO - Weighted Objectives Method (2)
28
 In WOM only G(x) is optimized
• we can use any method available for single-objective optimization
• graphically: the result is in the intersection of the feasible solutions region with
the hyperplane depending on the weights w
• applying different weights -> different results
• WOM is fast and very simple in implementation 
• Yet the biggest challenge is to determine the weights w 

Simplifying MOO – Lexicographic Optimization Method
29
 Lexicographic Optimization Method (LOM) always returns a Pareto optimal solution
 Instead of specifying weights we re-order the objective functions
• f1() the most important one
• …
• fn() the least important one
 We proceed with finding optimum just for single objective function!
• we find optimum only for the f1() objective function
• for the next (i-th) objective we find optimum just for the fi() and add new constraints for
optimum values for all preceding objective functions
• Stop the procedure if for step the solution is just a single point
𝑓𝑗 𝑥 ≤ 𝑓𝑗 𝑥𝑗
∗
, 𝑤ℎ𝑒𝑟𝑒 𝑗 = 1, . . , 𝑖 − 1

Simplifying MOO – LOM, example (1)
30
 Objectives:
• f1=4x1+x2 -> min
• f2=2x1+x2 -> min
• f3=x1+x2 -> min
 Constraints:
• C1: 7−x1−5x2 ≤0
• C2: 10−4x1−x2 ≤0
• C3: −7x1+6x2−9 ≤0
• C4: −x1+6x2−24 ≤0

Simplifying MOO – LOM, example (2)
31
 Step 1
• f1=4x1+x2 -> min
• result: BC line segment, f1
*=10.02
 Step 2
• f2=2x1+x2 -> min
• new constraint C5: f1≤ 10.02
• result: point B (x1*=2.26; x2*=0.947)
• f1
*=10.02; f2
*=5.47; f3
*=3.21
 END

Simplifying MOO – Hierarchical Optimization Method
32
 HOM extends LOM in the way the newly introduced constraints are built
• LOM
• HOM
• 𝜙𝑗 is the variance value allowed for the j-th objective function (if 𝜙𝑗=0 then LOM)
𝑓𝑗 𝑥 ≤ 𝑓𝑗 𝑥𝑗
∗
, 𝑤ℎ𝑒𝑟𝑒 𝑗 = 1, . . , 𝑖 − 1
𝑓𝑗 𝑥 ≤ (𝟏 − 𝝓𝒋)𝑓𝑗 𝑥𝑗
∗
, 𝑤ℎ𝑒𝑟𝑒 𝑗 = 1, . . , 𝑖 − 1

Simplifying MOO – LOM & HOM: pros & cons
33
 Advantages 
• quite handy: we can use it instead of specifying weights
• does not require normalized objective functions
• always provides a Pareto optimal solution
 Disadvantages 
• several single-objective problems have to be solved to obtain just one solution
point
• additional constraints have to be imposed
• when we change the order of objective functions then a different result
(Pareto optimal solution) will be returned

Simplifying MOO – Epsilon-Constrained Method
34
 Again, as in LOM/HOM, we select just one (r-th) objective to be optimized
• all the other objectives (i=1, .., n, i≠r) are turned into (n-1) new constraints
• i (i=1, .., n, i≠r) defines threshold for the i-th objective
 Threshold values i that can be accepted for i-th objective functions are established
• limits the objective space
 Method simple in implementation and fast 
 But we lose possibility to really optimize most of the objectives 

Simplifying MOO - Epsilon-Constrained, example
35
Original MOO
 Objectives:
• f1=4x1+x2 -> min
• f2=2x1+x2 -> min
• f3=x1+x2 -> min
 Constraints:
• C1: 7−x1−5x2 ≤0
• C2: 10−4x1−x2 ≤0
• C3: −7x1+6x2−9 ≤0
• C4: −x1+6x2−24 ≤0
Trade-off MOO
 Objectives:
• f1=4x1+x2 -> min
 Constraints:
• C1: 7−x1−5x2 ≤0
• C2: 10−4x1−x2 ≤0
• C3: −7x1+6x2−9 ≤0
• C4: −x1+6x2−24 ≤0
• C5(new): f2≤  2
• C6(new): f3≤  3

How in general can we solve MOO?
36
 Deterministic MOO methods
• limited to specific types of models e.g. purely linear ones
 Simplification of MOO (MOO SOO)
• cannot provide a complete nondominated solution set
• requires some kind of preferences by DM
 Thus meta-heuristic approach can be a good solution for MOO
• good approximation of Pareto front 
• universal - able to handle any MOO 
• not as fast or as simple as some deterministic/simplified approach 
• by definition cannot guarantee to provide a solution at all 

Meta-heuristics for MOO
37
 Multi-Objective Meta-Heuristics (MOMH) are AI methods able to solve real-life
multi-objective problems i.e. by approximating Pareto Fronts
• Genetic algorithm (GA)/ Evolutionary algorithm (EA)
• Ant Colony Optimization (ACO)
• Artificial Bee Colony (ABC)
• Grey Wolf Optimization (GWO)
• Particle Swarm Optimization (PSO)
• Differential Evolution (DE)
• Memetic Algorithm (MA)
• Local Search (LS)
• Harmony Search (HS)
• Simulated Annealing (SA)

Meta-heuristics in AI world
38
Artificial Intelligence (AI)
Machine Learning (ML)
Deep Learning (DL)
Reinforcement
Learning (RL)
Meta-heuristics
GA & EA ACO
PSO SA & TS
Generative AI

AI timeline
39
1943
Model of binary
neuron
1951
First neural
network computer
(SNARC)
1958
Perceptron
& early MLP
1959
ADALINE
& MADALINE
1972
Early RNN
1979 - 80
Neocognitron
& early CNN
1986
„Deep Learning”
1992
Early
Transformer
2017
Transformer
2018 - 2024
GPT-1, …, GPT-4
1952
stochastic
optimization
1963
Random search
1966
Evolutionary
programming
1975
GA
1986
TS
1983
SA
1981
RL
1991
EA
1992
ACO
1995
PSO
1996-2024
MOMH
ANN perspective
M-H + RL perspective
Place of Multi-Objective Meta-Heuristics (MOMH) in AI

Thank you for your attention
joanna.szlapczynska@pg.edu.pl

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