ZEIT4700 – S1, 2016
Mathematical Modeling and Optimization
School of Engineering and Information Technology
Optimization - basics
Maximization or minimization of given objective function(s), possibly
subject to constraints, in a given search space
Minimize f1(x), . . . , fk(x)
Subject to
gj(x) < 0, i = 1, . . . ,m
hj(x) = 0, j = 1, . . . , p
Xmin1 ≤ x1 ≤ Xmax1
Xmin2 ≤ x2 ≤ Xmax2
.
.
(objectives)
(inequality constraints)
(equality constraints)
(variable / search space)
Evolutionary Algorithms (EA)
Initialization
(population of
solutions)
Parent selection
Recombination /
Crossover
No
Output best
solution
obtained
Yes
Termination
criterion met ?
Mutation
Evaluate childpop
Reduction
Ranking (parent+child
pop)
“Evolve”
childpop
Constraint handling
x2
x2
x1
x1
Optimum
Feasible
Infeasible
-
Search space is reduced
Disconnected/constricted feasible regions possible
Feasibility of solutions to be considered in ranking
Constraint handling - Penalty function method
Minimize 𝑓1 (x)
(Constrained)
Subject to
𝑔𝑗
𝑥
≤ 0, 𝑗 = 1, 2 … 𝑚
Minimize 𝑓1
𝑥
+ Σ 𝜆𝑗 (max(0, 𝑔𝑗 𝑥 )2
(Unconstrained)
𝜆1 , 𝜆2 , … 𝜆𝑚 are penalty parameters
-
Performance is sensitive to choice of parameters
No fixed way to generate penalty parameters
Scaling between different terms
Constraint handling – feasibility first techniques
During the ranking, enforce the following relations:
1.
Between two feasible solutions, the one with superior objective value is better.
2.
Between a feasible and an infeasible solution, feasible is better
3.
Between two infeasible solutions, the one with lower constraint violation is better.
=> All feasible solutions are ranked above infeasible solutions
Optimization – Multi-objective
For a problem to be multi-objective, the objectives must be conflicting, i.e, maximization
of one of them must lead to minimization of other.
The true optimum for this case is called Pareto Optimal Front (POF)
A heuristic algorithm delivers a non-dominated set which preferably as close as
possible to the POF.
F2 (Minimize)
•
•
The final set of non-dominated solutions should be:
1. Converged (close to the Pareto optimal front)
2. Diverse (should span entire range of solutions,
preferably uniformly)
F1 (minimize)
Multiobjective Optimization – Scalarization approach
𝑀𝑖𝑛𝑖𝑚𝑖𝑧𝑒 𝑓1 , 𝑓2
𝑀𝑖𝑛𝑖𝑚𝑖𝑧𝑒 𝑤1 𝑓1 + 𝑤2 𝑓2
Where 𝑤1 , 𝑤2 𝜖 0,1 , 𝑤1 + 𝑤2 = 1
f2
f1
-
One solution per optimization search
Can only achieve convex fronts
Multiobjective Optimization – 𝜀 – constraint method
𝑀𝑖𝑛𝑖𝑚𝑖𝑧𝑒 𝑓1 , 𝑓2
𝑀𝑖𝑛𝑖𝑚𝑖𝑧𝑒 𝑓1
𝑆𝑢𝑏𝑗𝑒𝑐𝑡 𝑡𝑜 𝑓2 ≤ 𝑐
(for different values of c )
f2
f1
-
One solution per optimization search
Difficult to estimate c values
Multiobjective Optimization – Non-dominated sorting
Minimize f1, f2
Convergence
(nd-sort)
f2
f2
f1
f1
Diversity
(crowdingdistance sort)
d2
f2
d1
f1
Evolutionary Algorithm (cntd)
Minimize f(x) = (x-6)^2
0 ≤ x ≤ 31
Binary GA
Real Parameter GA
Representation
Binary
Real
Parent selection
Binary tournament/
Roulette wheel
Binary tournament/
Roulette wheel
Crossover
One point/multi-point
SBX,PCX …
Mutation
Binary flip
Polynomial
Ranking
Sort / ND
Sort / ND / CD
Resources
Course material and suggested reading can be
accessed at
http://www.mdolab.net/Hemant/design-2.html
Rank 1
Rank 3
f2
Rank 2
f1