June 8 Lecture Outline EnvE 320

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June 8 Lecture Outline EnvE 320

• Midterm Exam
• Begin now and you have until 920
• Rch 306, 230-400
• Solutions to my 3 practice problems on the web at
  945 am
• I will be available from 1130 1200 today
• Multi-objective optimization introduction
• this lecture is mainly about general
  multi-objective optimization and is not specific
  to linear or non-linear optimization formulations

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Multi-Objective (MO) Optimization

• we have so far dealt with single-objective
  optimization problems
• Max (or Min) Z Z(x1, x2,....,xn)
• g1(x1, x2,....,xn) , ?, ? b1
• g2(x1, x2,....,xn) , ?, ? b2
•   gm(x1, x2,....,xn) , ?, ? bm
• In an MO formulation, we have more than 2
  objectives, e.g.
• Max Z1 Z1(x1, x2,....,xn)
• Max Z2 Z2(x1, x2,....,xn)
• Min Z3 Z3(x1, x2,....,xn)
• Subject to same constraints as above

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Multi-Objective (MO) Optimization

• Examples
• Min Cost and Max environmental quality
• Max car speed and Max safety
• Min Cost and max Reliability of meeting WQ
  standard
• Min Cost, Min Cleanup time, Max Reliability
• Units of the objectives are typically not the
  same
• eg. dollars, probability, units of time
• Interesting MO problems requiring special
  solution techniques have conflicting objectives
  (non-commensurate)
• optimizing one objective and ignoring all other
  objectives does not yield the optimal values of
  the ignored objectives
• thus, more than just single objective opt.
  techniques needed

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Objective Space

• We know what graphs of decision space look like
• With 2 or more objectives, we can also graph
  objective space
• Consider the 2-objective non-linear example
• DECISION SPACE OBJECTIVE SPACE
• Each MO problem feasible region can be mapped in
  decision or objective space
• each objective is a function of one or more
  decision variables and the feasible region in
  objective space can be displayed for problems
  with any number of decision variables

Feasible region
min Z2
x2
Feasible region
min Z1
x1
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Objective Space

• Consider the 2-objective non-linear example in
  objective space
• Assume Z1 Z2 are to be minimized (cost
  failure probability)
• NOTE This picture is conceptual for a given
  problem, we do not know what the feasible region
  in objective space looks like it is often
  difficult to determine short of complete
  enumeration

Like the feasible region in decision space, the
feasible region in objective space contains all
solutions that simultaneously satisfy all
constraints in the problem
Feasible region
Z2
Z2
Z1
Z1
Ideal point defined by the best possible
objective function values if each one was
optimized independent of the other, it is not
feasible or achievable Z1 and Z1 have
different solutions or different optimal decision
variable values
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Definition of Inferiority or Dominance

• This concept is used to determine what feasible
  solutions are clearly worse than others and
  therefore not good solutions
• Demonstrate with same example min Z1 cost min
  Z2 failure probability
• solutions with characteristics like D, are called
  tradeoff, Pareto optimal, Non-inferior or
  Non-dominated solutions

1. Compare Solutions A B which one would you
choose?

• Choose B smaller cost smaller failure prob.
  B is said to dominate A.

C
A
Z2
2. Compare Solutions B C which one would you
choose?
B

• depends on preference - cost or failure prob.
  more important? Neither solution dominates the
  other no clear winner.

D
3. Consider Solution D any solutions that have
clearly better Z1 Z2 values?
Z2
Z1
Z1

• no! D dominates C dominates B, nothing
  dominates D.

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Definition of Inferiority or Dominance

• Same example min Z1 cost min Z2 failure
  probability
• The tradeoff curve (surface) can be continuous or
  discrete depends on specific problem
• Pareto Optimal Solution one in which no
  objective can be improved without degrading at
  least one other objective.

E
C
This line is the set of all non-dominated
solutions (tradeoff curve, Pareto Optimal
solutions). Convince yourself that Solutions E
F are inferior do not belong on tradeoff curve
A
Z2
B
D
F
Z2
Z1
Z1
Numerical Example Sol Z1 Z2 Z3
Sol 1 dominated by Sol 2
(assume max all 3)
1 5 8 4 Sol 3
dominated by Sol 2
2 10 9 5 Sol 2
is non-dominated
3 6
6 5 Ignoring Sol 2, Sol
1 Sol 3 are non- dominated
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General Goal of Multi-Objective Optimization

• For this course we are interested in determining
  the tradeoff curve
• Feasible region in objective space little or no
  interest
• Ideal (often unrealistic) goal
• identify all non-dominated solutions
• Typical (more realistic) goal
• approximate the set of non-dominated solutions
  (min-min e.g. below)
• more time optimizing, better the approximation
  becomes

approximation of tradeoff curve using only three
non-dominated solutions
Z2
true but unknown tradeoff curve
Z1
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Schematic of Multi-objective Optimization
Procedure
MO optimization problem Min Z1, Max Z2, St
constraints
Ideal multi-objective optimizer
Choose a single, non-inferior solution
Information from decision maker
Multiple tradeoff solutions found
Z2
Z1
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Classifying Tradeoff Curves

• A tradeoff curve is convex if only if a
  straight line drawn from any tradeoff solution to
  any other lies completely above the tradeoff
  curve
• A tradeoff curve is concave if only if a
  straight line drawn from any tradeoff solution to
  any other lies completely below the tradeoff
  curve
• Examples (in bi-objective space)

Convex
min
min
Convex
min
max
max
max
Concave
Concave
min
max