Pareto Optimality

1
Pareto Optimality
2
Review Background

• The typical role of a design engineer is to
  resolve conflicting objectives and arrive at a
  design that represents an acceptable or desired
  balance of all objectives. (Mattson Messac
  2002)
• Classical examples of conflicting objectives
• Truss Design Weight versus Strength
• Flywheel design Kinetic Energy stored versus
  Weight
• Finite Element Meshes Aspect Ratio versus
  Distortion Parameter
• Standard problem definition (Textbooks
  notation)
• Minimize f f1(x), f2(x), , fm(x) ,
• where each fi is an objective function
• Subject to x ?O (constraints on space of design
  variables)

Note We will use the terms objective, goal,
and criterion interchangeably.
3
Review Methods for Trading Off Across Objectives

• So far, we have employed different techniques to
  achieve multi-objective optimization
• Weighting of objectives (Archimedean)
• minimize f w1f1(x) w2f2(x) subject to x
  ?O where wi gt 0 and S wi 1.
• Lexicographic minimum preemptive ranking of
  objectives
• A slight twist Picking one objective as primary,
  transforming remaining objectives into
  constraints (p. 373)
• minimize f1(x)
• subject to f2(x) ? c2, f3(x) ? c3, and
  fm(x) ? cm where ci is a limit
• x ?O
• These all provide point solutions (x) based on
  an assignment of preferences among objectives.

Yes, weve seen this before.
4
The Need Globally Viewing Tradeoffs in Optimality

• Thus far in class, preferences, weights, limits
  were all chosen by engineering judgment trial
  and error, experience, etc.
• Varying weights preferences to explore goal
  tradeoffs is manually intensive.
• How can we visualize a global picture of the
  tradeoffs in optimum solutions over a wide range
  of weights?
• Answer Transform graphical solutions from
  design (variable) space to criterion space
  (also called objective space).

x2
f2
criterion space
f1
f2
O'
O
x1
f1
design space
See page 374-375
5
The Pareto Optimality Curve

• In criterion space, we can identify a special
  trade-off curve on the boundary where
• No point is better than any other point on the
  line with respect to both objectives.
• No improvements can be made in any objective
  without trading off (worsening) the other.
• Changing the weights in an Archimedean
  (weighted)
  objective function traces out
  the curves
  path.
• This part of the boundary is called the
  Pareto Curve (or
  Pareto Frontier)
• Or, the functionally efficient solution set
• There are Pareto curves in both the design
  variable space
  and the criterion space.
• Pareto curves contain Pareto points (solutions)
• Bold lines in the pictures (right) represent
  Pareto
  curves when maximizing objectives.

Pareto
Maximization Problem
6
Formal Definitions

• Strong Pareto Optimality
• A system variable vector x ? O is Pareto optimal
  iff there is no vector x ? O with the
  characteristics
• fi(x) ? fi(x) for all i
• and
• fi(x) lt fi(x) for at least one i (one objective)
• If only the 2nd condition above holds, x is
  weakly Pareto optimal
• The Pareto curve is the set of x where there are
  no other solutions for which simultaneous
  improvement in all objectives can occur.
• Dominance
• A vector x in O' is said to be dominated if
  
  other vectors of system variables can be found
  that have
  improved values for any fi without
  creating a lower
  value in any other objectives in f.
• Thus, the Pareto optimal set curve represents
  the set
  of all non-dominated points.

Dominating Points
Minimization Problem
7
Generating the Pareto Frontier

• Several different approaches
• Alter objective function weighting, plot results
  in criterion space.
• Will not generate a complete Pareto Optimal set
  for nonconvex problems.
• Genetic algorithms
• Non-dominated (Pareto) points are identified and
  mated to find new ones.
• Adjustments are made to fitness values to avoid
  clustering
• New approaches are the focus of recent research
• Normal Boundary Intersection Method
• Physical Programming
• Normal Constraint Method
• In instances where non-Pareto or locally Pareto
  solutions are accidentally generated, a Pareto
  Filter algorithm can eliminate dominated
  solutions.

8
Schemes for Picking a Best Solution Along the
Frontier

• The best solution, of course, depends on your
  preference.
• There are not any really rational ways to
  automate picks.
• The min-max (or ideal point) method uses the
  distance between an efficient design and a
  pre-defined ideal design as the representation of
  the designers overall preferences.
• First an ideal target point can be selected in
  the objective space, outside of the feasible
  portion.
• Min-max attempts to find a point on the Pareto
  front where the maximum deviation from the ideal
  point is minimized.
• Deviation is defined as zi fi(x) - fimin
  (x)
• Solve the min-max optimization problem min
  maxz1, z2

9
Minmax Concept Graphed

• Find the point Q in the space O' that minimizes
  the distance from the demand or ideal point to
  the Pareto front.

Minimization Problem
10
Emerging Research

• Traditionally, application of Pareto Optimality
  principles have been applied in the detailed
  design phase of engineering design.
• However, Mattson and Messac (2002) are using
  Pareto fronts to aid concept selection.
• Pareto curves are generated for
  concept alternatives,
  which exist
  within feasible regions.
• Depending on your aspiration
  levels for your
  objectives,
  different design concepts may
  
  be selected or eliminated.
• If one design has more uncertainty,
  its fronts may be
  shifted
  accordingly.

Minimization Problem