Pareto Optimality

1. Pareto Optimality

2. “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 (Textbook’s notation): Minimize f = [ f1(x), f2(x), … , fm(x) ], where each fi is an objective function Subject to xΩ (constraints on space of design variables) Review: Background Note: We will use the terms “objective”, “goal”, and “criterion” interchangeably.

3. 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Ω; where wi > 0 and Σ 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Ω These all provide point solutions (x*) based on an assignment of preferences among objectives. Review: Methods for Trading Off Across Objectives Yes, we’ve seen this before.

4. 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”). The Need Globally Viewing Tradeoffs in Optimality x2 f2 criterion space f1 f2 Ω' Ω x1 f1 design space See page 374-375

5. 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 curve’s 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. f2 f2 f1 f1 f2 f2 f1 f1 The Pareto Optimality Curve Pareto Maximization Problem

6. Strong Pareto Optimality A system variable vector x* Ω is Pareto optimal iff there is no vector x Ω with the characteristics: fi(x)  fi(x*) for all i and fi(x) <fi(x*) for at least one i (one objective) If only the 2nd condition above holds, x* is weaklyPareto 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 Ω' 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. Formal Definitions Dominating Points Minimization Problem

7. 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. Generating the Pareto Frontier

8. 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 designer’s 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[ max{z1, z2 }] Schemes for Picking a “Best” Solution Along the Frontier

9. Find the point Q in the space Ω' that minimizes the distance from the demand or ideal point to the Pareto front. Minmax Concept Graphed Minimization Problem

10. 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. Emerging Research Minimization Problem