Computation of Equispaced Pareto Fronts for Multiobjective Optimization

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Computation of Equispaced Pareto Fronts for
Multiobjective Optimization

• V. Pereyra
• Weidlinger Associates Inc. (Retired)
• Mountain View, CA
• Smart Fields
• Stanford University
• 2008

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Plan

• Multiobjective Optimization
• Pareto Equilibrium Optimality Conditions
• Pareto Set and Pareto Manifold
• Calculating the Pareto Front by Continuation
  Shooting Approach
• Numerical Examples
• Constrained Problems
• Variational Approach to Mesh Generation

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Multiobjective Optimization

• The problem
• minx?D F(x)
• where x ? R n ,
• F(x) ? f1 (x), ..., fk (x) ? R k and
• D ? R n is defined by a set of constraints.
• F
• Input Design
  Output Goal

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An Example Optimal Aircraft Design
Aircraft design optimization. J.J. Alonso,
P.LeGresley,V.Pereyra. To appear in Mathematics
and Computers in Simulation (2009).
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Pareto Equilibrium

• It is generally unlikely that a single point x
  will be a common minimizer for all objectives.
• In fact, these problems are characterized by the
  requirement of a subjective trade-off between
  conflicting objectives.
• The optimality concept here is known as Pareto
  equilibrium, which in words establishes that "x
  is a global Pareto equilibrium point if there is
  no other point that is dominated by x".

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Pareto Optimality

• A point x is dominated by a point y iff
• fi (y) fi (x), with strict inequality for
  at least one of the objectives.
• Thus, a global Pareto equilibrium point is such
  that no improvement for all objectives can be
  achieved by moving to any other feasible point.
• A local version of this concept is obtained if we
  limit the movement to an open neighborhood around
  the optimal point.
• The set of all Pareto points is called the Pareto
  manifold. Usually, for k objectives this manifold
  has dimension k-1. Thus, for two objectives it is
  a curve (segment or segments).

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Optimality Condition

• For differentiable convex objectives we can use
  the usual concepts of single objective
  optimization to arrive to an analytical and
  geometrical characterization of the local Pareto
  points.
• This is easily seen first for the unconstrained
  case of two objectives and two independent
  variables. Let x1 , x2 be local minima of f1
  and f2 respectively.
• The first observation is that these are Pareto
  points since moving away from them would increase
  at least one of the functionals.

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KKT Condition

• A segment of the Pareto manifold is a curve that
  joins these two points.
• It is defined as the parametric set of solutions
  x(?) of the Karush-Kuhn-Tucker optimality
  condition
• G(x(?) ?) (1- ?) ?f1 (x) ??f2 (x) 0, 0?
  ? ?1.
• The image of the Pareto manifold by the goal
  functionals is called the Pareto front.

f2
Pareto front (F space)
Pareto manifold (x space)
f1
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Geometrical Interpretation

• Geometrically the KKT condition says that a point
  is Pareto optimal if the contours of the two
  objectives are tangent at it, with gradients
  pointing in opposite directions i.e., the two
  functionals have no descent directions in common
  and the weights act as scalings.

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Constrained Problems

• The concept is the same the Pareto equilibrium
  points are those for which there are no feasible
  descent directions in common to all the
  objectives.
• In other words, there are no possible directions
  of movement that will improve all the objectives.
• Now the Lagrangian includes the active
  constraints.

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Numerical Calculation of Pareto Fronts

• Pareto fronts can be calculated in many ways. A
  popular one uses genetic or evolution algorithms.
  Starting with an initial population they evolve
  it through many generations to approximate the
  Pareto manifold.
• An important requirement is that the front be
  sampled as uniformly as possible the algorithm
  of Deb et al attempts to do just that.
• This is what it was used in the aircraft
  optimization problem. The potentially killing
  computing time was palliated by using surrogates
  for the most expensive parts of the calculations.
• K. Deb, A. Pratap, S. Agarwal, and T.
  Meyarivan, "A Fast and Elitist Multi-Objective
  Genetic Algorithm NSGA-II". KanGAL Report
  200001, Indian Institute of Technology, Kanpur
  (2000)
• ?V. Pereyra, Fast Computation of Equispaced
  Pareto Manifolds and Pareto Fronts for
  Unconstrained Multi-Objective Optimization
  Problems. To appear in Math. And Comp. in
  Simulation (2009).

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Newton Continuation

• A problem with genetic algorithms is that they
  usually require many function evaluations. If
  these function evaluations are expensive then the
  algorithms are not practical.
• We offer here an alternative in the case that
  derivatives of the goal functional are available.
  We consider first the unconstrained case.
• For a fixed ? we solve the KKT nonlinear
  equations by Newton's method and use continuation
  on the parameter ? (starting from x1, ? 0) to
  generate the Pareto front.
• Usually, such a procedure will not produce an
  uniform sampling of the Pareto front.

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Equispaced Continuation

• In order to obtain a well sampled front we will
  add an extra set of constraints that explicitly
  requests just that
• Li F(xi) - F(xi-1) 22 c . ()
• The constant c corresponds to the desired spacing
  on the discrete front. Ideally c Total Arc
  Length/points, but since the Total Arc Length
  (TAL) of the front is not known a priori it has
  to be estimated.
• Thus this method starts from x1 (?0), and
  solves for both xi and
• ?i for each i from the KKT conditions and
  () by using Newton's method.
• Similar work can be found in a recent publication
  by Gabriele Eichfelder Adaptive Scalarization
  Methods in Multiobjective Optimization. Springer
  (2008). This work includes an interesting
  application to radiology (planning of tumor
  irradiation).

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A global method

• Methods such as the evolutionary algorithms that
  attempt to generate the whole discrete Pareto
  front at once are called global, as opposite to
  the continuation type methods.
• One important difference is that continuation
  methods are inherently sequential, while the
  global methods can be parallelized.
• In our unconstrained paper we proposed a global
  method that essentially solves simultaneously all
  the optimization problems, which are coupled
  through the equidistance constraints. In the
  bi-objective case this method is akin to a
  two-point boundary value solver (bending), while
  the continuation methods would correspond to a
  shooting or marching type method.
• S. Leyffer has proposed also a global type method
  in
• A note on multiobjective optimization and
  complementarity constraints. ANL/MCS-P1290-0905,
  Argonne Nat. Lab. (2005).

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Examples

• We consider some test cases from the literature
• Problem SCH (trivial) (from Deb et al paper)
• f1 (x) x2 , f2 (x) (x - 2)2 .
• x1 0, x2 2,
• Pareto manifold 0,2.

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Problem SCH Pareto Front

• 30 points, less than 10 Newton iterations per
  point, i.e. lt 300 function evaluations.

Global Continuation
Distance Between Points
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Results For Genetic Algorithms

• NSGA-II used a population of 100 points and
  evolved them for 250 generations, for a total of
  25,000 function evaluations!

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Constrained Problems

• The method for constrained problems is the same.
• Instead of developing our own nonlinear
  programming code we use Gill, Murray and Saunders
  SNOPT.
• The additional equispacing constraint is added to
  the given problem constraints.
• SNOPT works with or without explicitly defined
  derivatives.

Fast Algorithms for Convex Constrained
Multiobjective Optimization V. Pereyra, J.
Castillo and M. A. Saunders. Submitted to Comp.
Opt. Appl. (2008).
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Numerical Examples

• First we consider a problem from Deb et al
• f1 (x1 -2)2 (x2 -1)2 2
• f2 9x1 -(x2 -1)2
• g1 x12 x22 lt225
• g2 x1 - 3x2 lt -10
• x1, x2 ? -20, 20.
• We use 100 points in our continuation.
• Deb et al require 50,000 function evaluations in
  their generic algorithm.
• Our calculation took 20 msec for a total of 565
  evaluations.

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Results for problem DEB
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Results for problem DEB (scaled to aspect ratio
1)
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Results for genetic algorithm (Deb et al)

Old algorithm (Chye)
New algorithm (Deb)
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Variational Grid Generation

• There are variational methods proposed by Thomson
  long ago to produce automatically good
  computational meshes for complicated regions for
  the solution of PDEs.
• Unfortunately when the optimizing criterion is
  based only on the length or area of the cells,
  there are regions for which the results are not
  so good (folding, cramping).
• Jose Castillo discovered a few years ago that
  combining these criteria gave better results.
• Since what results is a bi-objective problem we
  applied the method described here to obtain a
  good sampling of the Pareto front.

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Variational Grid Generation (cont.)

• We chose a 2D region from a problem on electric
  field calculations for lasers discussed by ?P. J.
  Roache, W. M. Moeny and S. Steinberg in 1984,
  which is known to be problematic.
• We start with a logically rectangular mesh. The
  variables of the problem are the Cartesian
  coordinates of the interior mesh nodes and the
  two criteria attempt to minimize the sum of all
  the lengths and the sum of all the areas of the
  mesh elements. It has been proven that this leads
  to meshes with almost constant lengths (in each
  coordinate direction) or almost constant areas.
• We apply our method to this problem for a 17x17
  mesh, of which only the interior points are
  allowed to move. Thus the problem has 450
  variables. We use bound constraints to avoid the
  mesh
• straying away from the region and of course
  our equispacing constraint.

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Pareto Front for Problem GenMesh
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Results for Problem GenMesh
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Conclusions

• A fast method to calculate equispaced Pareto
  fronts for multiobjective problems has been
  described and its performance has been shown in
  some simple examples from the literature and in
  some more challenging practical applications.
• The novel idea of adding an equispacing
  constraint extends to more than 2 objectives and
  problems with constraints.
• Newton's method can be replaced by your favorite
  nonlinear programming code. Derivatives are not
  essential.

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Data Fusion

• In 1989 we considered the problem of what it was
  called at the time Cooperative Inversion for
  Petroleum Exploration.
• The idea was that if one has multiple sources of
  data (seismic, electromagnetic, potential,
  well-logs, etc.) for a given play, it would be
  desirable to use that data in a cooperative
  fashion, in order to recover better material
  properties and in general, to reduce the
  ambiguity of the inversion process.
• We proposed to use techniques of multiobjective
  optimization and developed a system for combining
  seismic and magneto-telluric data. This was too
  ambitious an endeavor for the time, but we
  demonstrated its feasibility
• We are discussing now with David Echeverria the
  possibility of doing something similar for
  reservoir characterization, taking advantage of
  these more modern and better understood tools and
  the much improved computing capabilities.