Computation of Equispaced Pareto Fronts for Multiobjective Optimization

1. Computation of Equispaced Pareto Fronts for Multiobjective Optimization V. Pereyra Weidlinger Associates Inc. (Retired) Mountain View, CA Smart Fields Stanford University 2008

2. 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

3. 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

4. An Example: Optimal Aircraft Design

5. 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".

6. 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).

7. 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 x*1 , x*2 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.

8. 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.

9. 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.

10. 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.

11. 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).

12. 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 x*1, ? =0) to generate the Pareto front. Usually, such a procedure will not produce an uniform sampling of the Pareto front.

13. 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 x*1 (?=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).

14. 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).

15. 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 . x*1 = 0, x*2 = 2, Pareto manifold [0,2].

16. Problem SCH Pareto Front 30 points, less than 10 Newton iterations per point, i.e. < 300 function evaluations.

17. 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!

18. 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.

19. 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 <225 g2 = x1 - 3x2 < -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.

20. Results for problem DEB

21. Results for problem DEB (scaled to aspect ratio = 1)

22. Results for genetic algorithm (Deb et al)

23. 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 PDE�s. 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.

24. 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.

25. Pareto Front for Problem GenMesh

26. Results for Problem GenMesh

27. 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.

28. 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.