Methods and Applications of Multiobjective Optimization. Introduction. Terminology (1) Terminology(2) Multiobjective or Multicriteria optimization

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1 Methods and Applications of Multiobjective Optimization A review of The Normal Boundary Intersection Approach of Indraneel Das and John E. Dennis and some variations or extensions by Trond Steihaug Department of Informatics University of Bergen, Norway and Humboldt Universität zu Berlin Introduction Multiobjective or Multicriteria optimization f (x) min F (x). x V. f m (x) where m and V = {x R n c i (x) =i E,c i (x) i I}. The constraints should not be more difficult than the available algorithm can handle the problem of solving the (single objective) problem min f i(x), i =,...,m. x V May 4, 5 Slide Slide Terminology () A point x V is said to be locally Pareto optimal if and only if f i (x) f i (x ) for all i m and x V N(x ) x = x. A point x Cis said to be globally Pareto optimal if and only if f i (x) f i (x ) for all i m and x V x = x. Terminology() The shadow minimum or utopia point F is defined as the vector of the individual global (single objective) fi f i(x i ), F =(f,...,fm) T where x i = argmin{f i (x) x V} f(x) Feasible set x F = [f, f] F(x*) O Objective Space f(x) F(x*) Typically there is an entire curve or surface of Pareto points Slide 3 Slide 4

2 Terminology (3) The set of obtainable vectors F = {F (x) x V} R m. F is the objective space. F is the boundary of F. The set of all Pareto optimal points P F. Basic Idea of the method The intersection point between the boundary F and the normal pointing toward F emanating from any point in the CHIM is a Pareto optimal point f (x) B O C A f (x) AisF (x ), B is F (x ), C is a Pareto (global) optimal point, O is F (from now assumed to be ). The points on the line A to B is the convex hull of individual minima (CHIM) also called the Utopia line. unless it happens.... Slide 5 Slide 6... unless it happens unless it happens... that the algorithm returns a local solution of min{f i (x),x V}and not the global. N P... to lay on a sufficiently concave part of the boundary. Claim by D&D: The Pareto optimal surface (in the objective space) is convex in almost every application found in the literature....but why use when it is convex and a convex combination of objectives will work... The computed convex hull of the individual local minima is the not the CHIM. Slide 7 Time to do some mathematics Slide 8

3 Formulation of the Problem LetΦbeanm m matrix where column i is F (x i ) F. Note that Φ ij and Φ ii =. CHIM is now {y =Φβ R m m i= β i =,β i }. Let ˆn be the unit normal to the CHIM simplex pointing toward the origin. For a given point y in the CHIM the (half) line is y + tˆn, (t ). The point on this line and F closest to the origin is the follow subproblem max x,t t s.t. y + tˆn = F (x) x V Slide 9 points A and G are called anchor points. Arcs AB and FG: Global Pareto points. Arcs BC and DE: Local Pareto points Arcs CD and EF are neither R is a local Pareto point and a point. For every Pareto optimal point there exists a point unless.... Slide that m 3... unless it happens... f3 F(x*) Choosing the y in max{t y + tˆn = F (x),x V} F(x*) f F(x3*) f Slide Slide

4 The Optimizer Most (all) optimization routines for general nonlinear problems will in general find only local solutions. The type of algorithm will determine what kind of problems that can be solved. Matlab fmincon will handle linear and box constraints separately: Local v.s. global solution of max{t y + tˆn = F (x),x V} min x f(x) s.t. c i (x) =,i E c i (x),i I S R Q Ax = b, Âx ˆb l x u. O Slide 3 Slide 4 Local v.s. global solution of min{f i (x),x V} f Example (N.Kroll) Only box constraints m =,n =,l = π x i π = u F (x,x )= +(φ (, ) φ (x,x )) +(φ (, ) φ (x,x )) (x +3) +(x +) P f The computed convex hull of the individual local minima is the not the CHIM. Here φ (x,x )=/ sin(x ) cos(x )+sin(x ) 3/ cos(x ) and φ (x,x )= 3/ sin(x ) cos(x ) + sin(x ) / cos(x ). Solved with using a very fine mesh of the CHIM. First plot shows the computed points (some are not Pareto point) Second plot shows the corresponding values in the feasible set (design space) Slide 5 Slide 6

5 f Pareto points in F space example (Kroll) f x Domain space points x example m =,n = 5, linear and nonlinear constraints. F (x) = x 3x +x x3 3 + (x 4 x 5 ) The constraints are x +x x 3 x4 + x 5 = 4x x x x 4 + x 5 = x Slide 7 Slide 8 example () f Pareto Front in F space Scaling - Filters - A Reformulation Several papers from the Multidisciplinary Design and Optimization Laboratory raise concern on scaling and generating only Pareto solutions for the. Rescale F so that f i (x i ) = (the anchor points). Pareto filter Reformulation of the sub problem. Coarse mesh on CHIM f A Pareto filter is a database of the smallest partially ordered computed y in the objective space y F= {F (x) x V}. Slide 9 Slide

6 Normalized Normal Constraint Method Let y i = F (x i ) and pick reference corner on the CHIM, say y r. Let y be any (interior)point on the CHIM. min x,t t s.t. (F (x) y) T (y r y k ), k r t = f r (x) x V Scaling In principle scaling of the anchor points should not effect the computation of the points max{t y + tˆn = F (x),x V}except for the distribution of the points since Dy + tˆn = DF(x) where D is the scale. If... then and NC(Messac 3) reformulation will generate the same point Proof: By handwaving. Slide Slide Warmstarts The number of subproblems to be solved for the and NC depends on the gridding or mesh of the CHIM. It is important to utilize that the subproblems will be near each other (in domain). Normalize and consider y = Φβ, β,...β m Issues not covered: The Road Ahead No need for the exact normal and D&D use ˆn = Φe, where e is a column vector of all ones. The relationship between and NC and minimization a convex combination of the objectives. Possible improvements: More advanced warm-starts possibilities for the optimizer (more than just the startingpoint) Use derivative information (Jacobian of F ) to eliminate non-pareto point. Slide Other scalar minimization sub-problems. Slide 4

7 $ ' The Road Ahead & Slide 5 % Slide 6