Mlesa Objective Function Behavior, E0 Mles Hessian Condition Number 505 direction of first The condition number of the Hessian is given by direction of second When K(H)=1, the objective function contours are circulal AS K(H)increases, the contours become elongated I K(H)>>1, the change in the objective function due to a contours of constantΦ small change in x will vary radically depending on the direction of perturbation If n2=M1, the contours are circular As 72/M, gets very small, the ellipsoids get more and more stretched k(H) can be computed via a Cholesky factorization If any eigenvalue is very close to zero, p will change very little when (H=LDL) moving along that eigenvector e Massachusetts Institute of Technology -Prof de Weck and Prof Willcox G Massachusetts Insttute of Technology .Prof de Weck and Prof Willcox Engineering Systems Division and Dept of Aeronautics and Astronautics Engineering Systems Division and Dept of Aeronautics and Astronautics MIest Scal In theory we should be able to choose any scaling of min X,+3x2+ X3 mn10x2+30x2+10x3+28 the design variables, constraints and objective st.5x,+2x3≥1 objective s t.5x2+2x3≥1 functions without affecting the solution 6x2-3x2≥2 ≥2 In practice, the scaling can have a large effect on the solution X2≥0 > numerical accuracy, numerical conditioning From Papalambros, p. 352: "scaling is the single constraint most important, but simplest, reason that can make min Xf+ 3x2+X3 min x+3x +10X the difference between success and failure of i design optimization algorithm st50x1+20x210 st5X+2x3≥1 6x2-3x≥2 6Xx2-30×3≥2 0 X2≥0 e Massachusetts Institute of Technology- Prof de Weck and Prof Willcox e Massachusetts Insttute of Technology. Prof de Weck and Prof Willcox Engineening Systems Division and Dept of Aeronautics and Astronautics Engineering Systems DiMsion and Dept of Aeronautics and Astronautics21 © Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Engineering Systems Division and Dept. of Aeronautics and Astronautics Objective Function Behavior Objective Function Behavior v1 v2 contours of constant Φ direction of first eigenvector • If λ2=λ1, the contours are circular • As λ2/λ1 gets very small, the ellipsoids get more and more stretched • If any eigenvalue is very close to zero, Φ will change very little when moving along that eigenvector direction of second eigenvector Φ(x) x 1 x 2 22 © Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Engineering Systems Division and Dept. of Aeronautics and Astronautics Hessian Condition Number Hessian Condition Number • The condition number of the Hessian is given by • When κ(H)=1, the objective function contours are circular • As κ(H) increases, the contours become elongated • If κ(H)>>1, the change in the objective function due to a small change in x will vary radically depending on the direction of perturbation • κ(H) can be computed via a Cholesky factorization (H=LDLT) n λ κ λ 1 ( )= Η 23 © Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Engineering Systems Division and Dept. of Aeronautics and Astronautics Scaling Scaling • In theory we should be able to choose any scaling of the design variables, constraints and objective functions without affecting the solution • In practice, the scaling can have a large effect on the solution ⇒ numerical accuracy, numerical conditioning • From Papalambros, p. 352: “scaling is the single most important, but simplest, reason that can make the difference between success and failure of a design optimization algorithm” 24 © Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Engineering Systems Division and Dept. of Aeronautics and Astronautics Scaling Scaling 1 2 1 23 3 1 2 2 3 min 3 s.t. 5 2 1 632 i 0 x xx x x x x x + + + ≥ − ≥ ≥ 1 2 1 23 3 1 2 2 3 min 10 30 10 28 s.t. 5 2 1 632 i 0 xxx x x x x x +++ + ≥ − ≥ ≥ 1 2 1 23 3 1 2 2 3 min 3 s.t. 50 20 10 632 i 0 x xx x x x x x + + + ≥ − ≥ ≥ 1 2 12 3 3 1 2 2 3 min 3 10 s.t. 5 2 1 6 30 2 i 0 xx x x x x x x + + + ≥ − ≥ ≥   ≡ ≡ scale objective scale design variable scale constraint