MITESD_77S10_lec16

# MITESD_77S10_lec16 - Multidisciplinary System Design...

This preview shows pages 1–8. Sign up to view the full content.

1 © Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Engineering Systems Division and Dept. of Aeronautics and Astronautics Post-Optimality Analysis Lecture 16 Olivier de Weck Karen Willcox Multidisciplinary System Design Optimization (MSDO)

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
2 © Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Engineering Systems Division and Dept. of Aeronautics and Astronautics Today’s Topics • Optimality Conditions & Termination – Gradient-based techniques – Heuristic techniques • Objective Function Behavior • Scaling
3 © Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Engineering Systems Division and Dept. of Aeronautics and Astronautics Standard Problem Definition 1 2 min ( ) s.t. ( ) 0 1,. ., ( ) 0 ., ., j k u i i i J g j m h k m x x x i n x x x For now, we consider a single objective function, J( x ) . There are n design variables, and a total of m constraints ( m = m 1 + m 2 ). The bounds are known as side constraints.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
4 © Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Engineering Systems Division and Dept. of Aeronautics and Astronautics Karush-Kuhn-Tucker Conditions If x * is optimum, these conditions are satisfied: 1. x * is feasible 2. j g j ( x *) = 0, j =1,. ., m 1 and j 0 3. The KKT conditions are necessary and sufficient if the design space is convex. sign in ed unrestrict 0 0 ) ( ) ( ) ( 1 2 1 1 1 * 1 * * k m j m k k k m m j j j x h x g x J
5 © Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Engineering Systems Division and Dept. of Aeronautics and Astronautics Karush-Kuhn-Tucker Conditions: Interpretation Condition 1: the optimal design satisfies the constraints Condition 2: if a constraint is not precisely satisfied, then the corresponding Lagrange multiplier is zero the j th Lagrange multiplier represents the sensitivity of the objective function to the j th constraint can be thought of as representing the “tightness” of the constraint if j is large, then constraint j is important for this solution Condition 3: the gradient of the Lagrangian vanishes at the optimum

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
6 © Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Engineering Systems Division and Dept. of Aeronautics and Astronautics Optimization for Engineering Problems • Most engineering problems have a complicated design space, usually with several local optima • Gradient-based methods can have trouble converging to the global optimum, and sometimes fail to find even a local optimum • Heuristic techniques offer no guarantee of optimality, neither global nor local • Your post-optimality analysis should address the question: – How confident are you that you have found the global optimum? – Do you actually care?
7 © Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Engineering Systems Division and Dept. of Aeronautics and Astronautics Optimization for Engineering Problems • Usually cannot guarantee that absolute optimum is found – local optima – numerical ill-conditioning

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 35

MITESD_77S10_lec16 - Multidisciplinary System Design...

This preview shows document pages 1 - 8. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online