MITESD_77S10_lec16

MITESD_77S10_lec16 - Multidisciplinary System Design...

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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)
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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
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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.
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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
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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
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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?
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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
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MITESD_77S10_lec16 - Multidisciplinary System Design...

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