MIest 16888 Today's Topics Multidisciplinary System Optimality Conditions Termination Design Optimization(MSDO) Gradient-based techniques Post-Optimality Analysis Heuristic techniques Lecture 14 · Lagrange multipliers 17 March 2004 Objective Function Beh Scaling Olivier de Weck Karen willcox 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 Standard Problem Definition Kuhn-Tucker Conditions min J(x) If x is optimum, these conditions are satisfied st.g(x)≤0j=1…m 1.x is feasib h2(x)=0k=1m2 239(x)=0,=1…m1and320 V(x)+∑4g(x)+∑mnVh(x)=0 X≤X≤X=1…,n A:≥0 For now, we consider a single objective function, J(/x) Am.k unrestricted in sign There are n design variables, and a total of m constraints(m=m,+m2) The Kuhn-tucker conditions are necessary and The bounds are known as side constraints sufficient if the design space is convex e Massachusetts Institute of Technology- Prof de Weck and Prof Willcox etts Institute of Technology .Prof de Weck and Prof Willcox Engineening Systems Division and Dept of Aeronautics and Astronautics ngineering Systems Divsion and Dept of Aeronautics and Astronautics1 © Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Engineering Systems Division and Dept. of Aeronautics and Astronautics Post-Optimality Analysis Lecture 14 17 March 2004 Olivier de Weck Karen Willcox Multidisciplinary System Multidisciplinary System Design Optimization (MSDO) Design Optimization (MSDO) 2 © Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Engineering Systems Division and Dept. of Aeronautics and Astronautics Today’s Topics Today’s Topics • Optimality Conditions & Termination – Gradient-based techniques – Heuristic techniques • Lagrange Multipliers • 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 Standard Problem Definition 1 2 min ( ) s.t. ( ) 0 1,.., ( ) 0 1,.., 1,.., j k u i ii J g jm h km x x xi n ≤ = = = ≤≤ = x x x A For now, we consider a single objective function, J(x). There are n design variables, and a total of m constraints (m=m1+m2). The bounds are known as side constraints. 4 © Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Engineering Systems Division and Dept. of Aeronautics and Astronautics Kuhn-Tucker Conditions Tucker Conditions If x* is optimum, these conditions are satisfied: 1. x* is feasible 2. λj gj(x*) = 0, j=1,..,m1 and λj ≥ 0 3. The Kuhn-Tucker conditions are necessary and sufficient if the design space is convex. unrestrictedin sign 0 ( ) ( ) ( ) 0 1 2 1 1 1 * 1 * * m k j m k m k k m j j j J x g x h x + = + = ≥ ∇ + ∑ ∇ +∑ ∇ = λ λ λ λ