16888 eSd ES077 Multidisciplinary System Design Optimization(MSDo) Numerical Optimization ll Lecture 7 25 February 2004 Karen willcox o Massachusetts Institute of Technology -Prof de Weck and Prof Willcox Engineering Systems Division and Dept of Aeronautics and Astronautics
1 © Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Engineering Systems Division and Dept. of Aeronautics and Astronautics Multidisciplinary System Multidisciplinary System Design Optimization (MSDO) Design Optimization (MSDO) Numerical Optimization II Lecture 7 25 February 2004 Karen Willcox
Today,'s Topics 16888 eSd ES077 Sequential Linear Programming Penalty and Barrier Methods Sequential Quadratic Programming Mixed Integer Programming o Massachusetts Institute of Technology -Prof de Weck and Prof Willcox Engineering Systems Division and Dept of Aeronautics and Astronautics
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 • Sequential Linear Programming • Penalty and Barrier Methods • Sequential Quadratic Programming • Mixed Integer Programming
Technique Overview 16888 ES077 Steepest Descent UNCONSTRAINED Conjugate Gradient Quasi-Newton Newton Simplex -linear CONSTRAINED SLP -often not effective SQP-nonlinear, expensive, common in engineering applications Exterior Penalty-nonlinear, discontinuous design spaces Interior Penalty -nonlinear Generalized Reduced Gradient-nonlinear Method of Feasible Directions -nonlinear Mixed Integer Programming o Massachusetts Institute of Technology -Prof de Weck and Prof Willcox Engineering Systems Division and Dept of Aeronautics and Astronautics
3 © Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Engineering Systems Division and Dept. of Aeronautics and Astronautics Technique Overview Technique Overview Steepest Descent Conjugate Gradient Quasi-Newton Newton Simplex – linear SLP – often not effective SQP – nonlinear, expensive, common in engineering applications Exterior Penalty – nonlinear, discontinuous design spaces Interior Penalty – nonlinear Generalized Reduced Gradient – nonlinear Method of Feasible Directions – nonlinear Mixed Integer Programming UNCONSTRAINED CONSTRAINED
Mesd Standard Problem Definition 16888 min J(x st.g/(x)≤0j=1…,m h(x)=0k=1.,m2 ≤X1≤j=1 For now, we consider a single objective function, J(x) There are n design variables and a total of m constraints(m=,+m2) For now we assume all x are real and continuous o Massachusetts Institute of Technology -Prof de Weck and Prof Willcox Engineering Systems Division and Dept of Aeronautics and Astronautics
4 © 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 i i J g j m h k m x x x i 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 = m 1 + m 2). For now we assume all xi are real and continuous
Optimization Process 16888 eSd q=0 Calculate VJ(xg) Calculate sq Perform 1-D search xq=xq-1+a sq no Converged? es Done o Massachusetts Institute of Technology -Prof de Weck and Prof Willcox Engineering Systems Division and Dept of Aeronautics and Astronautics
5 © Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Engineering Systems Division and Dept. of Aeronautics and Astronautics Optimization Process Optimization Process x 0 , q=0 Calculate ∇ J ( x q ) Calculate S q Perform 1-D search x q = x q-1 + α S q Converged? q=q+1 no yes Done