MITESD_77S10_lec08

MITESD_77S10_lec08 - 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 Multidisciplinary System Design Optimization (MSDO) Numerical Optimization II Lecture 8 Karen Willcox
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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 • Sequential Linear Programming • Penalty and Barrier Methods • Sequential Quadratic Programming
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3 © Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Engineering Systems Division and Dept. of Aeronautics and Astronautics 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
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4 © 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 ). For now we assume all x i are real and continuous.
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5 © Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Engineering Systems Division and Dept. of Aeronautics and Astronautics Optimization Process Calculate J ( x q ) Calculate S q Perform 1-D search x q = x q -1 + S q x 0 , q =0 Converged? Done yes no q=q +1
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6 © Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Engineering Systems Division and Dept. of Aeronautics and Astronautics Constrained Optimization Definitions: • Feasible design: a design that satisfies all constraints • Infeasible design: a design that violates one or more constraints • Optimum design: the choice of design variables that minimizes the objective function while satisfying all constraints In general, constrained optimization algorithms try to cast the problem as an unconstrained optimization and then use one of the techniques we looked at in Lecture 7.
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7 © Massachusetts Institute of Technology - Prof. de Weck and Prof. Willcox Engineering Systems Division and Dept. of Aeronautics and Astronautics Linear Programming Most engineering problems of interest are nonlinear • Can often simplify nonlinear problem by linearization • LP is often the basis for developing more complicated NLP algorithms Standard LP problem: All LP problems can be converted to this form.
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MITESD_77S10_lec08 - Multidisciplinary System Design...

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