Summary: Optimization with Constraints Sensitivity Analysis Boundary Value Problems (BVPs)10.34, Numerical Methods Applied to Chemical Engineering Professor William H. Green Lecture #18: Optimization. Sensitivity Analysis. Introduction: Boundary Value Problems (BVPs). Summary: Optimization with Constraints minx f(x) such that cm(x) - sm = 0 sm ≥ 0 m = 1 … Ninequalities minx f(x) + ξ(c-s)2 sm = 0 m > Ninequalities penalty method, second term ξ(c-s)2 is optional KKT conditions: at constrained (local) minimum: Augmented Vxf – Σm(λm Vxcm) = 0 Æ Lagrangian cm – sm = 0 (LA) λmcm = 0 {see book} sm ≥ 0 m = 1 … Ninequalities sm = 0 equalities ⎟⎟⎟⎟⎟⎠⎞⎜⎜⎜⎜⎜⎝⎛−∇−∇==⎟⎟⎟⎠⎞⎜⎜⎜⎝⎛=∑mmmmmmmscsccfxFscxλλλ0)( Newton Æ SQP If everything is linear: Æ SIMPLEX (i.e. many business problems) g(x) = 0 Æ xN = G(x1, …, xN-1) Unconstrained Æ trust region Newton-type BFGS gigantic Æ conjugate gradient In Chemical Engineering, the problems often involve models with differential equations: cost return f(x) = ()∑−ifioiixtYxtYw );();( knobs what we need what we produce (can adjust) feed composition Need Jacobian of G with respect to Y; need in stiff solver to solve. )()();(00xYtYxYGdtYd== Cite as: William Green, Jr., course materials for 10.34 Numerical Methods Applied to Chemical Engineering, Fall 2006. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].Need gradient and f. To use all of our methods, we need to be able to compute: ∑⎟⎟⎠⎞⎜⎜⎝⎛∂∂−∂∂=∂∂ijfijiijxtYxYwxf)(,σ how do you compute this? jinjnnijxGxYYGtYx ∂∂+⎟⎟⎠⎞⎜⎜⎝⎛∂∂∂∂=⎟⎠⎞⎜⎝⎛∂∂∂∂∑ {“sensitivity of Yi(tf) to xj”} chain rule ⎟⎟⎠⎞⎜⎜⎝⎛∂∂∂∂jixYt sij jinnjniijxGsYGsdtd∂∂+⎟⎟⎠⎞⎜⎜⎝⎛∂∂=∑ Î Vf (for every x we get an Vf that can be used for optimization) solve this with initial conditions J {Jacobian of G} Have n2 differential equations; stiff; linear in s. Sensitivity Analysis Programs to do this: DASPK SOLVE for s and f simultaneously DAEPACK DSL485 DASAC jinnjniijxGsYGsdtd∂∂+⎟⎟⎠⎞⎜⎜⎝⎛∂∂=∑ Initial Conditions What is sji(t0)? sji(t0) = 0 {most knobs} sji(t0) = 1 {for adjustment of Y0} Professor Barton teaches an advanced course in optimization. 10.34, Numerical Methods Applied to Chemical Engineering Lecture 18 Prof. William Green Page 2 of 4 Cite as: William Green, Jr., course materials for 10.34 Numerical Methods Applied to Chemical Engineering, Fall 2006. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].Boundary Value Problems (BVPs) Conservation Laws: ∂φ/∂t = -V10.34, Numerical Methods Applied to Chemical Engineering Lecture 18Prof. William Green Page 3 of 4 Cite as: William Green, Jr., course materials for 10.34 Numerical Methods Applied to Chemical Engineering, Fall 2006. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY]. ·(φv) - V·JD + S(φ) convection diffusion reaction JD = Γ+Vφ isotropic: JD = -cVφ for steady-state, isotropic: 0 = -V·(φv) - cV2φ + S(φ) ∀x Laplacian Boundary conditions: Dirichlet φ(boundary) = number von Neumann Vφ(boundary) = number or 0 Symmetry 0=∂∂jxφ φ(x) infinite {rare to find exact} φapprox(x) = f(x; c) adjust: large finite number (104) Basis function expansions φapprox = ∑=ΨbasisNnnnxc1)( ()ioninterpolat somegridmesh }{)(}){;(0)()()()(,1),,(2..===+∇+⋅∇−Ψ∫=ℜiiapproxixapproxNmcxapproxapproxapproxmxxxfscvxCBbasisapproxφφφφφφφcalled “Residual” 1111−+−+−−≈∂∂iiiixxxxiφφφ State how you did approximation because there are many ways to do it Dirichlet φ(boundary) {φ} i = 1,N Finite difference approximation to differential equation ..02021CBxxxxφφφφ−−=∂∂ von Neumann0xx∂∂φ given φ0? Usual Æ 2nd order polynomials 202200)(21)()()(00xxxxxxxxxx−∂∂+−∂∂+=φφφφ unknown known unknown 9 9 02212210),(),(1xxfxfx−−=∂∂=φφφφφφφ ...)(21)()(20122010100xxxxxxxxx−∂∂+−∂∂+=φφφφ ...)(21)()(20222020200xxxxxxxxx−∂∂+−∂∂+=φφφφ 0for 1)()()()(201202220120210xxxxxxxxxx∂∂−−−−−−=φφφφ=0 If Δx uniform, 34210φφφ−= This is how you find out B.C. with second order polynomial schemes and a finite difference approximation. 10.34, Numerical Methods Applied to Chemical Engineering Lecture 18 Prof. William Green Page 4 of 4 Cite as: William Green, Jr., course materials for 10.34 Numerical Methods Applied to Chemical Engineering, Fall 2006. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month
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