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Unformatted text preview: MIT OpenCourseWare http://ocw.mit.edu 16.323 Principles of Optimal Control Spring 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms . 16.323 Lecture 2 Nonlinear Optimization • Constrained nonlinear optimization • Lagrange multipliers • Penalty/barrier functions also often used, but will not be discussed here. Figure by MIT OpenCourseWare. Spr 2008 16.323 2–1 Constrained Optimization • Consider a problem with the next level of complexity: optimization with equality constraints min F ( y ) y such that f ( y ) = 0 a vector of n constraints • To simplify the notation, assume that the pstate vector y can be separated into a decision mvector u and a state nvector x related to the decision variables through the constraints. Problem now becomes: min F ( x , u ) u such that f ( x , u ) = 0 – Assume that p > n otherwise the problem is completely specified by the constraints (or over specified). • One solution approach is direct substitution , which involves – Solving for x in terms of u using f – Substituting this expression into F and solving for u using an unconstrained optimization. – Works best if f is linear (assumption is that not both of f and F are linear.) June 18, 2008 Spr 2008 16.323 2–2 • Example: minimize F = x 1 2 + x 2 2 subject to the constraint that x 1 + x 2 + 2 = 0 – Clearly the unconstrained minimum is at x 1 = x 2 = 0 – Substitution in this case gives equivalent problems: min F ˜ 2 = ( − 2 − x 2 ) 2 + x 2 2 x 2 or min F ˜ 1 = x 1 2 + ( − 2 − x 1 ) 2 x 1 for which the solution ( ∂F ˜ 2 /∂x 2 = 0 ) is x 1 = x 2 = − 1 x 1 x 221.510.5 0.5 1 1.5 221.510.5 0.5 1 1.5 2 Figure 2.8: Simple function minimization with constraint. • Bottom line : substitution works well for linear constraints, but pro cess hard to generalize for larger systems/nonlinear constraints. June 18, 2008 Spr 2008 16.323 2–3 Lagrange Multipliers • Need a more general strategy using Lagrange multipliers. • Since f ( x , u ) = 0 , we can adjoin it to the cost with constants λ T = λ 1 ... λ n without changing the function value along the constraint to create Lagrangian function L ( x , u , λ ) = F ( x , u ) + λ T f ( x , u ) • Given values of x and u for which f ( x , u ) = , consider differential changes to the Lagrangian from differential changes to x and u : ∂L ∂L dL = d x + d u ∂ x ∂ u where ∂L = ∂L ∂L (row vector) ∂ u ∂u 1 ∂u m ··· Since u are the decision variables it is convenient to choose λ so that • ∂L ∂F + λ T ∂ f = ∂ x ≡ (2.1) ∂ x ∂ x ∂F ∂ f − 1 ⇒ λ T = − ∂ x ∂ x (2.2) • To proceed, must determine what changes are possible to the cost keeping the equality constraint satisfied....
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This note was uploaded on 11/07/2011 for the course AERO 16.323 taught by Professor Jonathanhow during the Spring '08 term at MIT.
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