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Sequential Quadratic Programming 1 Lecture 17 ME EN 575 Andrew - - PDF document
Sequential Quadratic Programming 1 Lecture 17 ME EN 575 Andrew - - PDF document
Sequential Quadratic Programming 1 Lecture 17 ME EN 575 Andrew Ning aning@byu.edu Outline Equality Constrained Quadratic Programming Equality Constrained Sequential Quadratic Programming Quadratic Programming (QP) Equality Constrained
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Pull out a sheet of paper, and try to solve the following analytically. Don’t solve it by variable
- substitution. Assume that there are lots of
variables and equality constraints and so you couldn’t use that approach. minimize x2 − 8x + y2 − 12y + 48 subject to x + y = 8 Consider a general equality-constrained QP: minimize 1 2xTQx + xTc subject to Ax = b
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We can directly find the global minimum of any equality-constrained QP just by solving a linear system.
∗ As long as Q is positive definite.
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Example: Controls
Linear Dynamical System: xt+1 = Axt + But xt : state at time t ut : control input at time t Define a cost function: J =
N
- xT
t Qxt + uT t Rut
(cost often represents deviations in the states from target values plus energy expended by control action)
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Example: Aerodynamics
Consider a variation on the wing example I showed last time. Let’s optimize the lift distribution γ.
γ
Equality Constrained Sequential Quadratic Programming
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Now assume we have a general nonlinear equality-constrained optimization problem. minimize f(x) subject to ˆ cj(x) = 0, j = 1, . . . , ˆ m We locally approximate our problem as: minimize 1 2pT∇xxLkp + ∇xLT
k p + Lk
with respect to p subject to ∇xˆ ckp + ˆ ck = 0
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After simplification: ∇xxLk [∇xˆ ck]T ∇xˆ ck pk ˆ λk+1 = −gk −ˆ ck
SQP Procedure
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