Chapter 3 Quadratic Programming

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Optimization I; <strong>Chapter</strong> 3 63 where B ∈ lR n×n is symmetric positive definite, C ∈ lR m×n , A ∈ lR p×n , and b ∈ lR n , c ∈ lR m , d ∈ lR p . We write the matrices A and C in the form A = ⎡ ⎣ ⎤ a 1 · ⎦ , a i ∈ lR n , C = a p ⎡ ⎣ ⎤ c 1 · ⎦ , c i ∈ lR n . (3.30) c m The inequality constraints (3.29c) can be equivalently stated as a T i x ≤ d i , 1 ≤ i ≤ m . (3.31) The primal active set strategy is an iterative procedure: Given a feasible iterate x (ν) , ν ≥ 0, we determine an active set I ac (x (ν) ) ⊂ {1, ..., p} (3.32) and consider the corresponding constraints as equality constraints, whereas the remaining inequality constraints are disregarded. Setting we find p = x (ν) − x , b (ν) = Bx (ν) − b , (3.33) f(x) = f(x (ν) − p) = 1 2 pT Bp − (b (ν) ) T p + g , where g := 1 2 (x(ν) ) T Bx (ν) − b T x (ν) . The equality constrained QP problem to be solved at the (ν+1)-st iteration step is then: 1 minimize 2 pT Bp − (b (ν) ) T p (3.34a) over p ∈ lR n subject to Cp = 0 (3.34b) a T i p = 0 , i ∈ I ac (x (ν) ) , (3.34c) We denote the solution of (3.34a)-(3.34c) by p (ν) . The new iterate x (ν+1) is then obtained according to x (ν+1) = x (ν) − α ν p (ν) , α ν ∈ [0, 1] , (3.35) where α ν is chosen such that x (ν+1) stays feasible. In particular, for i ∈ I ac (x (ν) ) we have a T i x (ν+1) = a T i x (ν) − α ν a T i p (ν) = a T i x (ν) ≤ d i .
Optimization I; <strong>Chapter</strong> 3 64 On the other hand, if a T i p (ν) ≥ 0 for some i /∈ I ac (x (ν) ), it follows that a T i x (ν+1) = a T i x (ν) − α ν a T i p (ν) ≤ a T i x (ν) ≤ d i . Finally, if a T i p (ν) < 0 for i /∈ I ac (x (ν) ), we have a T i x (ν+1) = a T i x (ν) − α ν a T i p (ν) ≤ d i ⇐⇒ α ν ≤ aT i x (ν) − d i a T i p(ν) . Consequently, in order to guarantee feasibility, we choose We define Clearly, a T i x (ν) − d i α ν := min (1, min ) . (3.36) i/∈I ax (x (ν) ) a T i p(ν) a T i p(ν)
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Chapter 3 Quadratic Programming

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