Chapter 3 Quadratic Programming

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Optimization I; <strong>Chapter</strong> 3 69 The complementarity conditions can be equivalently stated as a T i x ∗ − d i ≤ 0 , µ ∗ i ≥ 0 , µ ∗ i (a T i x ∗ − d i ) = 0 , 1 ≤ i ≤ p µ ∗ ∈ ∂I K (v ∗ ) , v ∗ i := a T i x ∗ − d i , 1 ≤ i ≤ p , (3.50) where ∂I K denotes the subdifferential of the indicator function I K as given by µ ∈ ∂I K (v) ⇐⇒ I K (v) + µ T (w − v) ≤ I K (w) , w ∈ lR p . (3.51) Using the generalized Moreau-Yosida approximation of the indicator function I K , (3.50) can be replaced by the computationally more feasible condition µ ∗ = σ [ v ∗ + σ −1 µ ∗ − P K (v ∗ + σ −1 µ ∗ ) ] , (3.52) where σ is an appropriately chosen positive constant and P K denotes the projection onto K as given by P K (w) := { wi , w i < 0 0 , w i ≥ 0 , 1 ≤ i ≤ p . Note that (3.52) can be equivalently written as µ ∗ i = σ max ( 0, a T i x ∗ − d i + µ∗ ) i . (3.53) σ Now, given startiterates x (0) , λ (0) , µ (0) , the primal-dual active set strategy proceeds as follows: For ν ≥ 1 we determine the set I ac (x (ν) ) of active constraints according to We define and set I ac (x (ν) ) := {1 ≤ i ≤ p | a T i x (ν) − d i + µ(ν) i σ > 0} . (3.54) I in (x (ν) ) := {1, ..., p} \ I ac (x (ν) ) (3.55) p (ν) := card I ac (x (ν) ) , (3.56) µ (ν+1) i := 0 , i ∈ I in (x (ν) ) . (3.57) We compute (x (ν+1) , λ (ν+1) , ˜µ (ν+1) ) ∈ lR n × lR m × lR p(ν) as the solution of the KKT system associated with the equality constrained quadratic programming
Optimization I; <strong>Chapter</strong> 3 70 problem minimize Q(x) := 1 2 xT Bx − x T b (3.58a) over x ∈ lR n subject to c T i x = c i , 1 ≤ i ≤ m , (3.58b) a T i x = d i , i ∈ I ac (x (ν) ) . (3.58c) Since feasibility is not required, any startiterate (x (0) , λ (0) , µ (0) ) can be chosen. The following results show that we can expect convergence of the primal-dual active set strategy, provided the matrix B is symmetric positive definite. Theorem 3.3 Reduction of the objective functional Assume B ∈ lR n×n to be symmetric, positive definite and refer to ‖ · ‖ E := ((·) T B(·)) 1/2 as the associated energy norm. Let x (ν) , ν ≥ 0, be the iterates generated by the primal-dual active set strategy. Then, for Q(x) := 1 2 xT Bx−x T b and ν ≥ 1 there holds Q(x (ν) ) − Q(x (ν−1) ) = (3.59) = − 1 2 ‖x(ν) − x (ν−1) ‖ 2 E − ∑ µ (ν) i (d i − a T i x (ν−1) ) ≤ 0 . i∈I ac(x (ν) i/∈I ac(x (ν−1) ) Proof. Observing the KKT conditions for (3.58a)-(3.58c), we obtain Q(x (ν) ) − Q(x (ν−1) ) = = − 1 2 ‖x(ν) − x (ν−1) ‖ 2 E + (x (ν) − x (ν−1) ) T (Bx (ν) − b) = = − 1 ∑ 2 ‖x(ν) − x (ν−1) ‖ 2 E − µ (ν) i a T i (x (ν) − x (ν−1) ) . i∈I ac (x (ν) ) For i ∈ I ac (x (ν) ) we have a T i x (ν) = d i , whereas a T i x (ν−1) = d i for i ∈ I ac (x (ν−1) ). we thus get Q(x (ν) ) − Q(x (ν−1) ) = = − 1 2 ‖x(ν) − x (ν−1) ‖ 2 E − ∑ i∈I ac(x (ν) ) i/∈I ac (x (ν−1) ) µ (ν) i (d i − a T i x (ν−1) ) . But µ (ν) i ≥ 0, i ∈ I ac (x (ν) ) and a T i x (ν−1) ≤ d i , i /∈ I ac (x (ν−1) ) which gives the assertion. • Corollary 3.4 Convergence of a subsequence to a local minimum
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Chapter 3 Quadratic Programming

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