By Sven O. Krumke
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Extra info for Interior Point Methods with Applications to Discrete Optimization [Lecture notes]
21) equivalently before linearizing! We can write the condition XS − µI = 0 equivalently at least in three different ways: XS − µI = 0 SX − µI = 0 XS + SX − 2µI = 0 All these three equivalent ways lead to different Newton steps. In the case of Linear Programs, all ways were the same, because the diagonal matrices occuring there commutated. Now, however, we are faced with the situation that in general XS = SX! 21) is overdetermined: For y ∈ Rm and symmetric X, S ∈ Rn×n we have m+n(n+1) unknowns.
1 Set µ0 := X(0) , S(0) /n. 21) with µ = σk µk 5 Set (X(k+1) , y(k+1) , S(k+1) ) := (X(k) , y(k) , S(k) ) + αk (∆X, ∆y, ∆S) for some appropriate step length αk > 0 such that X(k+1) ≻ 0 and S(k+1) ≻ 0. 21d) Suppose that X ≻ 0, y ∈ Rm and S ≻ 0 are given and that we want to make a Newton step. 32c) We have alredy mentioned the issue that in general the above system might lead to nonsymmetric ∆X and ∆S. 32) by means of a symmetrization operator. Let P be a nonsingular n × n-matrix. We define SP : Rn×m → Sn to be the symmetrization operator, defined by SP (U) := 1 PUP−1 + (PUP−1 )T 2 Observe that, if P = I is the identity matrix, then for symmetric X and S we File: ÒÐÔ¹Ð ØÙÖ ºØ Ü Revision: ½º¿¼ Date: ¾¼¼ »¼ »¾¼ ½ ¼ ÅÌ 42 Semidefinite Programs have SP (XS) = (XS + SX)/2.
4) which we assumed to exist. Since vT (X + αB)v = vT Xv + αvT Bv for all v ∈ Rn we can conclude that A 0. Moreover, A(B) = 0. Since we have shown that L≤d is compact for all d ∈ R we have X + αB ∈ / L≤d for all large α which implies that limα→ ∞ C, X + αB = +∞. 27) which by assumption is at most θ for all α > 0. 27) goes to infinity at linear rate with α. Since for any matrix D we have det D = i λi (D) the second term can go to infinity only at logarithmic rate and thus for large α the term can never be bounded from above by θ wich is a contradiction.
Interior Point Methods with Applications to Discrete Optimization [Lecture notes] by Sven O. Krumke