From 3171067744cb6ca6f247f8aa3c0f0085c2724547 Mon Sep 17 00:00:00 2001 From: Claudio Maggioni Date: Mon, 12 Jul 2021 22:25:52 +0200 Subject: [PATCH] Uzzawa --- Claudio_Maggioni_5/Claudio_Maggioni_5.md | 31 +++------------------- Claudio_Maggioni_5/Claudio_Maggioni_5.pdf | Bin 241359 -> 232672 bytes 2 files changed, 3 insertions(+), 28 deletions(-) diff --git a/Claudio_Maggioni_5/Claudio_Maggioni_5.md b/Claudio_Maggioni_5/Claudio_Maggioni_5.md index 31262e4..4866abf 100644 --- a/Claudio_Maggioni_5/Claudio_Maggioni_5.md +++ b/Claudio_Maggioni_5/Claudio_Maggioni_5.md @@ -121,12 +121,6 @@ central path C converges to anything then $\tau$ approaches zero, therefore leading the path to the constrained minimizer while keeping $x$ and $s$ positive but at the same time minimizing their pairwise products to zero. -In practice, most central path implementation use Newton steps toward points on -$\mathcal{C}$ for which $\tau > 0$, rather than pure Newton steps for $F$. This -is usually possible since those newton steps are biased to stay in the -hyperspace region where $x > 0$ and $s > 0$ even without enforcing these -constraints. - # Exercise 2 ## Exercise 2.1 @@ -213,27 +207,8 @@ constrained minimizer. # Exercise 3 ## Exercise 3.1 - - -Yes, the problem can be solved with _Uzzawa_'s method since the problem can be +Yes, the problem can be solved with Uzawa's method since the problem can be reformulated as a saddle point system. The KKT conditions of the problem can be reformulated as a matrix-vector to vector equality in the following way: @@ -256,7 +231,7 @@ g\\ h \end{bmatrix}$$ -This is the system the _Uzzawa_ method will solve. Therefore, we need to check +This is the system the Uzawa method will solve. Therefore, we need to check if the matrix: $$K = \begin{bmatrix}G & A^T \\ A& 0\end{bmatrix} = \begin{bmatrix} @@ -279,7 +254,7 @@ $$\begin{bmatrix} 8.7663\end{bmatrix}$$ Therefore, the system is indeed a saddle point system and it can be solved with -_Uzzawa_'s method. +Uzawa's method. ## Exercise 3.2 diff --git a/Claudio_Maggioni_5/Claudio_Maggioni_5.pdf b/Claudio_Maggioni_5/Claudio_Maggioni_5.pdf index e795b235c97286d23e858d3809140d96e1f78934..90eef9d469bad95282d759ed7af128fa83a2cad9 100644 GIT binary patch delta 54228 zcmZUaV{@Pj(57SC*2K1L+nLz5@x-?6iH(VE8xz~kL=)|Kw`yy*YQOa_=o?q}+&0o) z7E%KzJUA;CXNpKPHL!v=k-W+Id{e)dRMm)bMRJ{8>GIQ=_I_GHgD3X1Kb<5BJxX2r z_vUr|+>j(1+H%Y_d0L9}-TwYHerU6Pcr&q~(f9l3#fayvRZ?0-s=7tBy(CK69jqm} z8V;taMKp>y#)RhBRo`1er@%!~_1en5Q=c}g$m3Uw_9EHZ3Q+IfO!78c)8w~q`{J~$ zp~l4rz3H%BT(=5e_0XeN3%#u&w#|8Cy4`g=+4KB#?{+sHj-*(T-J4(7rtj;fl(y-i z_Vly?EZzAXaxQJnS3Y1>gcc0^u*3e=`?PBpwA4$dC>H1aU0ntl&bvOOy>sT%W(YFH zm45%*^=cqv6@U@$V`n0G?w=aV8ImQckf}YMaKz@<}FRM+R 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