Write problem containing inverses as Semi Definite Programming
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Let $vinmathbb{R}^m$ and positive semidefinite $Minmathbb{S}^m$ be fixed. For all $xinmathbb{R}^n$, let $A(x)=A_0+sum_{i=1}^nx_iA_i$, where $A_0,...,A_ninmathbb{S}^m$ are fixed. Write $$begin{array}{rl}text{minimize}&v'A(x)^{-1}v+text{tr}(A(x)^{-1}M)\text{subject to}&A(x)succ 0end{array}$$ as an SDP.
The objective function is linear in $A(x)^{-1}$. I tried to use duality, but I couldn't minimize the Lagrangian over $x$ since $A(x)^{-1}$ and $A(x)$ were both present.
linear-algebra convex-optimization duality-theorems semidefinite-programming
edited Dec 5 '18 at 7:33
Jean Marie
28.9k41949
asked Dec 5 '18 at 7:14
L WheelsL Wheels
111
$endgroup$
add a comment |
$begingroup$
Let $vinmathbb{R}^m$ and positive semidefinite $Minmathbb{S}^m$ be fixed. For all $xinmathbb{R}^n$, let $A(x)=A_0+sum_{i=1}^nx_iA_i$, where $A_0,...,A_ninmathbb{S}^m$ are fixed. Write $$begin{array}{rl}text{minimize}&v'A(x)^{-1}v+text{tr}(A(x)^{-1}M)\text{subject to}&A(x)succ 0end{array}$$ as an SDP.
The objective function is linear in $A(x)^{-1}$. I tried to use duality, but I couldn't minimize the Lagrangian over $x$ since $A(x)^{-1}$ and $A(x)$ were both present.
linear-algebra convex-optimization duality-theorems semidefinite-programming
edited Dec 5 '18 at 7:33
Jean Marie
28.9k41949
asked Dec 5 '18 at 7:14
L WheelsL Wheels
111
$endgroup$
$begingroup$
Use Schur complement to model things like $tgeq v^TA^{-1}v$.
$endgroup$
– Michal Adamaszek
Dec 5 '18 at 8:08
add a comment |
$begingroup$
Let $vinmathbb{R}^m$ and positive semidefinite $Minmathbb{S}^m$ be fixed. For all $xinmathbb{R}^n$, let $A(x)=A_0+sum_{i=1}^nx_iA_i$, where $A_0,...,A_ninmathbb{S}^m$ are fixed. Write $$begin{array}{rl}text{minimize}&v'A(x)^{-1}v+text{tr}(A(x)^{-1}M)\text{subject to}&A(x)succ 0end{array}$$ as an SDP.
The objective function is linear in $A(x)^{-1}$. I tried to use duality, but I couldn't minimize the Lagrangian over $x$ since $A(x)^{-1}$ and $A(x)$ were both present.
linear-algebra convex-optimization duality-theorems semidefinite-programming
edited Dec 5 '18 at 7:33
Jean Marie
28.9k41949
asked Dec 5 '18 at 7:14
L WheelsL Wheels
111
$endgroup$
Let $vinmathbb{R}^m$ and positive semidefinite $Minmathbb{S}^m$ be fixed. For all $xinmathbb{R}^n$, let $A(x)=A_0+sum_{i=1}^nx_iA_i$, where $A_0,...,A_ninmathbb{S}^m$ are fixed. Write $$begin{array}{rl}text{minimize}&v'A(x)^{-1}v+text{tr}(A(x)^{-1}M)\text{subject to}&A(x)succ 0end{array}$$ as an SDP.
The objective function is linear in $A(x)^{-1}$. I tried to use duality, but I couldn't minimize the Lagrangian over $x$ since $A(x)^{-1}$ and $A(x)$ were both present.
linear-algebra convex-optimization duality-theorems semidefinite-programming
linear-algebra convex-optimization duality-theorems semidefinite-programming
edited Dec 5 '18 at 7:33
Jean Marie
28.9k41949
asked Dec 5 '18 at 7:14
L WheelsL Wheels
111
edited Dec 5 '18 at 7:33
Jean Marie
28.9k41949
asked Dec 5 '18 at 7:14
L WheelsL Wheels
111
edited Dec 5 '18 at 7:33
Jean Marie
28.9k41949
edited Dec 5 '18 at 7:33
Jean Marie
28.9k41949
edited Dec 5 '18 at 7:33
Jean Marie
28.9k41949
28.9k41949
asked Dec 5 '18 at 7:14
L WheelsL Wheels
111
asked Dec 5 '18 at 7:14
L WheelsL Wheels
111
asked Dec 5 '18 at 7:14
L WheelsL Wheels
111
111
$begingroup$
Use Schur complement to model things like $tgeq v^TA^{-1}v$.
$endgroup$
– Michal Adamaszek
Dec 5 '18 at 8:08
add a comment |
$begingroup$
Use Schur complement to model things like $tgeq v^TA^{-1}v$.
$endgroup$
– Michal Adamaszek
Dec 5 '18 at 8:08
$begingroup$
Use Schur complement to model things like $tgeq v^TA^{-1}v$.
$endgroup$
– Michal Adamaszek
Dec 5 '18 at 8:08
$begingroup$
Use Schur complement to model things like $tgeq v^TA^{-1}v$.
$endgroup$
– Michal Adamaszek
Dec 5 '18 at 8:08
add a comment |
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$begingroup$
Use Schur complement to model things like $tgeq v^TA^{-1}v$.
$endgroup$
– Michal Adamaszek
Dec 5 '18 at 8:08