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Yaroslav Bulatov
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Can someone give a workaround and/or explanation why Problem 1/Problem 2 fail to solve through SemidefiniteOptimization? Problem 3 works. (I'm using 12.1.0 on Mac). The main difference is that Problem 1+2 use diagonal matrix constraint, whereas Problem 3 matrix constraint has no 0's. I could solve them using other meanswithout calling SemidefiniteOptimization, but prefer to have a single toolsolution to cover a wide range of cases.

$$ \text{min}_{A,x} x $$ Subject to $$ I \succ A \succ -I\\ x I \succ -\sum_i^{d^2} V_i' A V_i \\ $$

Interestingly, substituting random matrices for $V_i$'s, I get the expected result out of SemidefiniteProgramming. For some reason Mathematica has more trouble with diagonal matrix constraints than random constraints

Can someone give a workaround and/or explanation why Problem 1/Problem 2 fail to solve through SemidefiniteOptimization? Problem 3 works. (I'm using 12.1.0 on Mac). I could solve them using other means, but prefer to have a single tool to cover a wide range of cases.

$$ \text{min}_{A,x} x $$ Subject to $$ I \succ A \succ -I\\ x I \succ -\sum_i^{d^2} V_i' A V_i \\ $$

Interestingly, substituting random matrices for $V_i$'s, I get the expected result out of SemidefiniteProgramming. For some reason Mathematica has more trouble with diagonal matrix constraints than random constraints

Can someone give a workaround and/or explanation why Problem 1/Problem 2 fail to solve through SemidefiniteOptimization? Problem 3 works. (I'm using 12.1.0 on Mac). The main difference is that Problem 1+2 use diagonal matrix constraint, whereas Problem 3 matrix constraint has no 0's. I could solve them without calling SemidefiniteOptimization, but prefer to have a single solution to cover a wide range of cases.

$$ \text{min}_{A,x} x $$ Subject to $$ I \succ A \succ -I\\ x I \succ -\sum_i^{d^2} V_i' A V_i \\ $$

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Yaroslav Bulatov
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I'm trying to calculate operator norms of linear transformations over spaces of matrices. For instance, forCan someone give a fixed matrix $X$, find norm of $f(A)=XA$ by optimizing following:

$$\max_{\|A\|=1} \|XA\|$$

The norm $\|\cdot\|$ above refersworkaround and/or explanation why Problem 1/Problem 2 fail to operator norm of matrixsolve through $A$SemidefiniteOptimization? Problem 3 works. Solving this kind of problem in Mathematica would give a nice numeric illustration of the Russo-Dye theorem (ie, section 2I'm using 12.5 of Bhatia's Positive Matrices book1.0 on Mac)

This looks like a semidefinite programme. I could solve them using other means, but I'm having trouble solving it with SemidefiniteOptimizationprefer to have a single tool to cover a wide range of cases. The example

Problem 1

Implementation below below fails with Stuck at the edge of dual feasibility. Any suggestions?

Problem 1

Problem 2

AnotherThis example below in $2$ dimension gives a different error: The matrix {{0.,1.},{2.,0.}} is not Hermitian or real and symmetric. I'm confused as whereWhere did this matrix camecome from?

Problem 2

Problem 3

FinallyFor this problem SemidefiniteOptimization works, ineven though the most general form I was curious if I can get numerical answer forproblem seems harder than the following formulation which covers every completely positive linear operator, as you can see in proof of Choi's theorem (also Theorem 3.1previous two.1 "Choi, Kraus" in Bhatia's Positive Matrices book)

Problem 3

I'm trying to calculate operator norms of linear transformations over spaces of matrices. For instance, for a fixed matrix $X$, find norm of $f(A)=XA$ by optimizing following:

$$\max_{\|A\|=1} \|XA\|$$

The norm $\|\cdot\|$ above refers to operator norm of matrix $A$. Solving this kind of problem in Mathematica would give a nice numeric illustration of the Russo-Dye theorem (ie, section 2.5 of Bhatia's Positive Matrices book)

This looks like a semidefinite programme, but I'm having trouble solving it with SemidefiniteOptimization. The example below fails with Stuck at the edge of dual feasibility. Any suggestions?

Problem 1

Another example below in $2$ dimension gives a different error: The matrix {{0.,1.},{2.,0.}} is not Hermitian or real and symmetric. I'm confused as where this matrix came from

Problem 2

Finally, in the most general form I was curious if I can get numerical answer for the following formulation which covers every completely positive linear operator, as you can see in proof of Choi's theorem (also Theorem 3.1.1 "Choi, Kraus" in Bhatia's Positive Matrices book)

Problem 3

Can someone give a workaround and/or explanation why Problem 1/Problem 2 fail to solve through SemidefiniteOptimization? Problem 3 works. (I'm using 12.1.0 on Mac). I could solve them using other means, but prefer to have a single tool to cover a wide range of cases.

Problem 1

Implementation below below fails with Stuck at the edge of dual feasibility.

Problem 2

This example in $2$ dimension gives a different error: The matrix {{0.,1.},{2.,0.}} is not Hermitian or real and symmetric. Where did this matrix come from?

Problem 3

For this problem SemidefiniteOptimization works, even though the problem seems harder than the previous two.

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SemidefiniteOptimization
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Yaroslav Bulatov
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SemidefiniteProgramming SemidefiniteOptimization for operator norms: Stuck at the edge of dual feasibility

d = 2;
ii = IdentityMatrix[d];
ClearAll[a];
extractVars[mat_] := DeleteDuplicates@Cases[Flatten@A, _a];
A = Array[a[Min[#1, #2], Max[#1, #2]] &, {d, d}];
vars = extractVars[A];
X = DiagonalMatrix@Range@d;
cons0 = VectorGreaterEqual[{A, -ii}, {"SemidefiniteCone", d}];
cons1 = VectorGreaterEqual[{ii, A}, {"SemidefiniteCone", d}];
cons2 = VectorGreaterEqual[{x ii, -X.A}, {"SemidefiniteCone", d}];
SemidefiniteOptimization[x, cons0 && cons1 && cons2, {x}~Join~vars]
(* Prints SemidefiniteOptimization::herm: The matrix {{0.,1.},{2.,0.}} is not Hermitian or real and symmetric. *)

SemidefiniteProgramming for operator norms: Stuck at the edge of dual feasibility

d = 2;
ii = IdentityMatrix[d];
ClearAll[a];
extractVars[mat_] := DeleteDuplicates@Cases[Flatten@A, _a];
A = Array[a[Min[#1, #2], Max[#1, #2]] &, {d, d}];
vars = extractVars[A];
X = DiagonalMatrix@Range@d;
cons0 = VectorGreaterEqual[{A, -ii}, {"SemidefiniteCone", d}];
cons1 = VectorGreaterEqual[{ii, A}, {"SemidefiniteCone", d}];
cons2 = VectorGreaterEqual[{x ii, -X.A}, {"SemidefiniteCone", d}];
SemidefiniteOptimization[x, cons1 && cons2, {x}~Join~vars]
(* Prints SemidefiniteOptimization::herm: The matrix {{0.,1.},{2.,0.}} is not Hermitian or real and symmetric. *)

SemidefiniteOptimization for operator norms: Stuck at the edge of dual feasibility

d = 2;
ii = IdentityMatrix[d];
ClearAll[a];
extractVars[mat_] := DeleteDuplicates@Cases[Flatten@A, _a];
A = Array[a[Min[#1, #2], Max[#1, #2]] &, {d, d}];
vars = extractVars[A];
X = DiagonalMatrix@Range@d;
cons0 = VectorGreaterEqual[{A, -ii}, {"SemidefiniteCone", d}];
cons1 = VectorGreaterEqual[{ii, A}, {"SemidefiniteCone", d}];
cons2 = VectorGreaterEqual[{x ii, -X.A}, {"SemidefiniteCone", d}];
SemidefiniteOptimization[x, cons0 && cons1 && cons2, {x}~Join~vars]
(* Prints SemidefiniteOptimization::herm: The matrix {{0.,1.},{2.,0.}} is not Hermitian or real and symmetric. *)
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clarify which norm to use
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Yaroslav Bulatov
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