# mathematical optimization – SemidefiniteProgramming for operator norms: Stuck at the edge of dual feasibility I’m trying to calculate operator norms of linear transformations over spaces of matrices. For instance, find norm of $$f(A)=XA$$ by optimizing following:

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

This looks like a semidefinite programme, but I’m having trouble solving it with SemidefiniteOptimization. Simplest failing example is to find operator norm of $$f(A)=5A$$ in 1 dimension. It fails with `Stuck at the edge of dual feasibility`. Any suggestions?

Constraints
$$A succ 0\ Isucc A \ x I succ -5 A$$
Objective

$$text{min}_{A,x} x$$

``````d = 1;
ii = IdentityMatrix(d);
(* Symbolic symmetric d-by-d matrix *)
ClearAll(a);
X = 5*ii;
A = Array(a(Min(#1, #2), Max(#1, #2)) &, {d, d});
vars = DeleteDuplicates(Flatten(A));

cons0 = VectorGreaterEqual({A, 0}, {"SemidefiniteCone", d});
cons1 = VectorGreaterEqual({ii, A}, {"SemidefiniteCone", d});
cons2 = VectorGreaterEqual({x ii, -X.A}, {"SemidefiniteCone", d});
SemidefiniteOptimization(x, cons0 && cons1 && cons2, {x}~Join~vars)
``````

Edit it seems I get the correct result if I drop cons0. Going to more dimensions still fails. It works when $$X$$ is a multiple of identity but when it’s a different diagonal matrix I get a message about some matrix not being symmetric, for instance the example below fails in this way.

``````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;
cons1 = VectorGreaterEqual({ii, A}, {"SemidefiniteCone", d});
cons2 = VectorGreaterEqual({x ii, -X.A}, {"SemidefiniteCone", d});
SemidefiniteOptimization(x, cons1 && cons2, {x}~Join~vars)
`````` Posted on Categories Articles