Convex Iteration
From Wikimization
Convex iteration is method for constraining rank or cardinality in an otherwise convex optimization problem.
A rank or cardinality constraint is replaced by a weighted linear regularization term added to the objective.
Then two convex problems are iterated until convergence where, ideally, solution to the original problem is found.
constraining cardinality of signed variable
(Excert from Semidefinite Programming).
Consider a feasibility problem equivalent to the classical problem from linear algebra
,
but with an upper bound
on cardinality
:
for vector
where means vector
has at most
nonzero entries;
such a vector is presumed existent in the feasible set.
Convex iteration works with a nonnegative variable;
absolute value is therefore needed.
We propose that problem (1) can be equivalently written
which moves the cardinality constraint to the objective.
To express this nonconvex problem as a convex iteration, we separate it into halves:
for a relatively small positive constant,
is iterated with
to find direction vector .
The term
in (2) is necessary to determine absolute value
because vector
can take zero values in its entries;
a good estimate of
is required for (3).
By initializing direction vector to
,
the first iteration of problem (2) is a 1-norm problem; i.e.,
Subsequent iterations of problem (2) engaging cardinality term
can be interpreted as corrections to this 1-norm problem leading to a 0-norm solution.
Iteration (2) (3) always converges to a locally optimal solution by virtue of
a monotonically nonincreasing objective sequence.
There can be no proof of global optimality, defined by an optimal objective euqal to 0.
Local solutions are therefore detected by nonzero optimal objective.
Heuristics for reinitializing direction vector can lead to a globally optimal solution.