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Home arrow Rank Constraint
Rank Constraint

A semidefinite feasibility problem is a convex optimization problem, over a subset of the positive semidefinite cone, having no objective function.  Constraining rank of a feasible solution can be thought of as introducing a linear objective function whose normal opposes the direction of search.  If one knows the proper search direction, then a solution of desired rank can be found.

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"In any SDP feasibility problem, an SDP feasible solution with the lowest rank must be an extreme point of the feasible set.  Thus, there must exist a linear objective function such that this lowest-rank feasible solution uniquely optimizes it."

In Chapter 4.4, we show how to determine what linear objective function will replace a rank constraint in a semidefinite program.  Finding the normal representing that linear function turns out to be a convex optimization problem over a Fantope.  Fantopes are introduced in Chapter 2.  Some semidefinite problems having a rank constraint can thereby be formulated as convex optimization problems.

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