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Semidefinite Lift

A semidefinite lift of a convex set C is an exact representation using linear equations in a positive semidefinite matrix followed by an affine function. A lift of size r has the form

C={pi(X):X>=0,A(X)=b},

where X is an r×r real matrix, A is a linear transformation, b is a fixed vector, and pi is an affine function. Here X>=0 requires X to be symmetric and positive semidefinite (Fawzi et al. 2015). The interval [0,1] has a size-2 lift using the diagonal matrices diag(x,1-x) with pi(X)=x_(11).

See also

Hyperbolicity Cone, Positive Semidefinite Matrix, Positive Semidefinite Rank, Projected Lax Conjecture, Semidefinite Programming, Spectrahedral Shadow

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References

Fawzi, H.; Gouveia, J.; Parrilo, P. A.; Robinson, R. Z.; and Thomas, R. R. "Positive Semidefinite Rank." Math. Program. 153, 133-177, 2015. https://doi.org/10.1007/s10107-015-0922-1.

Cite this as:

Weisstein, Eric W. "Semidefinite Lift." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/SemidefiniteLift.html

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