Why it matters

Ellipsoid enabled polynomial LP theory. Understanding shapes theory.

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The architecture

Start with large ellipsoid containing polytope.

Cut via separation oracle.

Shrink ellipsoid.

Ellipsoid methodEllipsoidcontaining feasibleSeparation oraclecut halfShrink ellipsoidhalf-space cutSlow in practice; theoretically important; separation-oracle-based (implicit constraints)
Ellipsoid.
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How it works end to end

Khachiyan 1979.

Grotschel-Lovasz-Schrijver equivalence: separation = optimization.

Theoretical foundation for combinatorial LP.