Convex Algebraic Geometry Cynthia Vinzant, North Carolina State University Cynthia Vinzant Convex Algebraic Geometry
What is convex algebraic geometry? Convex algebraic geometry is the study of convex semialgebraic objects, especially those arising in optimization and statistics. Cynthia Vinzant Convex Algebraic Geometry
What is convex algebraic geometry? Convex algebraic geometry is the study of convex semialgebraic objects, especially those arising in optimization and statistics. Many convex concepts have algebraic analogues. convex duality ↔ algebraic duality convex combinations ↔ secant varieties boundary of a projection ↔ branch locus Cynthia Vinzant Convex Algebraic Geometry
What is convex algebraic geometry? Convex algebraic geometry is the study of convex semialgebraic objects, especially those arising in optimization and statistics. Many convex concepts have algebraic analogues. convex duality ↔ algebraic duality convex combinations ↔ secant varieties boundary of a projection ↔ branch locus Algebraic techniques can help answer questions about these convex sets. Convexity provides additions tools and challenges. Cynthia Vinzant Convex Algebraic Geometry
Motivational Example: The elliptope The 3-elliptope is 1 x y ( x , y , z ) ∈ R 3 : � 0 x 1 z y z 1 Cynthia Vinzant Convex Algebraic Geometry
Motivational Example: The elliptope The 3-elliptope is 1 x y ( x , y , z ) ∈ R 3 : � 0 x 1 z y z 1 ◮ convex, semialgebraic (defined by polynomial ≤ ’s) Cynthia Vinzant Convex Algebraic Geometry
Motivational Example: The elliptope The 3-elliptope is 1 x y ( x , y , z ) ∈ R 3 : � 0 x 1 z y z 1 ◮ convex, semialgebraic (defined by polynomial ≤ ’s) ◮ a spectrahedron (feasible set of a semidefinite program) Cynthia Vinzant Convex Algebraic Geometry
Motivational Example: The elliptope The 3-elliptope is 1 x y ( x , y , z ) ∈ R 3 : � 0 x 1 z y z 1 ◮ convex, semialgebraic (defined by polynomial ≤ ’s) ◮ a spectrahedron (feasible set of a semidefinite program) appears in . . . ◮ statistics as set of correlation matrices ◮ combinatorial optimization Cynthia Vinzant Convex Algebraic Geometry
The elliptope: convex algebraic istructure Convex structure Cynthia Vinzant Convex Algebraic Geometry
The elliptope: convex algebraic istructure Convex structure ◮ ∞ -many zero-dim’l faces ◮ 6 one-dim’l faces ◮ 4 vertices Cynthia Vinzant Convex Algebraic Geometry
The elliptope: convex algebraic istructure Convex structure ◮ ∞ -many zero-dim’l faces ◮ 6 one-dim’l faces t ◮ 4 vertices Algebraic structure Bounded by a cubic hypersurface, { ( x , y , z ) ∈ R 3 : f = 2 xyz − x 2 − y 2 − z 2 + 1 = 0 } that has 4 nodes and contains 6 lines. Cynthia Vinzant Convex Algebraic Geometry
Low-rank matrices The elliptope exhibits general behavior for its size and dimension. For general A 0 , A 1 , A 2 , A 3 ∈ R 3 × 3 sym the set of matrices { A 0 + xA 1 + yA 2 + zA 3 : ( x , y , z ) ∈ R 3 or C 3 } contains 4 rank-one matrices over C and 0,2, or 4 over R . Cynthia Vinzant Convex Algebraic Geometry
Low-rank matrices The elliptope exhibits general behavior for its size and dimension. For general A 0 , A 1 , A 2 , A 3 ∈ R 3 × 3 sym the set of matrices { A 0 + xA 1 + yA 2 + zA 3 : ( x , y , z ) ∈ R 3 or C 3 } contains 4 rank-one matrices over C and 0,2, or 4 over R . Why? The set of matrices of rank ≤ 1 is variety of codimension 3 and sym ∼ degree 4 in R 3 × 3 = R 6 . Cynthia Vinzant Convex Algebraic Geometry
Low-rank matrices The elliptope exhibits general behavior for its size and dimension. For general A 0 , A 1 , A 2 , A 3 ∈ R 3 × 3 sym the set of matrices { A 0 + xA 1 + yA 2 + zA 3 : ( x , y , z ) ∈ R 3 or C 3 } contains 4 rank-one matrices over C and 0,2, or 4 over R . Why? The set of matrices of rank ≤ 1 is variety of codimension 3 and sym ∼ degree 4 in R 3 × 3 = R 6 . If A 0 ≻ 0, there will always be 2 or 4 matrices of rank-one over R . Cynthia Vinzant Convex Algebraic Geometry
Algebraic and convex duality We can use duality in algebraic geometry to calculate hypersurface bounding the dual of a convex body. Cynthia Vinzant Convex Algebraic Geometry
Algebraic and convex duality We can use duality in algebraic geometry to calculate hypersurface bounding the dual of a convex body. The dual of the elliptope is bounded by the union of a quartic surface and four planes. Cynthia Vinzant Convex Algebraic Geometry
Algebraic and convex duality We can use duality in algebraic geometry to calculate hypersurface bounding the dual of a convex body. The dual of the elliptope is bounded by the union of a quartic surface and four planes. Writing down the solution of a random linear optimization problem over the elliptope requires solving a degree four polynomial. Cynthia Vinzant Convex Algebraic Geometry
Moments and Sums of Squares The cone of nonnegative polynomials NN n , 2 d = { p ∈ R [ x 1 , . . . , x n ] ≤ 2 d : p ( x ) ≥ 0 for all R n } is convex, semialgebraic, and contains the cone of sums of squares SOS n , 2 d = { h 2 1 + . . . + h 2 r : h j ∈ R [ x 1 , . . . , x n ] ≤ d } . Cynthia Vinzant Convex Algebraic Geometry
Moments and Sums of Squares The cone of nonnegative polynomials NN n , 2 d = { p ∈ R [ x 1 , . . . , x n ] ≤ 2 d : p ( x ) ≥ 0 for all R n } is convex, semialgebraic, and contains the cone of sums of squares SOS n , 2 d = { h 2 1 + . . . + h 2 r : h j ∈ R [ x 1 , . . . , x n ] ≤ d } . The dual cone to NN n , 2 d is the cone of moments of degree ≤ 2 d : 1 , x 1 x 2 , . . . , x 2 d n ) : λ ∈ R , x ∈ R n } . NN ◦ n , 2 d = conv { λ (1 , x 1 , . . . , x n , x 2 Cynthia Vinzant Convex Algebraic Geometry
Moments and Sums of Squares The cone of nonnegative polynomials NN n , 2 d = { p ∈ R [ x 1 , . . . , x n ] ≤ 2 d : p ( x ) ≥ 0 for all R n } is convex, semialgebraic, and contains the cone of sums of squares SOS n , 2 d = { h 2 1 + . . . + h 2 r : h j ∈ R [ x 1 , . . . , x n ] ≤ d } . The dual cone to NN n , 2 d is the cone of moments of degree ≤ 2 d : 1 , x 1 x 2 , . . . , x 2 d n ) : λ ∈ R , x ∈ R n } . NN ◦ n , 2 d = conv { λ (1 , x 1 , . . . , x n , x 2 n , 2 d ⊆ SOS ◦ Duality reverses inclusion, so NN ◦ n , 2 d . Cynthia Vinzant Convex Algebraic Geometry
Moments and Sums of Squares The cone of nonnegative polynomials NN n , 2 d = { p ∈ R [ x 1 , . . . , x n ] ≤ 2 d : p ( x ) ≥ 0 for all R n } is convex, semialgebraic, and contains the cone of sums of squares SOS n , 2 d = { h 2 1 + . . . + h 2 r : h j ∈ R [ x 1 , . . . , x n ] ≤ d } . The dual cone to NN n , 2 d is the cone of moments of degree ≤ 2 d : 1 , x 1 x 2 , . . . , x 2 d n ) : λ ∈ R , x ∈ R n } . NN ◦ n , 2 d = conv { λ (1 , x 1 , . . . , x n , x 2 n , 2 d ⊆ SOS ◦ Duality reverses inclusion, so NN ◦ n , 2 d . Moreover SOS ◦ n , 2 d is a spectrahedron! Cynthia Vinzant Convex Algebraic Geometry
Sums of squares and the Goemans-Williamson relaxation MAXCUT: Given weights w e ∈ R to the edges of a graph G = ( V , E ), find a cut V → {± 1 } maximizing the summed weight of mixed edges. max � w ij (1 − x i x j ) s.t. x 2 1 = x 2 2 = x 2 3 = 1 w 13 w 23 w 12 Cynthia Vinzant Convex Algebraic Geometry
Sums of squares and the Goemans-Williamson relaxation MAXCUT: Given weights w e ∈ R to the edges of a graph G = ( V , E ), find a cut V → {± 1 } maximizing the summed weight of mixed edges. max � w ij (1 − x i x j ) s.t. x 2 1 = x 2 2 = x 2 3 = 1 w 13 w 23 = max � w ij (1 − y ij ) s.t. y belongs to w 12 C = conv { ( x 1 x 2 , x 1 x 3 , x 2 x 3 ) : x ∈ {± 1 } 3 } Cynthia Vinzant Convex Algebraic Geometry
Sums of squares and the Goemans-Williamson relaxation MAXCUT: Given weights w e ∈ R to the edges of a graph G = ( V , E ), find a cut V → {± 1 } maximizing the summed weight of mixed edges. max � w ij (1 − x i x j ) s.t. x 2 1 = x 2 2 = x 2 3 = 1 w 13 w 23 = max � w ij (1 − y ij ) s.t. y belongs to w 12 C = conv { ( x 1 x 2 , x 1 x 3 , x 2 x 3 ) : x ∈ {± 1 } 3 } The dual convex body is � C ◦ = { ( a 12 , a 13 , a 23 ) : a ij x i x j ≤ 1 for x ∈ {± 1 } 3 } � a ij x i x j is SOS mod � x 2 ⊆ { ( a 12 , a 13 , a 23 ) : 1 − j − 1 �} = S Cynthia Vinzant Convex Algebraic Geometry
Sums of squares and the Goemans-Williamson relaxation MAXCUT: Given weights w e ∈ R to the edges of a graph G = ( V , E ), find a cut V → {± 1 } maximizing the summed weight of mixed edges. max � w ij (1 − x i x j ) s.t. x 2 1 = x 2 2 = x 2 3 = 1 w 13 w 23 = max � w ij (1 − y ij ) s.t. y belongs to w 12 C = conv { ( x 1 x 2 , x 1 x 3 , x 2 x 3 ) : x ∈ {± 1 } 3 } The dual convex body is � C ◦ = { ( a 12 , a 13 , a 23 ) : a ij x i x j ≤ 1 for x ∈ {± 1 } 3 } � a ij x i x j is SOS mod � x 2 ⊆ { ( a 12 , a 13 , a 23 ) : 1 − j − 1 �} = S Then C ⊆ S ◦ = Cynthia Vinzant Convex Algebraic Geometry
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