The Algebraic Degree of Semidefinite Programming

Transcription

1 The Algebraic Degree ofsemidefinite Programming p. The Algebraic Degree of Semidefinite Programming BERND STURMFELS UNIVERSITY OF CALIFORNIA, BERKELEY joint work with and Jiawang Nie (IMA Minneapolis) Kristian Ranestad (Oslo, Norway) with a recent contribution by Hans-Christian Graf von Bothmer (Hannover, Germany)

2 The Algebraic Degree ofsemidefinite Programming p. Our Question Semidefinite programming is a numerical method in convex optimization. SDP is very efficient, both in theory and in practice. SDP is widely used in engineering and the sciences. Input: Several symmetric n n-matrices Output: One symmetric n n-matrix What is the function from the input to the output? How does the solution depend on the data? Let s begin with three slides on context...

3 The Algebraic Degree ofsemidefinite Programming p. Linear Programming is semidefinite programming for diagonal matrices The optimal solution of Maximize c x subject to A x = b and x 0 is a piecewise linear function of c and b. It is a piecewise rational function in the entries of A. To study the function data solution, one needs geometric combinatorics, namely matroids for the dependence on A and secondary polytopes for b,c.

4 The Algebraic Degree ofsemidefinite Programming p. Universality of Nash Equilibria Consider the following problem in game theory: Given payoff matrices, compute the Nash equilibria. For two players, this is a piecewise linear problem. For three and more players, it is non-linear. Datta s Universality Theorem (2003): Every real algebraic variety is isomorphic to the set of Nash equilibria of some three-person game. Corollary: The coordinates of the Nash equilibria can be arbitrary algebraic functions of the payoff matrices.

5 The Algebraic Degree ofsemidefinite Programming p. Maximum Likelihood Degree An optimization problem in algebraic statistics: Maximize p 1 (θ) u 1 p 2(θ) u 2 p m(θ) u m subject to θ Θ R d The p i (θ) are polynomial that sum to one. The u i are positive integers (these are the data). The optimal solution ˆθ is the maximum likelihood estimator. It is an algebraic function of the data: (u 1,...,u m ) ˆθ(u 1,...,u m ) This defines the ML degree of the statistical model.

6 The Algebraic Degree ofsemidefinite Programming p. Semidefinite Programming Given: A matrix C and an m-dimensional affine subspace U of real symmetric n n-matrices SDP Problem: Maximize trace(c X) subject to X U and X 0. Here X 0 means that X is positive semidefinite. The problem is feasible if and only if the subspace U intersects the cone of positive semidefinite matrices. The optimal solution ˆX is a (piecewise) algebraic function of the matrix C and of the subspace U.

7 The Algebraic Degree ofsemidefinite Programming p. Helton-Vinnikov Curves Let m = 2 and n = 3. Then X 0 defines a semialgebraic convex region in U R 2. It is bounded by the elliptic curve {det(x) = 0}.

8 The Algebraic Degree ofsemidefinite Programming p. Cayley s Cubic Surface Let m = n = 3. The cubic surface det(x) = 0 is a Cayley cubic, with four singular points...

9 The Algebraic Degree ofsemidefinite Programming p. Analytic Solution Let m = n = 3. The cubic surface det(x) = 0 is a Cayley cubic. Its dual is a quartic Steiner surface. SDP: Maximize a linear function over the convex region X 0 bounded by the Cayley cubic. We can express the optimal solution ˆX in terms of radicals using Cardano s formula: Either ˆX has rank one and is one of the four singular points of the Cayley cubic, or ˆX has rank two and is found by intersecting the Steiner surface with a line.

10 The Algebraic Degree ofsemidefinite Programming p. 1 The Rank Inequalities We always assume that the data C and U are generic. Theorem 1. (Alizadeh Haeberly Overton 1997; Pataki 2000) The rank r of the solution ˆX to the SDP problem satisfies the two inequalities ( ) ( ) r + 1 n + 1 m 2 2 ( ) n r + 1 m. 2 For fixed m and n, all ranks r in the specified range are attained for an open set of instances (U,C).

11 The Algebraic Degree ofsemidefinite Programming p. 1 Distribution of the optimal rank n m rank percent rank percent rank percent rank percent % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %

12 The Algebraic Degree ofsemidefinite Programming p. 1 Algebraic Degree of SDP Suppose that m, n and r satisfy the rank inequalities. The degree δ(m,n,r) of the algebraic function (C, U) ˆX is the algebraic degree of SDP. Small Examples: δ(2,n,n 1) = n(n 1) δ(3, 3, 1) = δ(3, 3, 2) = 4 Bigger Example: δ(105, 20, 10) = Duality: δ(m,n,r) = δ( ( ) n+1 2 m, n,n r).

13 The Algebraic Degree ofsemidefinite Programming p. 1 m r degree r degree r degree r degree r degree

14 The Algebraic Degree ofsemidefinite Programming p. 1 Determinantal Varieties Consider the complex projective space PU P m. Let D r U denote the variety of all matrices of rank r. Theorem 2. The codimension of DU r is ( ) n r+1 2. If m > ( ) n r+1 2 then D r U is irreducible. The singular locus of D r U degree(d r U) = equals Dr 1 U n r 1 j=0, and ) ( n+j n r j ( 2j+1 j )

15 The Algebraic Degree ofsemidefinite Programming p. 1 Projective Duality Let PU denote the dual projective space to PU. The points in PU correspond to hyperplanes in PU. Any variety V PU has a dual variety V PU. V is the Zariski closure of the set of all hyperplanes in PU that are tangent to V at a smooth point.

16 The Algebraic Degree ofsemidefinite Programming p. 1 Projection is Dual to Section Usually V is a hypersurface, but some varieties V are degenerate, in the sense that codim(v ) 2. Lemma 3. Suppose V PU is degenerate and W = V H is a general hyperplane section. Then W is the projection of V from the point H in PU, and W has the same dimension and degree as V.

17 The Algebraic Degree ofsemidefinite Programming p. 1 Connecting the Dots Lemma 4. If m = ( ) n+1 2 then the projective dual of equals the complementary determinantal variety: D r U (D r U) = D n r U Theorem 5. The variety DU r is non-degenerate if and only if the rank inequalities hold. The algebraic degree of SDP is the degree of the dual hypersurface: δ(m,n,r) = degree (D r U)

18 The Algebraic Degree ofsemidefinite Programming p. 1 Rows of the Table A conjectured formula expresses each row as a polynomial in n. It is based on work of Pragacz on degeneracy loci and Schur Q-functions. Many cases have been proved, for instance: ( ) ( ) n + 2 n + 2 δ(6,n,n 3) = ( ) ( ) n + 3 n + 2 δ(7,n,n 3) = ( ) ( ) ( ) n+4 n+3 n+2 δ(8,n,n 3) =

19 The Algebraic Degree ofsemidefinite Programming p. 1 m r degree r degree r degree r degree r degree

20 The Algebraic Degree ofsemidefinite Programming p. 2 Columns of the Table Theorem 6. Let Q {r} be the variety of pairs (X,Y ) of symmetric n n-matrices with X Y = 0, rank(x) = r and rank(y ) = n r. The bidegree of Q {r} equals the generating function for the algebraic degree of semidefinite programming: C(Q {r} ;s,t) = ( n+1 2 ) m=0 δ(m,n,r) s (n+1 2 ) m t m. Setting s = t = 1 we get the scalar degree of Q {r} : C(Q {3} ; 1, 1) = = 40 C(Q {2} ; 1, 1) = = 122

21 The Algebraic Degree ofsemidefinite Programming p. 2 Check with Macaulay 2 R = QQ[x11,x12,x13,x14,x22,x23,x24,x33,x34,x44, y11,y12,y13,y14,y22,y23,y24,y33,y34,y44]; X = matrix {{x11, x12, x13, x14}, {x12, x22, x23, x24}, {x13, x23, x33, x34}, {x14, x24, x34, x44}}; Y = matrix {{y11, y12, y13, y14}, {y12, y22, y23, y24}, {y13, y23, y33, y34}, {y14, y24, y34, y44}}; Q3 = minors(1,x*y) + minors(4,x) + minors(2,y); codim Q3, degree Q3 (10, 40) Q2 = minors(1,x*y) + minors(3,x) + minors(3,y); codim Q2, degree Q2 (10, 122)

22 NEW Bothmer s Formula NEW Define a skew-symmetric matrix (ψ ij ) 0 i<j n by ψ 0j = 2 j 1 and ψ ij = j 1 k=i ( ) i + j 2 k For any subset I = {i 1,...,i r } of {1,...,n} let ψ I denote the sub-pfaffian of (ψ ij ) indexed by I if I is even and by I {0} if I is odd. Theorem 7. δ(m,n,r) = I ψ I ψ I c where I = {i 1,...,i r } runs over all r-subsets of {1, 2,...,n} with i i r = ( ) n+1 2 m. The Algebraic Degree ofsemidefinite Programming p. 2

23 Reminder on Genericity In this talk the subspace U was always generic. For special instances, the algebraic degree is smaller. Example: The 3-ellipse is a Helton-Vinnikov octic What s the degree of its dual? Hint: < δ(2, 8, 7) = 56. The Algebraic Degree ofsemidefinite Programming p. 2

24 The Algebraic Degree ofsemidefinite Programming p. 2 Three Conclusions Conclusion for Optimization Experts: Algebraic Geometry might be useful. Conclusion for Pure Mathematicians: Optimization might be interesting. Conclusion for Search Committees: Hire Jiawang.