Convex algebraic geometry, optimization and applications

Transcription

1 Convex algebraic geometry, optimization and applications The American Institute of Mathematics The following compilation of participant contributions is only intended as a lead-in to the AIM workshop Convex algebraic geometry, optimization and applications. This material is not for public distribution. Corrections and new material are welcomed and can be sent to workshops@aimath.org Version: Wed Sep 16 02:13:

2 2 Table of Contents A. Participant Contributions Ahmadi, Amir Ali 2. De Loera, Jesus 3. Gouveia, Joao 4. Greene, Jeremy 5. Klep, Igor 6. Kuhlmann, Salma 7. Lasserre, Jean 8. Laurent, Monique 9. Lim, Lek-Heng 10. Marshall, Murray 11. McCormick, Megan 12. Nahas, Joules 13. Nelson, Christopher 14. Nie, Jiawang 15. Omar, Mohamed 16. Powers, Victoria 17. Putinar, Mihai 18. Rostalski, Philipp 19. Scheiderer, Claus 20. Schweighofer, Markus 21. Sturmfels, Bernd 22. Verschelde, Jan 23. Vinnikov, Victor 24. Vinzant, Cynthia 25. Zhang, Hongchao

3 3 A.1 Ahmadi, Amir Ali Chapter A: Participant Contributions I am interested in the interplay between sum of squares and convexity of polynomials. In a recent paper joint with my advisor Pablo Parrilo, we presented the first example of a convex polynomial that is not sos-convex. Since then, we have made more progress: we currently have a complete characterization of dimensions and degrees in which there is no gap between convexity and sos-convexity. Our results include two new counterexamples that are in fact minimal (one for the degree and one for the dimension). I am interested in the following problem (which to the best of my knowledge is open): What is the complexity of deciding convexity of polynomials? Finally, I would like to make a conjecture: Every convex (or perhaps strictly convex) nonnegative polynomial is a sum of squares. It would be great to discuss some of these problems at the workshop. A.2 De Loera, Jesus My main interest is the use of algebraic and geometric techniques to create efficient optimization algorithms. For the past four years, in joint work with various co-authors, I have proved that such non-standard tools yield new results in integer Programming and Combinatorial optimization. I believe no one could have obtained such complexity results otherwise, namely good math is the foundation of good algorithmics. Moreover it just happens to be that I already published work on real algebraic geometry (Hilbert 17th and topology of real hypersurfaces). With this in mind I want to say that my contribution to the workshop will be asking lots of questions and aiming to apply the algebraic theorems to obtain new results in optimization. I feel certain the work of Gouveia, Thomas and Parrilo will serve to this purpose and I hope to exchange ideas with them and other participants. I am thrilled to attend a workshop on my main research interest. A.3 Gouveia, Joao I m interested in the interplay between sums of squares and convexity, namely in how to get sums of squares descriptions of the convexification of real varieties and the fundamental question of SDP representability of convex sets. I m also interested in applications of these tools to combinatorial optimization. A.4 Greene, Jeremy I am a graduate student at UCSD working with Bill Helton. I am interested in noncommutative polynomials. In particular, I study noncommutative plurisubharmonic (nc plush) polynomials. We call a polynomial nc plush if it has a nc complex hessian that is positive semidefinite when evaluated on all tuples of n n matrices for every size n. Bill Helton, Victor Vinnikov, and I have classified all nc plush polynomials as convex polynomials with an analytic change of variables; ie, a nc polynomial p is nc plush if and only if it has the form where f j, k j, F are all nc analytic. p = f T j f j + k j k T j + F + F T (1)

4 4 The arguments of Helton and Vinnikov are a special analysis of polynomials while mine analyzes a representation of nc quadratics. Also, I have shown that the usual Gram type representation for the nc complex hessian has a very rigid structure. Certainly I would like feedback on this result and discussions of nc changes of variables. A.5 Klep, Igor Consider the ring R s s endowed with the usual transposition of matrices, denoted here by. It is easy to see that for any A = A R s s, we have (A) A 0 iff A is a sum of hermitian squares; (B) A 0 iff there exist B i R s s such that i B i AB i 1 is a sum of hermitian squares; (C) tra = 0 iff A is a sum of commutators; (D) tra 0 iff A is a sum of hermitian squares and commutators. I am interested in the free polynomial analogs of these statements. Let f = f(x, X ) R X, X be a polynomial in noncommuting variables. For each tuple of matrices A of the same size there is a well-defined evaluation f(a, A ). The free questions corresponding to the items (A) (D) from above are: (A) free when is f(a, A ) 0 for all A? (B) free when is f(a, A ) 0 for all A? (C) free when is trf(a, A ) = 0 for all A? (D) free when is trf(a, A ) 0 for all A? (A) free and (C) free are well understood (and have exact parallels to (A) and (C), respectively) due to results of Helton & McCullough, and Schweighofer & K., respectively. On the other hand, the understanding of (B) free, and more importantly, (D) free, is still somewhat limited. There are examples of trace positive polynomials which are not sums of hermitian squares and commutators. As a concrete, warm up problem for (D) free consider the following: suppose f R X, X (one variable) is degree 4 and trace positive. Is it a sum of hermitian squares and commutators? If not, is there a natural type of identity certifying this property? A.6 Kuhlmann, Salma In the last decade, I worked on the representation of positive polynomials by sums of squares and on the multidimensional moment problem. In particular, I have looked at this problem in special situations, for example when the positivity domains or the positive polynomials under consideration have special (geometric or combinatorial) properties such as sparsity or invariance under group actions. My interest in this workshop is along these lines: in this context, the positive polynomials/ the basic closed semialgebraic sets under consideration have the special property of being convex. I share interest in the questions formulated by several other participants. My main expectation from the workshop is the opportunity to listen and get updated on state of the art. I will also use the opportunity to resume discussions with some of the participants. In particular, I would be curious to find out more about the following, in increasing order of priority: 1. Describe the closure of the cone of SOS polynomials in the various convex topologies of the polynomial ring. This would give a quantitative description of how these topologies differ.

5 2. This question arose in an correspondence with Cynthia Vinzant: SMP: A subset C of R[X] is SMP if every linear functional L on R[X] that is nonnegative on C comes integration over a borel measure on the closed semialgebraic set K. MP: A subset C of R[x] is MP if every linear functional L on R[X] that is non-negative on C comes integration over a Borel measure on R n. SMP implies MP, but there exist preorderings that satisfy MP but not SMP. Question: Are there closed semialgebraic sets which admit a MP description but not an SMP one? In general, if K S is a of subset K S, then a linear functional L nonnegative on T S comes from a measure supported on K S iff P os(k S ) is contained in the linear closure of the preordering T S. So the question can be reformulated : Assume that K=K S andpos(r n ) is a subset of T S. Can one find an S* s.t. K = K S and P os(k) is a subset of the linear closure of T S? This seems to be unlikely. 3. Let P be a preordering in R[X]. Study the following condition on P: (LT) for all g R[X] : g P = LT (g) P Here LT (g) := g d =the homogeneous component of g of degree d, where d is the total degree of g. This property can be reformulated in terms of the residue map of the -deg valuation on R(X). Main Example: if P is the preordering of PSD polynomials on a cone C, the P has the property LP. Questions: Characterize preorderings with this property, what is the relation between this property and other properties of a preordering (such as closure, moment problem, saturation)? Can one generalize this set-up to other valuations on R(X)? Can one generalize this set-up to other monomial orderings (and corresponding (LT) operation)? A.7 Lasserre, Jean I am interested in using various results of real algebraic geometry for solving global (polynomial) optimization problems, and more generally, for solving the generalized problem of moments with polynomial data. In particular, I am interested in the following issues: - Do (and how) some results of real algebraic geometry specialize in the presence of convexity? - In polynomial optimization, is the moment-sos approach more efficient in the presence of convexity? - Testing convexity of a basic semi-algebraic set, and algebraic characterization of convex basic semi-algebraic sets in terms of the defining polynomials - Parametric polynomial optimization - Algorithms for real solving of systems of polynomial equations A.8 Laurent, Monique Polynomial equations and real radical ideals: Consider an ideal I in the ring R[x 1,..., x n ] of multivariate polynomials. Some basic questions include: computing the complex variety V C (I) (assuming it is finite), as well as the radical ideal sqrti of this ideal I. These questions have been extensively studied and are now well understood. Their real analogues are however much less understood: compute the real variety V R (I), as well as the real radical R (I) of I. When the real variety is finite a method has been proposed to compute these objects (by Lasserre, Laurent, Rostalski) which is based on using moment matrices (Hankel operators) combined with semidefinite programming. We 5

6 6 propose to investigate whether this method can be extended to the positive dimensional case. Tools might include using the Noether normalization theorem, and it might be necessary to prove new results about the moment problem (e.g. extending the known flat extension results of Curto and Fialkow). The above is one of the research topics on which I would like to work and collaborate; of course I will gladly take part to discussions about other research topics, like matrix completion problems and statistics, or SDP hierarchies for semialgebraic sets and real varieties. A.9 Lim, Lek-Heng Consider the following problems: (1) given a collection of p m n matrices, what is the smallest collection of rank-1 matrices whose span contains the given collection?; (2) given a point in projective space P pmn, what is the smallest r so that this point sits on an r-secant to the Segre variety Seg(p, m, n)?; (3) given a homogeneous noncommutative polynomial in p + m + n variables modulo multilinear relations, what is the minimum number of products of linear forms required to reconstitute it? These problems are one and the same, namely, determining the rank of a tensor; and it is NP-hard in general. The tensor rank problem and its variants appear in various guises in different applications. Over the complex, it arises as an algebraic measure of entanglement in quantum computing. Its symmetric variant arises as independent component analysis in signal processing and as principal directions of the Nie-Sturmfels m-ellipses. Its convex variant arises in naive Bayes classifiers in machine learning and in Markov models for taxon trees in phylogenetics. Its tropical variant arises in a max-plus algebraic approach to discrete-event systems in control theory. Its operator variant arises in the approximation of Schrodinger operators by Kronecker products of lower-dimensional operators in quantum chemistry. My interest is in the linear, convex, multilinear, and multiconvex decompositions (and approximations) of tensors, with a view towards applications. Complex, real, convex, and tropical algebraic geometry play natural roles in the studies of such problems. For example, it is known that the two semialgebraic sets defined by the sign of the Cayley hyperdeterminant may be used to characterize such decompositions over the reals and that convex approximations in the cone of nonnegative tensors satisfy certain nice properties not shared by their real and complex counterparts. The computational intractability of many such problems and their natural formulation as polynomial optimization problems also lead one to consider convex relaxations along the lines of Lasserre-Parrilo. A.10 Marshall, Murray My interest is in polynomial optimization using methods exploiting the relationship between positive polynomials and sums of squares. In particular, I would like to understand how the situation simplifies when the polynomial in question is assumed to be convex. For example, is it true that every positive convex polynomial is a sum of squares of polynomials? J. Cimpric and I have worked together on this a bit, but with no success so far. This question is related to the question of whether or not a convex polynomial is SOS convex. This latter question was answered negatively just recently by A. Ahmadi and P. Parrilo.

7 A.11 McCormick, Megan I am a graduate student at the University of Washington, and recently I ve started working with Rekha Thomas on some material involving convexity and algebraic geometry problems. So far I have focused my efforts on small problems hoping to build a stronger background and understanding of Convex Algebraic Geometry. Now I would like to start working on larger problems. During this workshop I hope to continue building my background in this area and also find some interesting problems that I can consider as possible thesis problems. I m looking forward to working with other mathematicians in my area so that I can possibly build connections for future collaboration. A.12 Nahas, Joules Question Let G be a real connected, simply connected Lie group, g the corresponding Lie algebra and let p, q 1,... q n be elements in the enveloping algebra of g (for such algebras there is a positivstellensatz). For a Hilbert space H, let (U, H) denote a representation U of G into Aut(H), and du the corresponding derived representation. In analogy with the optimization problem for real polynomials, we can consider the following problem. Find with the constraints that in H, min (U,H,φ) φ, du(p)φ du(q i ) 0. Is is possible to use an approximation technique for this problem that is similar to the approximation technique for real polynomial optimization? Backround For p, q 1,..., q n R[x 1,..., x d ] consider the problem of finding with the constraints min x R dp(x). (2) q i (x) 0. In the paper Global Optimization with Polynomials and the Problem of Moments, Lasserre showed that the solution can be approximated by turning it into a moment problem and solving Linear matrix inequalities. In particular, finding (2) is the same as finding min µ p dµ, (3) where µ is a positive measure that satisfies q i dµ 0. (4) For a d-tuple α, treat the moments a α x α dµ as variables. Then we want to minimize (3) in a finite subset of the variables a α. The a α must obey a positivity condition in order for them to be moments of a measure. By solving (3) using an increasing number of the variables a α with the positivity constraint, we obtain better approximations to (2). 7

8 8 A.13 Nelson, Christopher I am a graduate student at UCSD. Recently I have been studying NC polynomial solutions of differential equations. The directional derivative of an NC polynomial in the direction h is defined as D[p(x 1,..., x g ), x i, h] := d dt [p(x 1,..., (x i + th),..., x g )] t=0. From this NC derivative, one may define differential equations which take as solutions noncommutative polynomials. For example, consider the differential equation g l f = 0 (5) x l i i=1 for some positive integer l. The corresponding NC differential equation is g L(f)[h] = D[... [D[f, x i, h]..., x i, h] = 0. (6) i=1 I have classified all solutions to (6). The basic building blocks to characterizing said solutions are homogeneous polynomials which are invariant under permutation that is, invariant under the action of S d defined by σ[x a1... x ad ] = x aσ(1)... x aσ(d). Any polynomial which is invariant under permutation and which has a commutative collapse satisfying (5) also satisfies (6). Further, if the polynomials p 1,..., p k are all invariant under permutation and have no variables in common, then the product p 1... p k is called an independent product. The independent product p 1... p k satisfies (6) if and only if each of p 1,..., p k do. Theorem. An NC polynomial p R x is a solution to (6) if and only if it is a sum of permutations of independent products, each of which satisfy (6). This work extends results of Helton, McAllaster, and Hernandez on NC harmonic and subharmonic polynomials, which classified solutions for the l = 2 case in two variables x 1 and x 2. My questions include (i): A subharmonic noncommutative polynomial is a polynomial p such that g i=1 D[D[p, x i, h], x i, h] is a positive polynomial. How might one classify all such subharmonic polynomials (in two variables all homogeneous subharmonics are already classified). (ii): What are the nice properties of these noncommutative solutions to differential equations? A.14 Nie, Jiawang The basic question in semidefinite programming theory is what convex sets are expressible by direct or lifted linear matrix inequality. Clearly a clear necessary condition is the sets must be convex and semialgebraic. Is there any other further necessary conditions? Helton and Nie found the following sufficient conditions: the defining polynomials are sos-convex; the convex semialgebraic set is compact and its boundary is nonsingular and positively curved. The gaps between these sufficient conditions and the known necessary

9 conditions are the singularity and the zero curvature. The technique of perspective transformation, which is equivalent to the blow-up in complex algebraic geometry, can resolved the singularity partially. Here are some open questions: Is every convex semialgebraic set is representable by lifted LMI? If the boundary of a convex set is singular, can we lift it to a convex set whose boundary is smooth? A.15 Omar, Mohamed I am a student of Jesus De Loera at UC Davis. I am interested in convex algebraic geometry, particularly its application to combinatorial problems such as the famous graph automorphism problem. One particular problem I would like to discuss is finding necessary and sufficient conditions on vectors a R n, b R such that the solution set of the knapsack {x {0, 1} n a T x = b} is Theta-1 exact (as defined by Gouveia, Parrilo and Thomas). At the workshop, I would like to speak to these authors and others about this problem. A.16 Powers, Victoria My interest in the workshop comes mainly from the real algebraic geometry side, specifically, I am interested in representations of positive polynomials on semialgebraic sets and related areas, especially the noncommutative generalizations and the development of noncommutative semialgebraic geometry. I hope to learn more about this at the workshop. I know very little about the applications side and would like to learn more about these as well. Most of the results about representations of positive polynomials involve certificates of positivity using sums of squares. This leads to questions about the existence of rational certificates of positivity: If the input polynomials have rational coefficients, does there always exist a certificate involving output data with rational coefficients. Such questions are important given the applications and the use of numerical and seminumerical algorithms. A basic question is the following: Given a rational polynomial which is a sums of squares of real polynomials, is it always a sum of squares of rational polynomials? Partial results have been obtained, but this is still an open question. A.17 Putinar, Mihai A Positivestellensatz for measurable functions will be derived via spectral analysis techniques. An application to the stability of delay systems will be discussed. A.18 Rostalski, Philipp Theoretical aspects. Recently it was shown by D. Henrion, 2009 that the convex hull of certain rationally parameterizable varieties is (lifted) semidefinite representable. This representation can be achieved by performing a standard moment relaxation as it is used, e.g., in semidefinite relaxations for global polynomial optimization problems, see J.B. Lasserre, Following D. Henrion we define 9 ζ 2d (x) = [x d 0, x d 1 0 x1, x d 1 0, x2,..., x d 2 x 2 1,..., x d m] and y = [y α ] α 2d as a real-valued sequence indexed by ζ 2d (x). Any homogeneous polynomial can then be written as p(x) = vec(p) T ζ 2d (x), where vec(p) T is the coefficient vector. For

10 10 any sequence y we can define the linear mapping Λ y (p) = vec(p) T y, and the moment matrix M d (y) with entries [M d (y)] α,β = y α+β for all α, β N m with α + β 2d. Now one may ask the following general question: Given an ideal I = (h 1,..., h m ) with a positive dimensional real variety (and possibly certain additional semialgebraic constraints), when and how can we characterize this real variety (semialgebraic set) by a projection of a semialgebraic set of the form {y M d (y) 0, A y = 0, where A is a constant matrix derived by requiring that Λ y vanishes on polynomials in the ideal I. In this respect, interesting research questions include: For what other class of (positive dimensional) real varieties and semi-algebraic sets can we represent the convex hull using (lifted) semidefinite representations? Can this representability be detected automatically, e.g., by observing the rank sequence of low-order moment relaxations? How can one achieve such a representation more direct (possibly using a different dual space basis)? Computational aspects. In practice moment relaxations are usually obtained by iterations on the relaxation order d, that is, low order relaxations are solved first and the relaxation order is increased one by one. P. Parrilo and K. Gatermann, 2004 showed how to exploit sparsity in the describing data when setting up this moment relaxations. It would be interesting to see what additional information, e.g., from previous relaxations one can use to decrease the computational burden. Last but not least, the question of what basis to choose for semidefinite relaxations in different applications, both for representing sums of squares as well as the corresponding dual space elements, is an interesting and highly relevant question. It should also allow to better exploit the low displacement rank structure in the semidefinite programming problem, which is now hidden in the quasi-hankel structure of M d (y) in the monomial basis. A.19 Scheiderer, Claus Here are two suggestions for discussions. I may write more at a later point. Rationality questions for convex sets. Given a representation of a convex set C R n by rational data, what can be said about the finite extensions K/Q for which C contains points with coordinates in K. Rational data could mean inequalities with rational coefficients, like for example an LMI or lifted LMI representation of C. Study of the sos cone in the polynomial ring R[x 1,..., x n ]. Description of its boundary and its faces. Related to this should be the study of the compact convex set of psd Gram matrices of a given polynomial. A.20 Schweighofer, Markus I am interested in describing convex semialgebraic sets by linear matrix inequalities (LMIs) with or without additional variables (semidefinite and LMI representations). There is the question of pure existence of the representations: It is easy to see that each spectrahedron (solution set of an LMI) is rigidly convex in the sense of Helton & Vinnikov, and that each semidefinitely representable set (linear image of a spectrahedron) is convex and semialgebraic. Recent seminal works of Helton & Vinnikov as well as of Helton & Nie show that the converses are true to a surprisingly big extent. Are they always true? If not, how to find counterexamples?

11 There is also the question of how to compute the representations when they are known to exist. For example, how to compute effectively the determinantal representation of real zero polynomials in two variables whose existence is guaranteed by the proven Lax conjecture? A.21 Sturmfels, Bernd Convex algebraic geometry is concerned with the geometry of real algebraic varieties and semi-algebraic sets that arise in convex optimization, especially in semidefinite programming. A fundamental problem is to study convex sets that arise as linear sections and projections of the cone of positive definite matrices. This problem arises naturally in mathematical statistics, namely, when studying maximum likelihood estimation in linear concentration models for Gaussian random variables. In particular, the issue of estimating a covariance matrix from the sufficient statistics can be seen as an extension of the familiar semidefinite matrix completion problem (Barrett et al. 1993; Grone et al. 1984). My current interest is to develop an algebraic and geometric framework for systematically addressing such problems. There are many open problems arising from the explorations in my recent paper with Caroline Uhler, titled Multivariate Gaussians, Semidefinite Matrix Completion, and Convex Algebraic Geometry. I d like to invite the workshop participants to look at (the pictures in) that paper. It is posted at Here is a concrete open question: What is the number of isolated singular points (= corank 2 matrices) on the boundary of a three-dimensional spectrahedron defined by an LMI of size n x n. An upper bound coming from algebraic geometry says that the number of such singular points is at most n For instance, for n = 3 this counts the four familiar vertices of the pillow. But for n 4 I have no idea whether this bound is tight, i.e. whether there exists a 3-dimensional spectrahedron of degree n that has n + 13 singular points. Note that this bound grows cubic in n, while the number vertices of a 3-dimensional polytope with n facets is only linear in n. Are spectrahedra in dimension three so much more complex than polytopes? A.22 Verschelde, Jan Real solutions of polynomial system may occur as isolated points on positive dimensional solution sets. Therefore they are often much harder to compute. Recently, in joint work with Kathy Piret, we developed a path following method to sweep a complex algebraic curve for singular solutions. In the development of a sweep for singularities on general positive dimensional algebraic surface, we would like to find the distance and the path from a generic point to the nearest singular solution, or to compute a certificate in case the surface would be free of singularities. A.23 Vinnikov, Victor My background is operator theory and system theory, as well as algebraic geometry, and more recently, noncommutative algebra. My specific interest in the workshop can be divided into several items: A. Applications of linear matrix inequalities (LMIs) to problems in systems and control. B. LMI representations of convex semialgebraic sets. C. Lifted LMI representations of convex semialgebraic sets. 11

12 12 D. LMI representations in the (free) noncommutative setting. For items 1 and 3, I am going to be mostly an active listener. For item 2, I expect to discuss my views and ideas about possible generalizations of the Lax conjecture to the higher dimensional case. For item 4, I expect to engage in a discussion regarding both recent progress and future directions in free noncommutative real algebraic geometry and related subjects. A.24 Vinzant, Cynthia My interests include the geometry of semidefinite representable sets, connections between real and tropical geometry, and applications of semidefinite programming to combinatorial optimization problems. A.25 Zhang, Hongchao Suppose f(x), x R n is a convex function and only its function values are available, i.e. given x R n, only the function value f(x) is provided. How to design an efficient algorithm to minimize the function f(x)? How to design a polynomial with degree at most two to approximate the f(x) in some sense? If using polynomial interpolation models to approximate f(x), what are the best interpolation points?