## THEMATIC PROGRAMS

June 26, 2022

### Short courses (in alphabetical order):

Peter Burgisser - Universitat Paderborn, and Felipe Cucker - City University of Hong Kong
Condition

1) Condition Numbers
We describe the origin of condition numbers in numerical analysis and the various ways they occur in both finite-precision analysis and complexity estimates.

2) Probabilistic Analyses of Condition Numbers
Condition numbers can not be efficiently estimated a priori. To eliminate their occurrence in complexity estimates one randomizes data and performs a probabilistic analysis. We describe the notions of average and smoothed analysis and exemplify them with the classical condition number of a matrix.

3) On a Problem Posed by Steve Smale
Condition numbers play a central role in computing zeros of complex polynomial systems ---a task proposed by Steve Smale for the mathematicians of the 21st century. We describe a randomized continuation algorithm and analyze its average and smoothed complexity. We also mention a result for a deterministic algorithm providing a near-solution to Smale's problem.

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Jean Pierre Dedieu - Universite de Toulouse
Complexity of Bezout's Theorem and the Condition Number

1. Complexity of path following methods in terms of the condition number
2. Probability aspects
3. Complexity and the condition metric

Our aim is to establish general principles for solving efficiently systems of polynomial equations. Very roughly speaking, algorithms for solving such systems can be divided into two classes: one algebraic, with methods based on symbolic computation, the other one based on numerical analysis, and approximation. In these lectures our interest focuses on this numerical approach.

In the first lecture we describe the general principles of continuation methods, and their numerical counterparts: the predictor-corrector algorithms. We estimate the complexity of such methods via the condition number analysis.

The second lecture is devoted to the probabilistic aspects of such questions. What can be said for random polynomial systems ? What is the distribution of the condition number ? What is the complexity of path-following methods on the average ?

In the third lecture we emphasize on a new approach, based on the condition metric (to be defined), and we list some open problems.

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Askold Khovanski - University of Toronto
An interplay between Algebraic Geometry and Convex Geometry

1. Intersecion theoy on on complete algebraic varieties
2. Algebraic equations and convex bodies

The famous Bernstein-Kushniremko theorem from the Newton Polyhedra Theory (usually it is considered as a part of the theory of Toric Varieties) relates algebraic geometry with the theory of mixed volumes. Recently Kiumars Kaveh and I have found a far-reaching generalization of this theorem to generic systems of algebraic equations on any algebraic variety. In the lectures I will review these results and their applications to Algebraic and Convex Geometry. In the first lecture I will discuss a "naive" intersection theory applicable even to non complete algebraic varieties. This intersection theory is birationally invariant an enjoys all the properties of mixed volumes of convex bodies. In the second lecture I will discuss Newton convex body --- a far-reaching generalization of Newton polyhedron. Its applications to Algebra include new theorems about growth of the Hilbert function for wide class of graded algebras, a new approach to semi-groups of integral points, a generalization of Fujita approximation theorem, a new version of Hodge index inequality. Applications to Geometry include a simple proof of the famous Alexandrov- Fenchel inequality from the theory of mixed volumes. I will use the Hilbert Theorem on the degree of projective variety from Algebraic Geometry and the isoperimetric inequality from Classical Geometry. No other special knowledge will be needed.

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Gregoire Lecerf - Universite de Versailles
Symbolic deformation techniques for polynomial system solving

1. Introduction, motivation and first concepts
2. Incremental solving
3. The Kronecker solver

Nowadays polynomial system solvers are involved in sophisticated computations in algebraic geometry as well as in practical engineering. The most popular algorithms produce Grobner bases, made of multivariate polynomials in dense representation. The major drawback of these bases, but more generally whenever multivariate polynomials coming from an elimination process are in dense representation, is the exponential growth of the number of the monomials involved to represent positive dimensional solution sets. The Kronecker solver is an alternative approach that uses evaluation data structures to represent the polynomials as the functions that compute their values at any given point. These talks are devoted to a self-contained presentation of this solver. The only prerequisites concern elementary facts about the Zariski topology, the primary decomposition of ideals, and the theory of modules over principal rings.

Part 1: introduction, motivation and first concepts.
We will start with an overview of the algorithm with a short history and motivating examples. Then we will define the dimension of an algebraically closed set, and describe an efficient probabilistic algorithm for computing it.

Part 2: incremental solving.
We will see how positive dimensional solution sets can be handled as if they were zero-dimensional. This leads to an efficient representation in terms of birational maps to hypersurfaces, and then to a constructive definition of the degree of an algebraic variety. In case of a reduced and regular sequence of polynomials we will be able to analyze in details the way it can be solved incrementally.

Part 3: the Kronecker solver.
The last talk will start with the algorithmic details of the solver in the regular case, with the cost analysis. We shall then present extensions regarding to the structure of the multiplicities and to thecomputation of the equidimensional decomposition in the general case.

We will finish with implementation issues and open problems.

References:
C. Durvye and G. Lecerf. A concise proof of the Kronecker polynomial system solver from scratch, Expositiones Mathematicae, 26(2), 2008.
M. Giusti, G. Lecerf, and B. Salvy. A Grobner free alternative for polynomial system solving. Journal of Complexity, 17(1), 2001.

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Renato Monteiro - GeorgiaTech
Algorithms for large scale structured optimization problems

1. Overview of algorithms for large scale structured optimization problems.
2. Interior-point algorithms for cone programming and generalizations
3. First-order methods for large scale structured optimization problems.

In this series of lectures, we will discuss efficient algorithms for solving large scale structured optimization problems. More specifically, we will first discuss interior-point algorithms and specific classes of convex optimization problems that can be solved by these methods. These methods are based on variants of Newton method and possess the theoretically nice property that they converge in polynomial time. However, due to its second-order nature, their iteration are not cheap and these methods can only solve small-to-medium size convex optimization problems. We next turn our attention to first-order methods which are based on variants of the (projected) gradient methods applied to suitable reformulations of the original convex optimization problem. Compared to the above second-order methods, first-order algorithms tend to perform a large number of iterations and are able to obtain only moderately accurate solutions but have the advantage that their iteration is substantially cheaper. For these reasons, the first-order methods seem to be the only alternative for solving large scale convex optimization problems. We will present several first-order variants and discuss their iteration complexity bounds. We also discuss variants which have performed well in practice but for which no complexity bound has been developed so far.

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### Talks (in alphabetical order)

Saugata Basu - Purdue University
Polynomial hierarchy, Betti numbers and a real analogue of Toda's theorem

Toda proved in 1989 that the (discrete) polynomial time hierarchy, $\mathbf{PH}$, is contained in the class $\mathbf{P}^{\#\mathbf{P}}$, namely the class of languages that can be decided by a Turing machine in polynomial time given access to an oracle with the power to compute a function in the counting complexity class $\#\mathbf{P}$. This result which illustrates the power of counting is considered to be a seminal result in computational complexity theory. An analogous result in the complexity theory over the reals (in the sense of Blum-Shub-Smale real Turing machines) has been missing so far. We formulate and prove a real analogue of Toda's theorem. Unlike Toda's proof in the discrete case, which relied on sophisticated combinatorial arguments, our proof is topological in nature.
(Joint work with Thierry Zell).

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Carlos Beltran - Universidad de Cantabria
Path-following methods for solving Smale's 17th problem. Recent progress and open questions

Path-following methods are the most used and successful numerical tools for solving systems of polynomial equations. They are interesting from the point of view of Complexity of Numerical Analysis, providing a probabilistic answer to Smale's 17th problem Can a zero of n complex polynomial equations in n unknowns be found approximately, on the average, in polynomial time with a uniform algorithm?''.

The study of the complexity of path-following methods leads to a number of challenges in fields as Geometric Measure Theory and non--smooth Analysis. In this talk I will present some recent progress on this topic and related open questions.

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Alexandre D'Aspremont - Princeton University
Tractable performance bounds for compressed sensing

Recent results in compressed sensing show that, under certain conditions, the sparsest solution to an underdetermined set of linear equations can be recovered by solving a linear program. These results either rely on computing sparse eigenvalues of the design matrix or on properties of its nullspace. So far, no exact tractable algorithm is known to test these conditions and most current results rely on asymptotic properties of sparse eigenvalues of random matrices. Given a matrix A, we use semidefinite relaxation techniques to test the nullspace property on A and show on some numerical examples that these relaxation bounds can prove perfect recovery of sparse solutions with relatively high cardinality.

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Marc Giusti - Ecole Polytechnique Palaiseau
On the geometry of polar varieties

(Joint work with Bernd Bank, Joos Heintz, Mohab Safey El Din and Eric Schost)
The problem addressed here is real root finding in algebraic varieties, with intrinsic complexity bounds. The results exposed form the geometrical main ingredients for the computational treatment of singular hypersurfaces. In particular,
- we show the non-emptiness of suitable fully generic dual polar varieties of (possibly singular) real varieties
- we show that fully generic polar varieties may become singular at smooth points of the original variety, and give a sufficient criterion when this is not the case
- we give an intrinsic degree estimate for polar varieties and introduce the new concept of a sufficiently generic polar variety.
Our statements are illustrated by examples and a computer experiment.

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Bill Helton - UC San Diego
Convex Matrix Inequalities vs Linear Matrix Inequalities
(slides)

When is semidefinite programming applicable? Namely, which problems can be expressed as a linear matrix inequality (LMI). Any such problem is necessarily convex, so the issue is: How much more restricted are LMIs than Convex Matrix Inequalities?

There are several main branches of this pursuit. First there are two fundamentally different classes of linear systems problems. Ones whose statements do depend on the dimension of the system "explicitly" and ones whose statements do not. Dimension dependent systems problems lead to traditional semialgebraic geometry. problems, while dimensionless systems problems lead directly to a new area which might be called noncommutative semialgebraic geometry. The classic results of control lead to what we are calling noncommutative problems.

A completely different tool is that of lifting a given convex set to one which has an LMI representation.

In this talk after laying out the distinctions above we give results and conjectures on the answer to the LMI vs convexity question.

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Myong-Hi Kim (SUNY at Old Westbury)
The average cost of one variable root finding polynomial

We analyze a path-lifting algorithm for finding an approximate zero of a complex polynomial, and show that for any polynomial with distinct roots in the unit disk, the average number of iterates this algorithm requires is universally bounded by a constant times the log of the condition number. In particular, this bound is independent of the degree d of the polynomial. The average is taken over initial values z with |z| = 1+1/d using uniform measure.

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Bernard Mourrain - INRIA Sophia Antipolis
Isolation of real roots of polynomial systems, complexity and condition number

The presentation is about subdivision algorithms for isolating the real roots of a system of multivariate polynomials. We shall describe two methods using either the Bernstein basis representation or the monomial basis representation. We connect these methods with continued fraction approximation of the coordinates of the real roots. The representation of the subdivided domains is done through homographies, which allows us to use only integer arithmetic and to treat efficiently unbounded regions. We use univariate bounding functions, projection and preconditioning techniques to reduce the domain of search. The resulting boxes have optimized rational coordinates,
corresponding to the first terms of the continued fraction expansion of the real roots.

We analyse the complexity of these methods in terms of geometric quantities associated with the roots of the system, such as their condition number. An extension of Vincent's theorem to multivariate polynomials is used for the termination of the algorithm. This yields new complexity bounds for real root isolation.

Joint work with A. Mantzaflaris and E. Tsigaridas.

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Marie-Francoise Roy - Universite de Rennes 1
Certificates of positivity in the Bernstein basis

Certificates of positivity (on the positive orthant, or on the simplex) are based on results by Polya and Bernstein. Bernstein polynomials, for example, are positive on the simplex, and a positive combination of them is positive as well. A partial reciprocal is true, by increasing the degree of the Bernstein, or by subdividing the simplex. This gives two different kinds of certificates, the approach by subdivision being much more efficient.

These certificates are not uniform, since the size of the certificate output depends on the bitsize of the entries.

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Andrew Sommese - University of Notre Dame
Zebra Fish, Tumor Growth, and Algebraic Geometry

Problems of central importance in engineering and science often lead to systems of partial differential equations, for which the only hope of solution is to compute numerical solutions. Often the systems are intrinsically nonlinear with several solutions corresponding to the same set of physical conditions. Discretizations of such systems of differential equations often lead to large systems of polynomial equations whose solutions correspond to potential solutions of the system of differential equations. These naturally arising polynomial systems are well beyond the pale of systems previously investigated in numerical algebraic geometry. This talk will describe some of the recent work of Wenrui Hao, Jonathan Hauenstein, Bei Hu, Yuan Liu, Yong-Tao Zhang, and myself in successfully solving such systems. First I will give an overview of homotopy continuation methods and what they allow us to compute.

Second, I will discuss two applications to numerical solution of partial differential equations. The first is our work on finding steady state solutions of a reaction-diffusion model on zebrafish dorsal-ventral patterning. Here, using a new approach we found seven solutions with the boundary conditions, of which three are stable: only stable solutions have meaning in the physical world. The second is our work on the solution of several models for tumor growth. Here the boundary of the tumor is the most important part of the solution: the goal is to solve this free boundary problem as mu, a parameter of the model called the "tumor aggressiveness factor," varies. There is a family of easily computed radially symmetric solutions, which, for certain discrete values of mu, meets branches of solutions that are not radially symmetric away from the point where the branches meet. The problem is to compute the solutions on the nonradial branch. There are no standard ways of solving this sort of free boundary problem for any but very small distances from the radial solutions. Our successful solution of this problem required a new approach that came to grips with some of the numerical algebraic geometry underlying systems of several thousand polynomials in a like number of variables. Finally, I will also discuss the direction this research is going.

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Paul Tseng - Washington University
Problem reformulations and first-order gradient methods for stuctured convex optimization

We study how problem reformulations, say, using duality or regularization, affect the computational complexity of first-order gradient methods for large-scale structured convex optimization. Relevant applications include image denoising and multi-task learning.

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Joris van der Hoeven - Universit'e Paris-Sud
On the art of guessing

Current computer algebra systems usually work in a shell mode: the user asks a question and the system hopefully gives an answer. To what extent would it be possible to discover additional mathematical structure in the s problem in an automatic fashion ? For instance, if we encounter 2.094395102393195492308428922 in the output of a numerical computation, the system might suggest its replacement by 2/3. Even though guessing such additonal relations was not explicitly requested by the user, it may lead to interesting insights and does not necessarily lead to a big increase of the overall computation time. In our talk, we will address several classical guessing algorithms and present a few new ones. We will start with rational number and function recovery, the LLL-algorithm and Pade-Hermite approximation. We will next turn our attention to guessing possible asymptotic expansions for sequences and possible relations at singularities of analytic functions. If time permits it, we will also discuss some guessing techniques for symbolic expressions.

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Yinyu Ye - Stanford University
On the complexity of L-p norm minimization for p less than 1

Recently, L-p norm minimization for p less than 1 is used to compute a sparse solution for a system of linear equations. We show that the problem is NP-hard for any 0<p<1. On the positive side, we show that the interior-point affine scaling algorithm works well to compute a local minimizer. Computational results indicate that the model produces solutions sparser than that from the L-1 minimization model.