10:00am - 12:00pmPolynomial optimization and its applications
Chair(s): Timo de Wolff (Technische Universität Berlin, Germany), Simone Naldi (Université de Limoges, France), João Gouveia (Universidade de Coimbra, Portugal)
The importance of polynomial (aka semi-algebraic) optimization is highlighted by the large number of its interactions with different research domains of mathematical sciences. These include, but are not limited to, automatic control, combinatorics, and quantum information. The mini-symposium will focus on the development of methods and algorithms dedicated to the general polynomial optimization problem. Both the theoretical and more applicative viewpoints will be covered.
(25 minutes for each presentation, including questions, followed by a 5-minute break; in case of x<4 talks, the first x slots are used unless indicated otherwise)
The Geometry of SDP-Exactness in Quadratic Optimization
Diego Cifuentes1, Corey Harris2, Bernd Sturmfels3
1MIT, Cambridge, MA, USA, 2University of Oslo, Norway, 3MPI Leipzig, Germany
Consider the problem of minimizing a quadratic objective subject to quadratic equations. We study the semialgebraic region of objective functions for which this problem is solved by its semidefinite relaxation. For the Euclidean distance problem, this is a bundle of spectrahedral shadows surrounding the given variety. We characterize the algebraic boundary of this region and we derive a formula for its degree. This is joint work with Corey Harris and Bernd Sturmfels.
Semidefinite representations of the set of separable states
Hamza Fawzi
Cambridge University, United Kingdom
The convex set of separable states plays a fundamental role in quantum information theory and corresponds to the set of non-entangled states. In this talk I will discuss the question of (exact) semidefinite representations for this convex set. Using connections with nonnegative polynomials and sums of squares I will characterize the cases when this set has, or not, an SDP representation.
Noncommutative polynomial optimization and quantum graph parameters
Sander Gribling1, David de Laat2, Monique Laurent1
1CWI, Amsterdam, The Netherlands, 2Emory University, Atlanta, GA, USA
Graph parameters such as the stability number and chromatic number can be formulated in several ways. For example as polynomial optimization problems or using nonlocal games (in which two separated parties must convince a referee that they have a valid stable set/coloring of the graph of a certain size). After recalling these formulations, we show how they can be used in quantum information theory to study the power of entanglement. The formulation in terms of nonlocal games gives rise to quantum versions of these graph parameters. The polynomial optimization perspective provides hierarchies of semidefinite programming bounds on the classical parameters and we show how the framework of noncommutative polynomial optimization can be used to obtain analogous hierarchies on the quantum graph parameters. This approach unifies several existing bounds on the quantum graph parameters.
On Convexity of Polynomials over a Box
Georgina Hall1, Amir Ali Ahmadi2, Mihaela Curmei3
1INSEAD, Paris, France, 2Princeton University, NJ, USA, 3Microsoft
In the first and main part of this talk, I show that unless P=NP, there exists no polynomial time (or even pseudo-polynomial time) algorithm that can test whether a cubic polynomial is convex over a box. This result is minimal in the degree of the polynomial and in some sense justifies why convexity detection in nonlinear optimization solvers is limited to quadratic functions or functions with special structure. As a byproduct, the proof shows that the problem of testing whether all matrices in an interval family are positive semidefinite is strongly NP-hard. This problem, which was previously shown to be (weakly) NP-hard by Nemirovski, is of independent interest in the theory of robust control. I will explain the differences between weak and strong NP-hardness clearly and show how our proof bypasses a step in Nemirovski's reduction that involves "matrix inversion". Indeed, while this operation takes polynomial time, it can result in an exponential increase in the numerical value of the rational numbers involved.
In the second and shorter part of the talk, I present sum-of-squares-based semidefinite relaxations for detecting or imposing convexity of polynomials over a box. I do this in the context of the convex regression problem in statistics. I also show the power of this semidefinite relaxation in approximating any twice continuously differentiable function that is convex over a box.