Date: 02 October 2025
Venue: Room 301, A5, Institute of Mathematics

9:00 - 10:00: Jiao Liguo (Northeast Normal University, China)
Title: Optimality conditions at infinity in semialgebraic vector optimization

Abstract:  In this talk, we establish optimality conditions for semialgebraic vector optimization problems with a situation in which generalized (weakly) nondominated points do exist but are not attained as image points of (weakly) efficient solutions. To this end, cones associated with unbounded semialgebraic sets at infinity are introduced and studied. Then optimality conditions at infinity in terms of the Newton polyhedra of the objective mappings and of the cones associated with the constraint sets at infinity for the problems in question are proposed.

10:30-11:30: Jean Bernard Lasserre (LAAS-CNRS & Toulouse School of Economics, University of Toulouse, France)
Title: The Christoffel function: Some applications, connections and extensions

Abstract: Even though the Christoffel function (CF) is well-known in approximation theory and orthogonal polynomials, it is only recently that some of its remarkable properties have been shown to be useful in some other applications, like data analysis and mining (e.g. for outlier detection and support inference), and approximation of possibly discontinuous functions with no Gibbs phenomenon. So in this talk we will briefly introduce the CF and describe how some of its main features can be exploited in some applications. Moreover we will also describe connections of the CF with seemingly unrelated fields, like positive polynomials, Pell's equation and equilibrium measure of compact sets, and if time permits, we will introduce some variants with interesting additional properties.

12:00-14:00: Lunch

14:00-15:00:  Vũ Trung Hiếu (RIKEN Center for Advanced Intelligence Project, University of Tokyo, Japan)
Title: Deciding non-negativity of generic polynomials based on Sturm’s theorem

Abstract: We address the problem of deciding the non-negativity of a real polynomial $f$. To this end, we employ the critical value polynomial $\varphi_f,$ whose roots are the complex critical values of $f$. First, we show that, under a genericity assumption, this decision problem reduces to determining whether $\varphi_f$ has only non-negative roots. This reduction allows us to apply Sturm’s theorem to resolve the problem. Second, we propose a symbolic algorithm for deciding non-negativity and analyze its bit complexity in the case where the input polynomial has rational coefficients. The talk is based on joint work with Nguyen Hong Duc and Akiko Takeda.

15:30-16:30:  Jae Hyoung Lee (Pukyong National University, Korea)
Title: A bounded degree hierarchy with SOS relaxations for classes of polynomial optimization problems

Abstract: In this talk, we consider separable plus lower degree (SPLD) polynomials, by which we mean polynomials that have the decomposition of the sum of univariate polynomials (in different variables) and a polynomial whose degree is lower than the one of the separable polynomial. A type of bounded degree SOS hierarchy, referred to as BSOS-SPLD, is proposed to efficiently solve optimization problems involving SPLD polynomials. Numerical experiments on several benchmark problems indicate that the proposed method yields better performance than the standard bounded degree SOS hierarchy (Lasserre et al. in EURO J Comput Optim 5:87–117, 2017). An exact SOS relaxation for a class of convex SPLD polynomial optimization problems is also proposed. Finally, we present an application of SPLD polynomials to convex polynomial regression problems arising in statistics.