Related papers: The Saddle Point Problem of Polynomials
Polynomial system solving arises in many application areas to model non-linear geometric properties. In such settings, polynomial systems may come with degeneration which the end-user wants to exclude from the solution set. The…
In this paper we study various approaches for exploiting symmetries in polynomial optimization problems within the framework of semi definite programming relaxations. Our special focus is on constrained problems especially when the…
We study the problem of enumerating Tarski fixed points on finite lattices. We derive query complexity lower bounds for finding three or more Tarski fixed points of isotone maps and the subclasses of increasing and decreasing isotone maps.…
We analyse and compare the complexity of several algorithms for computing modular polynomials. We show that an algorithm relying on floating point evaluation of modular functions and on interpolation, which has received little attention in…
To compute solutions of sparse polynomial systems efficiently we have to exploit the structure of their Newton polytopes. While the application of polyhedral methods naturally excludes solutions with zero components, an irreducible…
Recent advances in symbolic dynamic programming (SDP) combined with the extended algebraic decision diagram (XADD) data structure have provided exact solutions for mixed discrete and continuous (hybrid) MDPs with piecewise linear dynamics…
We give a simple algorithm that determines whether a given post-critically finite topological polynomial is Thurston equivalent to a polynomial. If it is, the algorithm produces the Hubbard tree; otherwise, the algorithm produces the…
The goal of this paper is to provide computational tools able to find a solution of a system of polynomial inequalities. The set of inequalities is reformulated as a system of polynomial equations. Three different methods, two of which…
This article establishes a method to answer a finite set of linear queries on a given dataset while ensuring differential privacy. To achieve this, we formulate the corresponding task as a saddle-point problem, i.e. an optimization problem…
We propose an algorithm for determining the irreducible polynomials over finite fields, based on the use of the companion matrix of polynomials and the generalized Jordan normal form of square matrices.
High-index saddle dynamics (HiSD) is an effective approach for computing saddle points of a prescribed Morse index and constructing solution landscapes for complex nonlinear systems. However, for problems with ill-conditioned Hessians…
We consider the problem of approximating numerically the moments and the supports of measures which are invariant with respect to the dynamics of continuous- and discrete-time polynomial systems, under semialgebraic set constraints. First,…
Many problems of systems control theory boil down to solving polynomial equations, polynomial inequalities or polyomial differential equations. Recent advances in convex optimization and real algebraic geometry can be combined to generate…
In this paper, we extend the inexact Uzawa algorithm in [Q. Hu, J. Zou, SIAM J. Matrix Anal., 23(2001), pp. 317-338] to the nonsymmetric generalized saddle point problem. The techniques used here are similar to those in [Bramble \emph{et…
This paper studies convex Generalized Nash Equilibrium Problems (GNEPs) that are given by polynomials. We use rational and parametric expressions for Lagrange multipliers to formulate efficient polynomial optimization for computing…
Large-scale saddle-point problems arise in such machine learning tasks as GANs and linear models with affine constraints. In this paper, we study distributed saddle-point problems (SPP) with strongly-convex-strongly-concave smooth…
We propose general notions to deal with large scale polynomial optimization problems and demonstrate their efficiency on a key industrial problem of the twenty first century, namely the optimal power flow problem. These notions enable us to…
We develop a new tool, namely polynomial and linear algebraic methods, for studying systems of word equations. We illustrate its usefulness by giving essentially simpler proofs of several hard problems. At the same time we prove extensions…
This paper proposes a bilevel hierarchy of strengthened complex moment relaxations for complex polynomial optimization. The key trick entails considering a class of positive semidefinite conditions that arise naturally in characterizing the…
The paper deals with equations driven by a non-local integrodifferential operator $\mathcal L_K$ with homogeneous Dirichlet boundary conditions. These equations have a variational structure and we find a solution for them using the Saddle…