Given a polynomial function $f \colon \mathbb{R}^n \rightarrow \mathbb{R}$ and a unbounded basic closed semi-algebraic set $S \subset \mathbb{R}^n,$ in this paper we show that the conditions listed below are characterized exactly in terms of the so-called {\em tangency variety} of $f$ on $S$: (i) The $f$ is bounded from below on $S;$ (ii) The … Read more
Crouzeix’s conjecture states that for all polynomials p and matrices A, the inequality ||p(A)||
This paper is devoted to study the Lipschitzian/Holderian type global error bound for systems of many finitely convex polynomial inequalities over a polyhedral constraint. Firstly, for systems of this type, we show that under a suitable asymtotic qualification condition, the Lipschitzian type global error bound property is equivalent to the Abadie qualification condition, in particular, … Read more
Let $p(x_1,…,x_m) = \sum_{r_1 + \cdots + r_m = n} a_{r_1,…,r_m} \prod_{1 \leq i \leq m } x_i^{r_{i}}$ be a homogeneous polynomial of degree $n$ in $m$ variables. We call such polynomial {\bf H-Stable} if $p(z_1,…,z_m) \neq 0$ provided that the real parts $Re(z_i) > 0: 1 \leq i \leq m$. It can be assumed … Read more
We give a detailed numerical and theoretical analysis of a stabilization problem posed by V. Blondel in 1994. Our approach illustrates the effectiveness of a new gradient sampling algorithm for finding local optimizers of nonsmooth, nonconvex optimization problems arising in control, as well as the power of nonsmooth analysis for understanding variational problems involving polynomial … Read more
All isolated solutions of the cyclic-n polynomial equations are not known for larger dimensions than 11. We exploit two types of symmetric structures in the cyclic-n polynomial to compute all isolated nonsingular solutions of the equations efficiently by the polyhedral homotopy continuation method and to verify the correctness of the generated approximate solutions. Numerical results … Read more