Positive polynomials on unbounded domains

多项式优化问题的无界性证明证书

A Certificate of Unboundedness for Polynomial Optimization Problems

本文提出一种预处理算法，用于判断多项式优化问题是否从下无界，从而在求解前提供关于问题渐进行几何的信息。

无界集合上的稀疏多项式优化

Sparse Polynomial Optimization with Unbounded Sets

针对无界集合上的稀疏多项式优化问题，提出带扰动的稀疏齐次化Moment-SOS层级方法，证明其渐近收敛性，给出两种消去扰动的替代稀疏层级，得到新的稀疏正多项式正性定理，数值实验验证其可处理数千变量问题，还可应用于轨迹优化。

多项式优化中无界集的均质化

Homogenization for polynomial optimization with unbounded sets

本文研究了无界集的多项式优化问题，提出均质化方法和SOS松弛层次，证明在特定条件下该层次具有有限收敛性，并扩展了Putinar-Vasilescu类型的正定定理。

Minimum Restraint Functions for unbounded dynamics: general and control-polynomial systems

We consider an exit-time minimum problem with a running cost, $l\geq 0$ and unbounded controls. The occurrence of points where $l=0$ can be regarded as a transversality loss. Furthermore, since controls range over unbounded sets, the family of admissible trajectories may lack important compactness properties. In the first part of the paper we show that the existence of a $p_0$-minimum restraint function provides not only global asymptotic controllability (despite non-transversality) but also a state-dependent upper bound for the value function (provided $p_0>0$). This extends to unbounded dyna…

Minimizing Cubic and Homogeneous Polynomials over Integers in the Plane

We complete the complexity classification by degree of minimizing a polynomial over the integer points in a polyhedron in $\mathbb{R}^2$. Previous work shows that optimizing a quadratic polynomial over the integer points in a polyhedral region in $\mathbb{R}^2$ can be done in polynomial time, while optimizing a quartic polynomial in the same type of region is NP-hard. We close the gap by showing that this problem can be solved in polynomial time for cubic polynomials. Furthermore, we show that the problem of minimizing a homogeneous polynomial of any fixed degree over the integer points in a b…

GAC, savings, and unbounded inputs

Let a control system and a target be given on an open subset of an Euclidean space. The existence of a Control Lyapunov Function - namely a positive definite, semiconcave, solution of the Hamilton-Jacobi inequality corresponding to the control vector field -- guarantees Global Asymptotic Controllability (GAC). In this case, however, minimization is not an issue. Instead, if a Lagrangean with non-negative values is considered as well, an optimal control problem can be defined in relation to the corresponding integral functional. In the first part of the present paper we show that the existence …