We prove that if $$q_1,\dots ,q_m:{\mathbb {R}}^n \rightarrow {\mathbb {R}}$$ are quadratic forms in variables $$x_1,\dots ,x_n$$ such each $$q_k$$ depends on at most r and has common with other forms, then the average value of product $$(1+q_1)\cdots (1+q_m)$$ respect to standard Gaussian measure $${\mathbb {R}}^n$$ can be approximated within relative error $$\epsilon >0$$ quasi-polynomial $$n...

If f is a positive definite form, Reznick’s Positivstellensatz states that there exists [Formula: see text] such sum of squares polynomials. Assuming can be written as forms text], where each l depends on subset the initial variables, and assuming these subsets satisfy so-called running intersection property, we provide sparse version Positivstellensatz. Namely, σ polynomials, H uniform polynom...

Abstract We show that adjustable robust linear programs with affinely box data uncertainties under separable polynomial decision rules admit exact sums of squares (SOS) reformulations. These problems share the same optimal values and a one-to-one correspondence between solutions. A sum representation non-negativity non-convex over plays key role in reformulation. This reformulation allows us to...

(1) f = det(xM1 + yM2 + zM3), where M1,M2,M3 are Hermitian d × d matrices. The representation is definite if there is a point e ∈ R for which the matrix e1M1+e2M2+e3M3 is positive definite. This imposes an immediate condition on the projective curve VC(f). Because the eigenvalues of a Hermitian matrix are real, every real line passing through e meets this hypersurface in only real points. A pol...

The discrete-time bounded-real lemma for nonminimal discrete systems is presented. Based on this lemma, rigorous necessary and sufficient conditions for the existence of positive definite solutions to the Lyapunov equation for n-dimensional (n-D) digital systems are proposed. These new conditions can be applied to n-D digital systems with n-D characteristic polynomials involving factor polynomi...

Kernel-based data mining algorithms, such as Support Vector Machines, project data into high-dimensional feature spaces, wherein linear decision surfaces correspond to non-linear decision surfaces in the original feature space. Choosing a kernel amounts to choosing a high-dimensional feature space, and is thus a crucial step in the data mining process. Despite this fact, and as a result of the ...

In this paper we propose a novel semidefinite programming based method to compute robust domains of attraction for state-constrained perturbed polynomial systems. A domain is set states such that every trajectory starting from it will approach an equilibrium while never violating specified state constraint, regardless the actual perturbation. The semi-definite program constructed by relaxing ge...

Let ${\mathcal A}$ be a generalized $q$-Weyl algebra, it is generated by $u$, $v$, $Z$, $Z^{-1}$ with relations $ZuZ^{-1}=q^2u$, $ZvZ^{-1}=q^{-2}v$, $uv=P\big(q^{-1}Z\big)$, $vu=P(qZ)$, where $P$ Laurent polynomial. A Hermitian form $(\cdot,\cdot)$ on called invariant if $(Za,b)=\big(a,bZ^{-1}\big)$, $(ua,b)=(a,sbv)$, $(va,b)=\big(a,s^{-1}bu\big)$ for some $s\in {\mathbb C}$ $|s|=1$ and all $a,...

This work is concerned with finding a global optimization technique for a broad class of nonlinear optimization problems, including quadratic and polynomial optimization problems. The main objective of this paper is to investigate how the (hidden) structure of a given real/complexvalued optimization problemmakes it easy to solve. To this end, three conic relaxations are proposed. Necessary and ...

This paper presents an information retrieval system for the NTCIR-7 information retrieval for question answering task. This system is composed by three parts: (1) Query processing (2) Retrieval model (3) Re-rank module. Query processing filters stop-words and selects query terms to generate a required term set for further retrieval. Threes retrieval models from two famous retrieval systems are ...