Session: Systems of Quadratic Equations/Inequalities
Chair: Ruey-Lin Sheu
Cluster:
Talk 1: On the implementation issue of the non-homogeneous strict Finsler and the non-homogeneous Calabi theorems
Speaker: Min-Chi Wang
Abstract: Let f(x) and g(x) be two real quadratic functions defined on ℝⁿ. The existence of solutions to the system of (in)equality constraints [f(x) = 0, g(x) = 0] and [f(x) = 0, g(x) ≤ 0] has rarely been studied in the literature. Recently, new results have emerged, called the non-homogeneous strict Finsler Lemma and the non-homogeneous Calabi Theorem, which provide necessary and sufficient conditions for the solvability of the above two systems of (in)equality constraints. This talk aims to simplify the conditions to facilitate easier implementation. Numerical results show that our proposed method is efficient in computation.
Talk 2: A QP1QC approach for deciding whether or not two quadratic surfaces intersect
Speaker: Ting-Tsen Lin
Abstract: Given two $n$-variate quadratic functions $f(x)=x^TAx+2a^Tx+a_0, g(x)=x^TBx+2b^Tx+b_0$, we are interested in knowing whether or not the two hypersurfaces $\{x\in \mathbb{R}^n: f(x)=0\}$ and $\{x\in \mathbb{R}^n: g(x)=0\}$ intersect with each other. There are two aspects to look at this problem. In one respect, the famous Finsler-Calabi theorem (1936-1964) asserts that, if $n\ge3$ and $f,g$ are quadratic forms, $f=g=0$ has no common solution other than the trivial one $x=0$ if and only if there exists a positive definite matrix pencil $\alpha A+\beta B\succ0.$ The result is in general not true for non-homogeneous quadratic functions. On the other hand, Levin (c. late 1970) tried to directly solve the intersection curve of $\{x\in \mathbb{R}^n: f(x)=0\}$ and $\{x\in \mathbb{R}^n: g(x)=0\},$ but it turned out to be way too ambitious. In this paper, we show that, by incorporating with the information about the unboundedness and the un-attainability of several (at most 4) quadratic programming problems with one single quadratic constraint (QP1QC), the answer as to whether or not $f(x)=x^TAx+2a^Tx+a_0=0$ and $g(x)=x^TBx+2b^Tx+b_0=0$ intersect can be successfully determined. References: E. Calabi, Linear systems of real quadratic forms, Proceedings of the American Mathematical Society, 15 (1964), pp. 844–846. P. Finsler, Über das Vorkommen definiter und semidefiniter Formen in Scharen quadratischer Formen, Commentarii Mathematici Helvetici, 9 (1936), pp. 188–192. J. Z. Levin, A parametric algorithm for drawing pictures of solid objects composed of quadric surfaces, Communications of the ACM, 19 (1976), pp. 555–563. J. Z. Levin, Mathematical models for determining the intersections of quadric surfaces, Computer Graphics and Image Processing, 11 (1979), pp. 73–87.
Talk 3: Last two pieces of puzzle for un-solvability of a system of two quadratic (in)equalities
Speaker: Ruey-Lin Sheu
Abstract: Given two quadratic functions f(x) = x^{T}Ax+2a^{T}x+a_0 and g(x) = x^{T}Bx+ 2b^{T }x+b_0, each associated with either the strict inequality (< 0); non-strict inequality (≤ 0); or the equality (= 0), it is a fundamental question to ask whether or not the joint system {x ∈ R^n: f(x) ⋆ 0} and {x ∈ R^n: g(x)#0}, where ⋆ and # can be any of {
