Student Colloquium


DATE2019-12-10 12:00-13:00

PLACE數學館3176教室

SPEAKER朱雅琪 同學(成功大學數學系

TITLECheckable Conditions for Convexity of Joint Numerical Range of Two Quadratic Functions

ABSTRACT Given two quadratic functions $f(x) = x^T A x + a^T x + a_0$ and $g(x) = x^T B x + b^T x + b_0$ defined on $\mathbb{R}^n$, the convexity of the joint numerical range $C = \{\left(f(x),g(x)\right) | x \in \mathbb{R}^n\}$ is a useful tool for solving quadratic optimization problems. Although the necessary and sufficient conditions for the convexity of $C$ has been characterized by Flores-Bazán and Felipe Opazo (Minimax Theory Appl. 1, 2016), their conditions are still some distance away from being a checkable procedure for implementation. In this talk, by transforming the problem of the convexity of $C$ into a geometrical problem that checks whether there exist two constants $\alpha,~\beta$ for the level set $\{f=\alpha\}$ to separate $\{g=\beta\},$ we obtain a different set of necessary and sufficient conditions from that of Flores-Bazán and Felipe Opazo. The new insight enables us to develop a checkable procedure for determining the convexity of $C$. It requires to check whether $A$ has exactly one negative eigenvalue; whether $V^TAV$ is positive semi-definite with $V$ being the null space basis of $b$; and finally to check whether a system of $3n-1$ linear equations of $3n$ variables has a solution. All the above steps can be done by computers with a polynomial-time bound complexity. Examples are provided to illustrate the related geometrical pictures as well as the computation.
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