On the convexity for the range set of two quadratic functions
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866913715205439488 |
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| author | Nguyen, Huu-Quang Chu, Ya-Chi Sheu, Ruey-Lin |
| author_facet | Nguyen, Huu-Quang Chu, Ya-Chi Sheu, Ruey-Lin |
| contents | Given $n\times n$ symmetric matrices $A$ and $B$, Dines in 1941 proved that the joint range set $\{(x^TAx,x^TBx)|~x\in\mathbb{R}^n\}$ is always convex. Our paper is concerned with non-homogeneous extension of the Dines theorem for the range set $\mathbf{R}(f,g) = \{\left(f(x),g(x)\right)|~x \in \mathbb{R}^n \},$ $f(x) = x^T A x + 2a^T x + a_0$ and $g(x) = x^T B x + 2b^T x + b_0.$ We show that $\mathbf{R}(f,g)$ is convex if, and only if, any pair of level sets, $\{x\in\mathbb{R}^n|f(x)=α\}$ and $\{x\in\mathbb{R}^n|g(x)=β\}$, do not separate each other. With the novel geometric concept about separation, we provide a polynomial-time procedure to practically check whether a given $\mathbf{R}(f,g)$ is convex or not. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2503_01225 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the convexity for the range set of two quadratic functions Nguyen, Huu-Quang Chu, Ya-Chi Sheu, Ruey-Lin Optimization and Control 90C20, 90C22, 90C26 Given $n\times n$ symmetric matrices $A$ and $B$, Dines in 1941 proved that the joint range set $\{(x^TAx,x^TBx)|~x\in\mathbb{R}^n\}$ is always convex. Our paper is concerned with non-homogeneous extension of the Dines theorem for the range set $\mathbf{R}(f,g) = \{\left(f(x),g(x)\right)|~x \in \mathbb{R}^n \},$ $f(x) = x^T A x + 2a^T x + a_0$ and $g(x) = x^T B x + 2b^T x + b_0.$ We show that $\mathbf{R}(f,g)$ is convex if, and only if, any pair of level sets, $\{x\in\mathbb{R}^n|f(x)=α\}$ and $\{x\in\mathbb{R}^n|g(x)=β\}$, do not separate each other. With the novel geometric concept about separation, we provide a polynomial-time procedure to practically check whether a given $\mathbf{R}(f,g)$ is convex or not. |
| title | On the convexity for the range set of two quadratic functions |
| topic | Optimization and Control 90C20, 90C22, 90C26 |
| url | https://arxiv.org/abs/2503.01225 |