On the convexity for the range set of two quadratic functions

Huu-Quang Nguyen; Ya-Chi Chu; Ruey-Lin Sheu

doi:10.3934/jimo.2020169

Article Contents

2022, Volume 18, Issue 1: 575-592. Doi: 10.3934/jimo.2020169

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On the convexity for the range set of two quadratic functions

1.
Institute of Natural Science Education, Vinh University, Vinh, Nghe An, Vietnam
2.
Department of Mathematics, National Cheng Kung University, Tainan, Taiwan

^* Corresponding author: Ruey-Lin Sheu
^* Corresponding author: Ruey-Lin Sheu

In memory of Professor Hang-Chin Lai for his life contribution in Mathematics and Optimization

Received: June 30, 2020

Revised: August 31, 2020

Early access: November 2020

Published: January 2022

Huu-Quang, Nguyen's research work was supported by Taiwan MOST 108-2811-M-006-537 and Ruey-Lin Sheu's research work was sponsored by Taiwan MOST 107-2115-M-006-011-MY2

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Abstract

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) = \alpha\} $ and $ \{x\in\mathbb{R}^n|g(x) = \beta\} $, 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.

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Mathematics Subject Classification: Primary: 90C20; Secondary: 90C22, 90C26.

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Figure 1. The graph corresponds to Example 1

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Figure 2. Let $ f(x, y, z) = x^2+y^2 $ and $ g(x, y, z) = -x^2+y^2+z $

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Figure 3. For remark (c) and remark (e). Let $f(x, y) = -x^2 + 4 y^2$ and $g(x, y) = 2x-y$. The level set $\{g = 0\}$ separates $\{f<0\}, $ while $\{g = 0\}$ does not separate $\{f = 0\}$

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Figure 4. For remark (d). Let $f(x, y) = -x^2 + 4 y^2 - 1$ and $g(x, y) = x-5y$. The level set $\{g = 0\}$ separates $\{f = 0\}$ while $\{g = 0\}$ does not separate $\{f<0\}.$

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Figure 5. For remark (f) in which $ f(x, y) = -x^2 + 4 y^2 + 1 $ and $ g(x, y) = -(x-1)^2+4y^2+1 $

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Figure 6. Graph for Proof of Theorem 3.1

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Figure 7. For Example 2. Let $ f(x, y) = -\frac{\sqrt{3}}{2} x^2 + \frac{\sqrt{3}}{2} y^2 + x - \frac{1}{2} y $ and $ g(x, y) = \frac{1}{2} x^2 - \frac{1}{2} y^2 + \sqrt{3} x - \frac{\sqrt{3}}{2} y $

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Figure 8. The joint numerical range $ \mathbf{R}(f, g) $ in Example 3

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Table 1. Chronological list of notable results related to problem (P)

1941 (Dines [3])	(Dines Theorem) $ \left\{ \left. \left( x^T A x, x^T B x \right) \; \right\|\; x \in \mathbb{R}^n \right\} $ is convex. Moreover, if $ x^T A x $ and $ x^T B x $ has no common zero except for $ x=0 $, then $ \left\{ \left. \left( x^T A x, x^T B x \right) \; \right\|\; x \in \mathbb{R}^n \right\} $ is either $ \mathbb{R}^2 $ or an angular sector of angle less than $ \pi $.
1961 (Brickmen [1])	$ \mathbf{K}_{A, B} = \left\{ \left. \left( x^T A x, x^T B x \right) \; \right\|\; x \in \mathbb{R}^n\; , \; \\|x\\|=1 \right\} $ is convex if $ n \geq 3 $.
1995 (Ramana & Goldman [11]) Unpublished	$ \mathbf{R}(f, g) = \left\{ \left. \left( f(x), g(x) \right) \; \right\|\; x \in \mathbb{R}^n \right\} $ is convex if and only if $ \mathbf{R}(f, g) = \mathbf{R}(f_H, g_H) + \mathbf{R}(f, g) $, where $ f_H(x) = x^T A x $ and $ g_H(x) = x^T B x $.
1995 (Ramana & Goldman [11]) Unpublished	$ \mathbf{R}(f, g) = \left\{ \left. \left( f(x), g(x) \right) \; \right\|\; x \in \mathbb{R}^n \right\} $ is convex if $ n \geq 2 $ and $ \exists\; \alpha, \beta \in \mathbb{R} $ such that $ \alpha A + \beta B \succ 0 $.
1998 (Polyak [10])	$ \left\{ \left. \left( x^T A x, x^T B x, x^T C x \right) \; \right\|\; x \in \mathbb{R}^n \right\} $ is convex if $ n \geq 3 $ and $ \exists\; \alpha, \beta, \gamma \in \mathbb{R} $ such that $ \alpha A + \beta B + \gamma C \succ 0 $.
1998 (Polyak [10])	$ \left\{ \left. \left( x^T A_1 x, \cdots, x^T A_m x \right) \; \right\|\; x \in \mathbb{R}^n \right\} $ is convex if $ A_1, \cdots, A_m $ commute.
2016 (Bazán & Opazo [5])	$ \mathbf{R}(f, g) = \left\{ \left. \left( f(x), g(x) \right) \; \right\|\; x \in \mathbb{R}^n \right\} $ is convex if and only if $ \exists\; d=(d_1, d_2) \in \mathbb{R}^2 $, $ d \neq 0 $, such that the following four conditions hold: $ \bf{(C1):} $ $ F_L \left( \mathcal{N}(A) \cap \mathcal{N}(B) \right) = \{0\} $ $ \bf{(C2):} $ $ d_2 A = d_1 B $ $ \bf{(C3):} $ $ -d \in \mathbf{R}(f_H, g_H) $ $ \bf{(C4):} $ $ F_H(u) = -d \implies \langle F_L(u), d_{\perp}\rangle \neq 0 $ where $ \mathcal{N}(A) $ and $ \mathcal{N}(B) $ denote the null space of $ A $ and $ B $ respectively, $ F_H(x) = \left( f_H(x), g_H(x) \right) = \left( x^T A x , x^T B x \right) $, $ F_L(x) = \left( a^T x , b^T x \right) $, and $ d_{\perp} = (-d_2, d_1) $.

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