• Previous Article
    Stochastic comparisons of series-parallel and parallel-series systems with dependence between components and also of subsystems
  • JIMO Home
  • This Issue
  • Next Article
    Optimal lot-sizing policy for a failure prone production system with investment in process quality improvement and lead time variance reduction
doi: 10.3934/jimo.2020169

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

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

Received  July 2020 Revised  September 2020 Published  November 2020

Fund Project: 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

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.

Citation: Huu-Quang Nguyen, Ya-Chi Chu, Ruey-Lin Sheu. On the convexity for the range set of two quadratic functions. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2020169
References:
[1]

L. Brickmen, On the field of values of a matrix, Proceedings of the American Mathematical Society, 12 (1961), 61-66.  doi: 10.1090/S0002-9939-1961-0122827-1.  Google Scholar

[2]

K. Derinkuyu and M. Ç. Plnar, On the S-procedure and some variants, Mathematical Methods of Operations Research, 64 (2006), 55-77.  doi: 10.1007/s00186-006-0070-8.  Google Scholar

[3]

L. L. Dines, On the mapping of quadratic forms, Bulletin of the American Mathematical Society, 47 (1941), 494-498.  doi: 10.1090/S0002-9904-1941-07494-X.  Google Scholar

[4]

S.-C. FangD. Y. GaoG.-X. LinR.-L. Sheu and W. Xing, Double well potential function and its optimization in the n-dimensional real space–Part I, J. Ind. Manag. Optim., 13 (2017), 1291-1305.  doi: 10.3934/jimo.2016073.  Google Scholar

[5]

F. Flores-Bazán and F. Opazo, Characterizing the convexity of joint-range for a pair of inhomogeneous quadratic functions and strong duality, Minimax Theory Appl, 1 (2016), 257-290.   Google Scholar

[6]

H. Q. Nguyen, R. L. Sheu and Y. Xia, Solving a new type of quadratic optimization problem having a joint numerical range constraint, 2020. Available from: https://doi.org/10.13140/RG.2.2.23830.98887. Google Scholar

[7]

H.-Q. Nguyen and R.-L. Sheu, Geometric properties for level sets of quadratic functions, Journal of Global Optimization, 73 (2019), 349-369.  doi: 10.1007/s10898-018-0706-2.  Google Scholar

[8]

H.-Q. Nguyen and R.-L. Sheu, Separation properties of quadratic functions, 2020. Available from: https://doi.org/10.13140/RG.2.2.18518.88647. Google Scholar

[9]

I. Pólik and T. Terlaky, A survey of the S-lemma, SIAM Review, 49 (2007), 371-418.  doi: 10.1137/S003614450444614X.  Google Scholar

[10]

B. T. Polyak, Convexity of quadratic transformations and its use in control and optimization, Journal of Optimization Theory and Applications, 99 (1998), 553-583.  doi: 10.1023/A:1021798932766.  Google Scholar

[11]

M. Ramana and A. J. Goldman, Quadratic maps with convex images, Submitted to Math of OR. Google Scholar

[12]

H. Tuy and H. D. Tuan, Generalized S-lemma and strong duality in nonconvex quadratic programming, Journal of Global Optimization, 56 (2013), 1045-1072.  doi: 10.1007/s10898-012-9917-0.  Google Scholar

[13]

Y. XiaS. Wang and R.-L. Sheu, S-lemma with equality and its applications, Mathematical Programming, 156 (2016), 513-547.  doi: 10.1007/s10107-015-0907-0.  Google Scholar

[14]

Y. XiaR.-L. SheuS.-C. Fang and W. Xing, Double well potential function and its optimization in the n-dimensional real space–Part II, J. Ind. Manag. Optim., 13 (2017), 1307-1328.  doi: 10.3934/jimo.2016074.  Google Scholar

[15]

V. A. Yakubovich, The S-procedure in nolinear control theory, Vestnik Leninggradskogo Universiteta, Ser. Matematika, (1971), 62–77.  Google Scholar

[16]

Y. Ye and S. Zhang, New results on quadratic minimization, SIAM Journal on Optimization, 14 (2003), 245-267.  doi: 10.1137/S105262340139001X.  Google Scholar

show all references

References:
[1]

L. Brickmen, On the field of values of a matrix, Proceedings of the American Mathematical Society, 12 (1961), 61-66.  doi: 10.1090/S0002-9939-1961-0122827-1.  Google Scholar

[2]

K. Derinkuyu and M. Ç. Plnar, On the S-procedure and some variants, Mathematical Methods of Operations Research, 64 (2006), 55-77.  doi: 10.1007/s00186-006-0070-8.  Google Scholar

[3]

L. L. Dines, On the mapping of quadratic forms, Bulletin of the American Mathematical Society, 47 (1941), 494-498.  doi: 10.1090/S0002-9904-1941-07494-X.  Google Scholar

[4]

S.-C. FangD. Y. GaoG.-X. LinR.-L. Sheu and W. Xing, Double well potential function and its optimization in the n-dimensional real space–Part I, J. Ind. Manag. Optim., 13 (2017), 1291-1305.  doi: 10.3934/jimo.2016073.  Google Scholar

[5]

F. Flores-Bazán and F. Opazo, Characterizing the convexity of joint-range for a pair of inhomogeneous quadratic functions and strong duality, Minimax Theory Appl, 1 (2016), 257-290.   Google Scholar

[6]

H. Q. Nguyen, R. L. Sheu and Y. Xia, Solving a new type of quadratic optimization problem having a joint numerical range constraint, 2020. Available from: https://doi.org/10.13140/RG.2.2.23830.98887. Google Scholar

[7]

H.-Q. Nguyen and R.-L. Sheu, Geometric properties for level sets of quadratic functions, Journal of Global Optimization, 73 (2019), 349-369.  doi: 10.1007/s10898-018-0706-2.  Google Scholar

[8]

H.-Q. Nguyen and R.-L. Sheu, Separation properties of quadratic functions, 2020. Available from: https://doi.org/10.13140/RG.2.2.18518.88647. Google Scholar

[9]

I. Pólik and T. Terlaky, A survey of the S-lemma, SIAM Review, 49 (2007), 371-418.  doi: 10.1137/S003614450444614X.  Google Scholar

[10]

B. T. Polyak, Convexity of quadratic transformations and its use in control and optimization, Journal of Optimization Theory and Applications, 99 (1998), 553-583.  doi: 10.1023/A:1021798932766.  Google Scholar

[11]

M. Ramana and A. J. Goldman, Quadratic maps with convex images, Submitted to Math of OR. Google Scholar

[12]

H. Tuy and H. D. Tuan, Generalized S-lemma and strong duality in nonconvex quadratic programming, Journal of Global Optimization, 56 (2013), 1045-1072.  doi: 10.1007/s10898-012-9917-0.  Google Scholar

[13]

Y. XiaS. Wang and R.-L. Sheu, S-lemma with equality and its applications, Mathematical Programming, 156 (2016), 513-547.  doi: 10.1007/s10107-015-0907-0.  Google Scholar

[14]

Y. XiaR.-L. SheuS.-C. Fang and W. Xing, Double well potential function and its optimization in the n-dimensional real space–Part II, J. Ind. Manag. Optim., 13 (2017), 1307-1328.  doi: 10.3934/jimo.2016074.  Google Scholar

[15]

V. A. Yakubovich, The S-procedure in nolinear control theory, Vestnik Leninggradskogo Universiteta, Ser. Matematika, (1971), 62–77.  Google Scholar

[16]

Y. Ye and S. Zhang, New results on quadratic minimization, SIAM Journal on Optimization, 14 (2003), 245-267.  doi: 10.1137/S105262340139001X.  Google Scholar

Figure 1.  The graph corresponds to Example 1
Figure 2.  Let $ f(x, y, z) = x^2+y^2 $ and $ g(x, y, z) = -x^2+y^2+z $
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 $\{f = 0\}$
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 $\{f<0\}.$
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 $
Figure 6.  Graph for Proof of Theorem 3.1
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 $
Figure 8.  The joint numerical range $ \mathbf{R}(f, g) $ in Example 3
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 $.
$ \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 $.
$ \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) $.
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 $.
$ \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 $.
$ \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) $.
[1]

Kamran Jalilian, Kameleh Nasiri Pirbazari. Convex optimization without convexity of constraints on non-necessarily convex sets and its applications in customer satisfaction in automotive industry. Numerical Algebra, Control & Optimization, 2021  doi: 10.3934/naco.2021020

[2]

Meixia Li, Changyu Wang, Biao Qu. Non-convex semi-infinite min-max optimization with noncompact sets. Journal of Industrial & Management Optimization, 2017, 13 (4) : 1859-1881. doi: 10.3934/jimo.2017022

[3]

Yong Wang, Wanquan Liu, Guanglu Zhou. An efficient algorithm for non-convex sparse optimization. Journal of Industrial & Management Optimization, 2019, 15 (4) : 2009-2021. doi: 10.3934/jimo.2018134

[4]

Nurullah Yilmaz, Ahmet Sahiner. On a new smoothing technique for non-smooth, non-convex optimization. Numerical Algebra, Control & Optimization, 2020, 10 (3) : 317-330. doi: 10.3934/naco.2020004

[5]

Dan Zhu, Rosemary A. Renaut, Hongwei Li, Tianyou Liu. Fast non-convex low-rank matrix decomposition for separation of potential field data using minimal memory. Inverse Problems & Imaging, 2021, 15 (1) : 159-183. doi: 10.3934/ipi.2020076

[6]

Lipu Zhang, Yinghong Xu, Zhengjing Jin. An efficient algorithm for convex quadratic semi-definite optimization. Numerical Algebra, Control & Optimization, 2012, 2 (1) : 129-144. doi: 10.3934/naco.2012.2.129

[7]

Qilin Wang, Liu He, Shengjie Li. Higher-order weak radial epiderivatives and non-convex set-valued optimization problems. Journal of Industrial & Management Optimization, 2019, 15 (2) : 465-480. doi: 10.3934/jimo.2018051

[8]

Arezu Zare, Mohammad Keyanpour, Maziar Salahi. On fractional quadratic optimization problem with two quadratic constraints. Numerical Algebra, Control & Optimization, 2020, 10 (3) : 301-315. doi: 10.3934/naco.2020003

[9]

Yanqin Bai, Xuerui Gao, Guoqiang Wang. Primal-dual interior-point algorithms for convex quadratic circular cone optimization. Numerical Algebra, Control & Optimization, 2015, 5 (2) : 211-231. doi: 10.3934/naco.2015.5.211

[10]

Yanqin Bai, Lipu Zhang. A full-Newton step interior-point algorithm for symmetric cone convex quadratic optimization. Journal of Industrial & Management Optimization, 2011, 7 (4) : 891-906. doi: 10.3934/jimo.2011.7.891

[11]

Saeid Ansary Karbasy, Maziar Salahi. Quadratic optimization with two ball constraints. Numerical Algebra, Control & Optimization, 2020, 10 (2) : 165-175. doi: 10.3934/naco.2019046

[12]

Ye Tian, Qingwei Jin, Zhibin Deng. Quadratic optimization over a polyhedral cone. Journal of Industrial & Management Optimization, 2016, 12 (1) : 269-283. doi: 10.3934/jimo.2016.12.269

[13]

Yoon Mo Jung, Taeuk Jeong, Sangwoon Yun. Non-convex TV denoising corrupted by impulse noise. Inverse Problems & Imaging, 2017, 11 (4) : 689-702. doi: 10.3934/ipi.2017032

[14]

Tong Li, Jeungeun Park. Stability of traveling waves of models for image processing with non-convex nonlinearity. Communications on Pure & Applied Analysis, 2018, 17 (3) : 959-985. doi: 10.3934/cpaa.2018047

[15]

Tarik Aougab, Stella Chuyue Dong, Robert S. Strichartz. Laplacians on a family of quadratic Julia sets II. Communications on Pure & Applied Analysis, 2013, 12 (1) : 1-58. doi: 10.3934/cpaa.2013.12.1

[16]

Sun-Yung Alice Chang, Xi-Nan Ma, Paul Yang. Principal curvature estimates for the convex level sets of semilinear elliptic equations. Discrete & Continuous Dynamical Systems, 2010, 28 (3) : 1151-1164. doi: 10.3934/dcds.2010.28.1151

[17]

C. M. Elliott, B. Gawron, S. Maier-Paape, E. S. Van Vleck. Discrete dynamics for convex and non-convex smoothing functionals in PDE based image restoration. Communications on Pure & Applied Analysis, 2006, 5 (1) : 181-200. doi: 10.3934/cpaa.2006.5.181

[18]

Koh Katagata. Quartic Julia sets including any two copies of quadratic Julia sets. Discrete & Continuous Dynamical Systems, 2016, 36 (4) : 2103-2112. doi: 10.3934/dcds.2016.36.2103

[19]

Ye Tian, Wei Yang, Gene Lai, Menghan Zhao. Predicting non-life insurer's insolvency using non-kernel fuzzy quadratic surface support vector machines. Journal of Industrial & Management Optimization, 2019, 15 (2) : 985-999. doi: 10.3934/jimo.2018081

[20]

Mustapha Ait Rami, John Moore. Partial stabilizability and hidden convexity of indefinite LQ problem. Numerical Algebra, Control & Optimization, 2016, 6 (3) : 221-239. doi: 10.3934/naco.2016009

2019 Impact Factor: 1.366

Metrics

  • PDF downloads (45)
  • HTML views (247)
  • Cited by (0)

Other articles
by authors

[Back to Top]