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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 |
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.
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. |
[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. |
[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. |
[4] |
S.-C. Fang, D. Y. Gao, G.-X. Lin, R.-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. |
[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.
|
[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. |
[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. |
[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. |
[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. |
[13] |
Y. Xia, S. 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. |
[14] |
Y. Xia, R.-L. Sheu, S.-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. |
[15] |
V. A. Yakubovich, The S-procedure in nolinear control theory, Vestnik Leninggradskogo Universiteta, Ser. Matematika, (1971), 62–77. |
[16] |
Y. Ye and S. Zhang,
New results on quadratic minimization, SIAM Journal on Optimization, 14 (2003), 245-267.
doi: 10.1137/S105262340139001X. |
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. |
[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. |
[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. |
[4] |
S.-C. Fang, D. Y. Gao, G.-X. Lin, R.-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. |
[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.
|
[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. |
[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. |
[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. |
[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. |
[13] |
Y. Xia, S. 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. |
[14] |
Y. Xia, R.-L. Sheu, S.-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. |
[15] |
V. A. Yakubovich, The S-procedure in nolinear control theory, Vestnik Leninggradskogo Universiteta, Ser. Matematika, (1971), 62–77. |
[16] |
Y. Ye and S. Zhang,
New results on quadratic minimization, SIAM Journal on Optimization, 14 (2003), 245-267.
doi: 10.1137/S105262340139001X. |






1941 (Dines [3]) |
(Dines Theorem) is convex. Moreover, if |
1961 (Brickmen [1]) |
is convex if |
1995 (Ramana & Goldman [11]) Unpublished |
is convex if and only if |
is convex if |
|
1998 (Polyak [10]) |
is convex if |
is convex if |
|
2016 (Bazán & Opazo [5]) |
is convex if and only if where |
1941 (Dines [3]) |
(Dines Theorem) is convex. Moreover, if |
1961 (Brickmen [1]) |
is convex if |
1995 (Ramana & Goldman [11]) Unpublished |
is convex if and only if |
is convex if |
|
1998 (Polyak [10]) |
is convex if |
is convex if |
|
2016 (Bazán & Opazo [5]) |
is convex if and only if where |
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