On Local Convexity of Quadratic Transformations
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On Local Convexity of Quadratic Transformations Yong Xia
Received: 23 May 2014 / Revised: 1 August 2014 / Accepted: 4 August 2014 / Published online: 31 August 2014 Ó Operations Research Society of China, Periodicals Agency of Shanghai University, and SpringerVerlag Berlin Heidelberg 2014
Abstract In this paper, we improve Polyak’s local convexity result for quadratic transformations. Extension and open problems are also presented. Keywords
Convexity Quadratic transformation Joint numerical range
1 Introduction Let x 2 Rn and f ðxÞ ¼ ðf1 ðxÞ; ; fm ðxÞÞ, where 1 fi ðxÞ ¼ xT Ai x þ aTi x; i ¼ 1; ; m 2 are quadratic functions and Ai 2 Rnn (i ¼ 1; ; m) are symmetric. One interesting question is when the following joint numerical range Fm ¼ ff ðxÞ : x 2 Rn g Rm is convex.
This research was supported by Beijing Higher Education Young Elite Teacher Project (No. 29201442), and by the fund of State Key Laboratory of Software Development Environment (No. SKLSDE2013ZX13). Y. Xia (&) State Key Laboratory of Software Development Environment, LMIB of the Ministry of Education, School of Mathematics and System Sciences, Beihang University, Beijing 100191, China e-mail: [email protected]
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Though results on the convexity of complex quadratic functions were already there since 1918, see Toeplitz [19] and Hausdorff [8], the first such result for real case is due to Dines [4] in 1941. It states that if f1 ; f2 are homogeneous quadratic functions then the set F2 is convex. In 1971, Yakubovich [23, 24] used this basic result to prove the famous S-lemma, see [17] for a survey. Brickman [3] proved in 1961 that if f1 ; f2 are homogeneous quadratic functions and n > 3 then the set fðf1 ðxÞ; f2 ðxÞÞ : x 2 Rn ; kxk ¼ 1g R2 is convex. Fradkov [5] proved in 1973 that if matrices A1 ; ; Am commute and f1 ; ; fm are homogeneous, then Fm is convex. In 1995, it was showed by Ramana and Goldman [18] that the identification of the convexity of Fm is NP-hard. In the same paper, the quadratic maps, under which the image of every linear subspace is convex, was also investigated. Based on Brickman’s result, Polyak [14] proved in 1998 that if n > 3 and f1 ; f2 ; f3 are homogeneous quadratic functions such that l1 A1 þ l2 A2 þ l2 A3 0 (where notation A 0 means that A is positive definite) for some l 2 R3 , then the set F3 is convex. Moreover, as shown in the same paper, when n > 2 and there exists l 2 R2 such that l1 A1 þ l2 A2 0, the set F2 is convex. In 2007, Beck [1] showed that if m 6 n; A1 0 and A2 ¼ ¼ Am ¼ 0, then Fm is convex. However, if A1 0; A2 ¼ ¼ Anþ1 ¼ 0 and a2 ; ; anþ1 are linearly independent, then Fnþ1 is not convex. When m ¼ 2, Beck’s result reduces to be a corollary of Polyak’s result. Very recently, Xia et al. [22] used the new developed S-lemma with equality to establish the necessary and sufficient condition for the convexity of F2 for A2 ¼ 0 and arbitrary A1 . More generally, Polyak [15, 16] succeeded in proving a nonlinear image of a small ball
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