On Local Convexity of Quadratic Transformations

Authors: Yong Xia

Publish Date: 2014/08/31

Volume: 2, Issue: 3, Pages: 341-350

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Abstract

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 f 1 f 2 are homogeneous quadratic functions then the set F 2 is convex In 1971 Yakubovich 23 24 used this basic result to prove the famous Slemma see 17 for a survey Brickman 3 proved in 1961 that if f 1 f 2 are homogeneous quadratic functions and ngeqslant 3 then the set f 1x f 2x xin mathbbRn Vert xVert = 1 subseteq mathbbR2 is convex Fradkov 5 proved in 1973 that if matrices A 1 cdots A m commute and f 1cdots f m are homogeneous then F m is convex In 1995 it was showed by Ramana and Goldman 18 that the identification of the convexity of F m is NPhard 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 ngeqslant 3 and f 1 f 2 f 3 are homogeneous quadratic functions such that mu 1A 1+mu 2A 2+mu 2A 3succ 0 where notation Asucc 0 means that A is positive definite for some mu in mathbbR3 then the set F 3 is convex Moreover as shown in the same paper when ngeqslant 2 and there exists mu in mathbbR2 such that mu 1A 1+mu 2A 2succ 0 the set F 2 is convex In 2007 Beck 1 showed that if mleqslant n A 1succ 0 and A 2=cdots =A m=0 then F m is convex However if A 1succ 0 A 2=cdots =A n+1=0 and a 2cdots a n+1 are linearly independent then F n+1 is not convex When m=2 Beck’s result reduces to be a corollary of Polyak’s result Very recently Xia et al 22 used the new developed Slemma with equality to establish the necessary and sufficient condition for the convexity of F 2 for A 2=0 and arbitrary A 1More generally Polyak 15 16 succeeded in proving a nonlinear image of a small ball in a Hilbert space is convex provided that not only the derivative of the map is Lipschitz continuous on the ball but also the derivative at the center of the ball is surjective and the norm of its adjoint mapping is bounded from zero Later Uderzo 21 extended the result to a certain subclass of uniformly convex Banach spaces When focusing on quadratic transformations Polyak’s result reads as followsIn this paper we improve the above Polyak’s result for quadratic transformations ie Theorem 11 by strengthening the constant L Then Theorem 11 is extended to the image of the ball of the same radius varepsilon centered at any point a satisfying Vert aVert 2varepsilon varepsilon Furthermore we propose two new approaches for possible improvement of LThroughout the paper all vectors are column vectors Let vcdot denotes the optimal value of problem cdot Notation Asucceq 0 implies that the matrix A is positive semidefinite rm vecA denotes the vector obtained by stacking the columns of A one underneath the other The trace of A is denoted by traceA=sum i=1nA ii A is the matrix having entries A ij The Kronecker product and the inner product of the matrices A and B are denoted by Aotimes B and Abullet B=rm traceABrm T=sum ij=1na ijb ij respectively The identity matrix is denoted by I Vert xVert =sqrtxrm Tx is the ell 2norm of the vector x

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