A Characterization of Nonnegativity Relative to Proper Cones
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DOI: 10.1007/s13226-020-0442-4
A CHARACTERIZATION OF NONNEGATIVITY RELATIVE TO PROPER CONES Chandrashekaran Arumugasamy∗ , Sachindranath Jayaraman∗∗ and Vatsalkumar N. Mer∗∗ ∗ Department
of Mathematics, School of Mathematics and Computer Sciences,
Central University of Tamil Nadu, Neelakudi, Kangalancherry, Thiruvarur 610 101, Tamilnadu, India ∗∗ School
of Mathematics, Indian Institute of Science Education and Research Thiruvananthapuram, Maruthamala P.O., Vithura, Thiruvananthapuram 695 551, Kerala, India e-mails: [email protected]; [email protected]; [email protected] (Received 12 February 2019; accepted 13 May 2019)
Let A be an m × n matrix with real entries. Given two proper cones K1 and K2 in Rn and Rm , respectively, we say that A is nonnegative if A(K1 ) ⊆ K2 . A is said to be semipositive if there exists a x ∈ K1◦ such that Ax ∈ K2◦ . We prove that A is nonnegative if and only if A + B is semipositive for every semipositive matrix B. Applications of the above result are also brought out. Key words : Semipositivity of matrices; nonnegative matrices; proper cones; semipositive cone. 2010 Mathematics Subject Classification : 15B48, 90C33.
1. I NTRODUCTION , D EFINITION AND P RELIMINARIES We work with the field R of real numbers throughout. The following notations will be used. Mm,n (R) denotes the vector space of m × n matrices over R. When m = n, this space will be denoted by Mn (R). Let us recall that a subset K of Rn is called a convex cone if K + K ⊆ K and αK ⊆ K for all α ≥ 0. A convex cone K is said to be proper if it is topologically closed, pointed (K ∩−K = {0}) and has nonempty interior (denoted by K ◦ ). The dual, K ∗ , is defined as K ∗ = {y ∈ Rn : hy, xi ≥ 0 ∀x ∈ K}, where h., .i denotes the usual Euclidean inner product on Rn . Well known examples of proper cones that occur frequently in the optimization literature are the nonnegative orthant Rn+ in Rn , the Lorentz cone (also known as the ice-cream cone) Ln+ := {x = (x1 , . . . , xn )t ∈ Rn : xn ≥
936
CHANDRASHEKARAN ARUMUGASAMY et al.
0, x2n −
n−1 X
n of all symmetric positive semidefinite matrices in S n , the copositive x2i ≥ 0}, the set S+
i=1
(COP n ) and completely positive (CP n ) cones in S n and finally, the cone of squares K in a finite dimensional Euclidean Jordan algebra (see [7] for details). Recall that Mn (R) and S n have the trace inner product hX, Y i = trace(Y t X) and hX, Y i = trace(XY ), respectively. Assumptions : All cones in this paper are proper cones in appropriate finite dimensional real Hilbert spaces. Definition 1.1 — Let K1 and K2 be proper cones in Rn . A ∈ Mn (R) is 1. Nonnegative (positive) if A(K1 ) ⊆ K2 (A(K1 \ {0}) ⊆ K2◦ ). 2. Semipositive if there exists a x ∈ K1◦ such that Ax ∈ K2◦ . 3. Eventually nonnegative (positive) if there exists a positive integer k0 such that Ak is nonnegative (positive) for every k ≥ k0 . Let π(K1 , K2 ) and S(K1 , K2 ) denote, respectively, the set of all matrices that are nonnegative and semipositive relative to proper cones K1 and K2 . Whe
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