On the Spherical Quasi-convexity of Quadratic Functions on Spherically Subdual Convex Sets
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On the Spherical Quasi-convexity of Quadratic Functions on Spherically Subdual Convex Sets Orizon Pereira Ferreira1 · Sándor Zoltán Németh2 · Lianghai Xiao2 Received: 10 May 2020 / Accepted: 19 August 2020 © The Author(s) 2020
Abstract In this paper, the spherical quasi-convexity of quadratic functions on spherically subdual convex sets is studied. Sufficient conditions for spherical quasi-convexity on spherically subdual convex sets are presented. A partial characterization of spherical quasi-convexity on spherical Lorentz sets is given, and some examples are provided. Keywords Spherical quasi-convexity · Quadratic function · Subdual cone Mathematics Subject Classification 26B25 · 90C25
1 Introduction The aim of this paper is to study theoretical properties of spherical quasi-convexity of quadratic functions on spherically subdual convex sets. It is well known that quadratic functions play an important role in nonlinear programming theory. For instance, the minimization problem of quadratic functions on the sphere occurs as a subproblem in methods of nonlinear programming (see the background section of [1] for an extensive review of the literature on the subject). We are interested in the problem of minimizing a quadratic function, defined by the symmetric matrix Q, constrained to a subset C of the sphere. This problem is a quadratic constrained optimization problem on the sphere,
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Sándor Zoltán Németh [email protected] Orizon Pereira Ferreira [email protected] Lianghai Xiao [email protected]
1
Instituto Matemática e Estatística, Universidade Federal de Goiás, Goiânia, GO 74001-970, Brazil
2
School of Mathematics, University of Birmingham, Watson Building, Edgbaston, Birmingham B15 2TT, UK
123
Journal of Optimization Theory and Applications
and it is also a minimum eigenvalue problem in C. It is essential to emphasize that there exists a special case, when C is the intersection of the Lorentz cone with the sphere. This special case is of particular interest because the minimum eigenvalue of Q in C is non-negative, if and only if the matrix Q is Lorentz copositive; see [2,3]. In general, changing the Lorentz cone by an arbitrary closed and convex cone K would lead to a more general concept of K -copositivity, thus our study is anticipated to initialize new perspectives for investigating the general copositivity of a symmetric matrix. In general, exploiting the specific intrinsic geometric and algebraic structure of problems posed on the sphere can significantly lower down the cost of finding solutions; see [4–9]. We know that a strict local minimizer of a spherically quasi-convex quadratic function is also a strict global minimizer, which makes interesting and natural to refer the problem about characterizing the spherically quasi-convex quadratic functions on spherically convex sets. The aim of this paper is to introduce both sufficient conditions and necessary conditions for quadratic functions to be spherically quasi-convex on spherically subdual convex sets. In particular, several examples are presen
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