Abstract
In this article, we consider the numerical method for solving the system of inequalities under the order induced by a symmetric cone with the function involved being monotone. Based on a perturbed smoothing function, the underlying system of inequalities is reformulated as a system of smooth equations, and a smoothing-type method is proposed to solve it iteratively so that a solution of the system of inequalities is found. By means of the theory of Euclidean Jordan algebras, the algorithm is proved to be well defined, and to be globally convergent under weak assumptions and locally quadratically convergent under suitable assumptions. Preliminary numerical results indicate that the algorithm is effective.AMS subject classifications: 90C33, 65K10.
Highlights
Let V be a finite dimensional vector space over 4 with an inner product 〈·,·〉
A natural question is that how to develop a smoothing-type algorithm to solve the system of inequalities under the order induced by a symmetric cone
We introduce the smoothing function: φ(μ, y) = y + y2 + 4μ2e, ∀y ∈ V, μ ∈
Summary
Let V be a finite dimensional vector space over 4 with an inner product 〈·,·〉. If there exists a bilinear transformation from V × V to V, denoted by “○,” such that for any x, y, z Î V, x ◦ y = y ◦ x; x ◦ (x2 ◦ y) = x2 ◦ (x ◦ y); x ◦ y, z = x, y ◦ z , where x2 := x ○ x, (V, ○, 〈·,·〉) is called a Euclidean Jordan algebra. Investigation of (1.2) could provide a unified theoretical framework for studying the system of respective inequalities under the order induced by the nonnegative orthant, the second-order, and the positive semidefinite matrix cones. On one hand, smoothing-type algorithms have been developed to solve symmetric cone complementarity problems On the other hand, smoothing-type algorithms have been developed to solve the system of inequalities under the order induced by n +. [17,18,19]) From these recent studies, a natural question is that how to develop a smoothing-type algorithm to solve the system of inequalities under the order induced by a symmetric cone.
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