Conditional Quadratic Semidefinite Programming: Examples and Methods

Abstract

Abstract:The conditional Quadratic Semidefinite Programming (cQSDP) refers to a class of matrix optimization problems whose matrix variables are required to be positive semidefinite on a subspace and the objectives are quadratic. The chief purpose of this paper is to focus on two primal examples of cQSDP: the problem of matrix completion/approximation on a subspace and the Euclidean distance matrix problem.For the latter problem, we review some classical contributions and establish certain links among them. Moreover, we develop a semismooth Newton method for a special class of cQSDP and establish its quadratic convergence under the condition of constraint nondegeneracy.We also include an application in calibrating the correlation matrix in Libor market models. We hope this work will stimulate new research in cQSDP. Response to Reviewers:The author would like to thank the referees and the associate editor for their efforts in their speedy review. Powered by Editorial Manager® and ProduXion Manager® from Aries Systems Corporation Conditional Quadratic Semidefinite Programming: Examples and MethodsHou-Duo Qi * April 14, 2014; Revised May 3, 2014 AbstractThe conditional Quadratic Semidefinite Programming (cQSDP) refers to a class of matrix optimization problems whose matrix variables are required to be positive semidefinite on a subspace and the objectives are quadratic. The chief purpose of this paper is to focus on two primal examples of cQSDP: the problem of matrix completion/approximation on a subspace and the Euclidean distance matrix problem. For the latter problem, we review some classical contributions and establish certain links among them. Moreover, we develop a semismooth Newton method for a special class of cQSDP and establish its quadratic convergence under the condition of constraint nondegeneracy. We also include an application in calibrating the correlation matrix in Libor market models. We hope this work will stimulate new research in cQSDP.

Feasibility and a fast algorithm for Euclidean distance matrix optimization with ordinal constraints

“…Theorem 3.3 Suppose D * is an optimal solution of ( 24). Let D * σ be an optimal solution of the penalized problem (26). Let > 0 be given.…”

“…The characterization (4) was used, which was the key to the success of semismooth Newton's method for solving the dual problem of (6). A majorized penalty approach [26] was further proposed to deal with the low dimensional embedding, i.e., min D∈S n 1 2 D − ∆ 2 F s.t. diag(D) = 0, −D ∈ K n + , rank(JDJ) ≤ r, (7) where r was the prescribed embedding dimension.…”

“…Note that different from the approach proposed in [26], the rank constraint is represented by the rank cut of conditional positive semidefinite cone, based on which the following equivalent reformulation is proposed for −D ∈ K n + (r),…”

“…We refer to [5,21,15,31] just to name a few of outstanding SDP based approaches for EDM optimization in SNL and MC. The characterization (4) has fundamental differences from (3) [25] as it describes an EDM via the conditional positive semidefinite cone, based on which great progress has been made on numerical algorithms for EDM optimization [25,28,27,26,13,22,35,36], as we will detail below.…”

Parameter-related projection-based iterative algorithm for a kind of generalized positive semidefinite least squares problem

“…As described in [7], the symmetric positive semidefinite cone, the nonsymmetric positive semidefinite cone and the conditional symmetric positive semidefinite cone are all special cases of K. Hence, Problem (1) can be found in a wide range of application fields, and you can refer the papers [1,14,17,4,6,15,12] for the more details. Moreover, there are several methods can be used to solve some special cases of Problem (1), such as the gradient projection method [5], the proximal pointtype method [2,7], the predictor-corrector algorithm [1], the interior-point method [16,6], the GSVD method [9] and the semi-proximal ADMM [8], the Newton-type method [13], the semismooth newton-CG method [11].…”