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This document discusses the importance of optimization in various fields such as statistics, biology, finance, economics, and control. It highlights the advancements made in optimization theory and methods, including first-order methods and augmented Lagrangian methods. The document also introduces a special issue on "Symmetry in Optimization Theory, Algorithms and Applications," which features five papers on topics such as ensemble learning, ant colony system algorithms, subspace algorithms, consensus problems in multi-agent systems, and dynamic conditional correlation models. The authors express their gratitude to the contributors and reviewers of the special issue. [Extracted from the article]

The covariance matrix estimation plays an important role in portfolio optimization and risk management. It is well-known that portfolio is essentially a convex quadratic programming problem, which is also a special case of symmetric cone optimization. Accurate covariance matrix estimation will lead to more reasonable asset weight allocation. However, some existing methods do not consider the influence of time-varying factor on the covariance matrix estimations. To remedy this, in this article, we propose an improved dynamic conditional correlation model (DCC) by using nonconvex optimization model under smoothly clipped absolute deviation and hard-threshold penalty functions. We first construct a nonconvex optimization model to obtain the optimal covariance matrix estimation, and then we use this covariance matrix estimation to replace the unconditional covariance matrix in the DCC model. The result shows that the loss of the proposed estimator is smaller than other variants of the DCC model in numerical experiments. Finally, we apply our proposed model to the classic Markowitz portfolio. The results show that the improved dynamic conditional correlation model performs better than the current DCC models. [ABSTRACT FROM AUTHOR]

The estimation of the covariance matrix is an important topic in the field of multivariate statistical analysis. In this paper, we propose a new estimator, which is a convex combination of the linear shrinkage estimation and the rotation-invariant estimator under the Frobenius norm. We first obtain the optimal parameters by using grid search and cross-validation, and then, we use these optimal parameters to demonstrate the effectiveness and robustness of the proposed estimation in the numerical simulations. Finally, in empirical research, we apply the covariance matrix estimation to the portfolio optimization. Compared to the existing estimators, we show that the proposed estimator has better performance and lower out-of-sample risk in portfolio optimization. [ABSTRACT FROM AUTHOR]

A real n × n matrix M is said to be a P -matrix if all its principal minors are positive. In a recent paper Chua and Yi [2] described this property in terms of norm: There exists a γ > 0 such that for all nonnegative diagonal matrices D and vectors x , ‖ M x + D x ‖ ≥ γ ‖ x ‖ . In this paper, we introduce a block version of this property for a linear transformation defined on a product of normed or inner product spaces. In addition to relating this to (real) positive stability and positive principal minor properties, we study the invariance of this property by principal subtransformations and Schur complements. We also specialize this property to Z -transformations and to Euclidean Jordan algebras. [ABSTRACT FROM AUTHOR]

A real square matrix $$Q$$ is a bilinear complementarity relation on a proper cone $$K$$ in $$\mathbb{R }^n$$ if where $$K^*$$ is the dual of $$K$$ . The bilinearity rank of $$K$$ is the dimension of the linear space of all bilinear complementarity relations on $$K$$ . In this article, we continue the study initiated by Rudolf et al. (Math Prog Ser B 129:5-31, ). We show that bilinear complementarity relations are related to Lyapunov-like transformations that appear in dynamical systems and in complementarity theory and further show that the bilinearity rank of $$K$$ is the dimension of the Lie algebra of the automorphism group of $$K$$ . In addition, we correct a result of Rudolf et al., compute the bilinearity ranks of symmetric and completely positive cones, and state some Schur-type results for Lyapunov-like transformations. [ABSTRACT FROM AUTHOR]

Abstract: The low-rank solutions of continuous and discrete Lyapunov equations are of great importance but generally difficult to compute in control system analysis and design. Fortunately, Mesbahi and Papavassilopoulos (1997) [30] showed that with the semidefinite cone constraint, the lowest-rank solutions of the discrete Lyapunov inequality can be efficiently solved by a linear semidefinite programming. In this paper, we further show that the lowest-rank solutions of both the continuous and discrete Lyapunov equations over symmetric cone are unique and can be exactly solved by their convex relaxations, the symmetric cone linear programming problems. Therefore, they are polynomial-time solvable. Since the underlying symmetric cone is a more general algebraic setting which contains the semidefinite cone as a special case, our results also answer an open question proposed by Recht, Fazel and Parrilo (2010) [36]. [Copyright &y& Elsevier]

In this paper, we study interconnections between pseudomonotonicity, the column sufficiency property, and the globally uniquely solvable property in the setting of Euclidean Jordan algebras. [ABSTRACT FROM AUTHOR]

Abstract: In this article, we introduce and study ultra and super SSM-(cone GUS)-properties for linear transformations on Euclidean Jordan algebras. We show that the Jordan quadratic strict semimonotonicity (JQSSM)-property is equivalent to the completely-Q-property for linear transformations on any Euclidean Jordan algebra. Also, we investigate some interconnections between the SSM-property, the Jordan linear SSM (JLSSM)-property, and the completely-Q property on a Euclidean Jordan algebra. Moreover, we give a characterization of the completely-Q property on the Jordan spin algebra and study the completely-Q property for some special transformations on a Euclidean Jordan algebra. [Copyright &y& Elsevier]

Abstract: The P, , and properties of a linear transformation on a Euclidean Jordan algebra are generalizations of the corresponding properties of a square matrix on R n . Motivated by the equivalence of P and properties for a Z-matrix and a similar result for Lyapunov and Stein transformations on the space of real symmetric matrices , in this paper, we present two results supporting the conjecture that P and properties are equivalent for a Z-transformation on a Euclidean Jordan algebra. We show that the conjecture holds for Lyapunov-like transformations and Z-transformations satisfying an additional condition. [Copyright &y& Elsevier]

Abstract: In a recent paper [7], Gowda et al. extended Ostrowski–Schneider type inertia results to certain linear transformations on Euclidean Jordan algebras. In particular, they showed that whenever by the min–max theorem of Hirzebruch, where the inertia of an element x in a Euclidean Jordan algebra is defined bywith , , and denoting, respectively, the number of positive, negative, and zero eigenvalues, counting multiplicities. In this paper, we present a Peirce decomposition version of Wimmer’s result [13] and show that it is equivalent to the above result. In addition, we extend Higham and Cheng’s result ([8], Lemma 4.2) to the setting of Euclidean Jordan algebras. [Copyright &y& Elsevier]

Abstract: This paper deals with some inertia theorems in Euclidean Jordan algebras. First, based on the continuity of eigenvalues, we give an alternate proof of Kaneyuki’s generalization of Sylvester’s law of inertia in simple Euclidean Jordan algebras. As a consequence, we show that the cone spectrum of any quadratic representation with respect to a symmetric cone is finite. Second, we present Ostrowski–Schneider type inertia results in Euclidean Jordan algebras. In particular, we relate the inertias of objects and in a Euclidean Jordan algebra when or , where and denote Lyapunov and Stein transformations, respectively. [Copyright &y& Elsevier]

Motivated by the similarities between the properties of Z-matrices on $$R^{n}_+$$ and Lyapunov and Stein transformations on the semidefinite cone $$\mathcal {S}^n_+$$ , we introduce and study Z-transformations on proper cones. We show that many properties of Z-matrices extend to Z-transformations. We describe the diagonal stability of such a transformation on a symmetric cone by means of quadratic representations. Finally, we study the equivalence of Q and P properties of Z-transformations on symmetric cones. In particular, we prove such an equivalence on the Lorentz cone. [ABSTRACT FROM AUTHOR]

In matrix theory, Fu and Markham showed using majorization technique that if a Hermitian matrix satisfies certain conditions, then the matrix must be block-diagonal. In this paper, we extend this result to the setting of simple Euclidean Jordan algebras by using the Cauchy interlacing theorem and the Schur complement Cauchy interlacing theorem. [ABSTRACT FROM AUTHOR]

This paper is concerned with the problem of sparse recovery on Euclidean Jordan algebra (SREJA), which includes the sparse signal recovery problem and the low-rank symmetric matrix recovery problem as special cases. We introduce the notions of restricted isometry property (RIP), null space property (NSP), and s -goodness for linear transformations in s -SREJA, all of which provide sufficient conditions for s -sparse recovery via the nuclear norm minimization on Euclidean Jordan algebra. Moreover, we show that both the s -goodness and the NSP are necessary and sufficient conditions for exact s -sparse recovery via the nuclear norm minimization on Euclidean Jordan algebra. Applying these characteristic properties, we establish the exact and stable recovery results for solving SREJA problems via nuclear norm minimization. [ABSTRACT FROM AUTHOR]

Abstract: Given a closed cone in , we consider the corresponding completely positive (convex) cone generated by in . Under certain conditions on , we describe the automorphism group of and its corresponding Lie algebra in terms of those of and/or . In particular, we show that when is a (closed convex) proper cone, the automorphism groups of and are isomorphic and their corresponding Lie algebras are isomorphic. [Copyright &y& Elsevier]