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We study analyticity, differentiability, and semismoothness of Löwner's operator and spectral functions under the framework of Euclidean Jordan algebras. In particular, we show that many optimization-related classical results in the symmetric matrix space can be generalized within this framework. For example, the metric projection operator over any symmetric cone defined in a Euclidean Jordan algebra is shown to be strongly semismooth. The research also raises several open questions, whose answers would be of strong interest for optimization research.

Based on an inverse function theorem for a system of semismooth equations, this paper establishes several necessary and sufficient conditions for an isolated solution of a complementarity problem defined on the cone of symmetric positive semidefinite matrices to be strongly regular/stable. We show further that for a parametric complementarity problem of this kind, if a solution corresponding to a base parameter is strongly stable, then a semismooth implicit solution function exists whose directional derivatives can be computed by solving certain affine problems on the critical cone at the base solution. Similar results are also derived for a complementarity problem defined on the Lorentz cone. The analysis relies on some new properties of the directional derivatives of the projector onto the semidefinite cone and the Lorentz cone. (1980) and the related concept of a strongly stable stationary point of a differentiable nonlinear program (NLP) introduced by Kojima (1980) are two of the most important ideas in contemporary perturbation analysis of mathematical programming problems. Beginning with Jongen et al. (1987), many authors have established the equivalence between these two concepts for the Karush-Kuhn-Tucker (KKT) system of a nonlinear program with finitely many twice differentiable functions. The article by Klatte and Kummer (1999) presents a unified framework that handles both concepts simultaneously and contains a brief bibliographical note. For an excellent survey of perturbation analysis of optimization problems, see the review by Bonnans and Shapiro (1998) and their comprehensive monograph (Bonnans and Shapiro 2000). Introduction. The concept of a strongly regular solution to a generalized equation introduced by RobinsonExtending the seminal work of Robinson and Kojima mentioned above, many authors have investigated the solution stability of variational inequalities (VIs); see, e.g., Dontchev and Rockafellar (1996), which characterizes strong stability in linearly constrained VIs in terms of the "Aubin property." For a comprehensive treatment of the subject of solution stability of VIs, we refer the reader to Facchinei and Pang (2003, Chapter 4). As explained in Liu (1995), there are substantial differences between the sensitivity and stability analysis of an NLP and that of a VI. Most importantly, the lack of symmetry in the defining function of a VI invalidates a straightforward optimization approach for such analysis. Focusing on an NLP, Kojima was the first person to utilize degree theory on a nondifferentiable system of equations to derive stability results in mathematical programming. Kojima's equation approach turns out to be very fruitful for the sensitivity and stability study of the VI and the related complementarity problem (CP). The forthcoming monograph (Facchinei and Pang 2003) contains a long chapter on this subject, which is developed based on the equation approach and degree theory; there are many references in the bibliography therein. Among its many applications, the strong stability...

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Abstract. We show that the Fischer-Burmeister complementarity functions, associated to the semidefinite cone (SDC) and the second order cone (SOC), respectively, are strongly semismooth everywhere. Interestingly enough, the proof stems in a relationship between the singular value decomposition of a nonsymmetric matrix and the spectral decomposition of a symmetric matrix.