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Register a new userCommutation principles for optimization problems on spectral sets in Euclidean Jordan algebras
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Muddappa Gowda
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The commutation principle of Ramirez, Seeger, and Sossa proved in the setting of Euclidean Jordan algebras says that when the sum of a real valued function $h$ and a spectral function $Phi$ is minimized/maximized over a spectral set $E$, any local optimizer $a$ at which $h$ is Fr{e}chet differentiable operator commutes with the derivative $h^{prime}(a)$. In this paper, assuming the existence of a subgradient in place the derivative (of $h$), we establish `strong operator commutativity relations: If $a$ solves the problem $underset{E}{max},(h+Phi)$, then $a$ strongly operator commutes with every element in the subdifferential of $h$ at $a$; If $E$ and $h$ are convex and $a$ solves the problem $underset{E}{min},h$, then $a$ strongly operator commutes with the negative of some element in the subdifferential of $h$ at $a$. These results improve known (operator) commutativity relations for linear $h$ and for solutions of variational inequality problems. We establish these results via a geometric commutation principle that is valid not only in Euclidean Jordan algebras, but also in the broader setting of FTvN-systems.
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The commutation principle of Ramirez, Seeger, and Sossa cite{ramirez-seeger-sossa} proved in the setting of Euclidean Jordan algebras says that when the sum of a Fr{e}chet differentiable function $Theta(x)$ and a spectral function $F(x)$ is minimized over a spectral set $Omega$, any local minimizer $a$ operator commutes with the Fr{e}chet derivative $Theta^{prime}(a)$. In this paper, we extend this result to sets and functions which are (just) invariant under algebra automorphisms. We also consider a similar principle in the setting of normal decomposition systems.
Let V be a Euclidean Jordan algebra of rank n. The eigenvalue map from V to R^n takes any element x in V to the vector of eigenvalues of x written in the decreasing order. A spectral set in V is the inverse image of a permutation set in R^n under the eigenvalue map. If the permutation set is also a convex cone, the spectral set is said to be a spectral cone. This paper deals with connectedness and arcwise connectedness properties of spectral sets. By relying on the result that in a simple Euclidean Jordan algebra, every eigenvalue orbit is arcwise connected, we show that if a permutation invariant set is connected (arcwise connected), then the corresponding spectral set is connected (respectively, arcwise connected). A related result is that in a simple Euclidean Jordan algebra, every pointed spectral cone is irreducible.
Motivated by Horns log-majorization (singular value) inequality $s(AB)underset{log}{prec} s(A)*s(B)$ and the related weak-majorization inequality $s(AB)underset{w}{prec} s(A)*s(B)$ for square complex matrices, we consider their Hermitian analogs $lambda(sqrt{A}Bsqrt{A}) underset{log}{prec} lambda(A)*lambda(B)$ for positive semidefinite matrices and $lambda(|Acirc B|) underset{w}{prec} lambda(|A|)*lambda(|B|)$ for general (Hermitian) matrices, where $Acirc B$ denotes the Jordan product of $A$ and $B$ and $*$ denotes the componentwise product in $R^n$. In this paper, we extended these inequalities to the setting of Euclidean Jordan algebras in the form $lambdabig (P_{sqrt{a}}(b)big )underset{log}{prec} lambda(a)*lambda(b)$ for $a,bgeq 0$ and $lambdabig (|acirc b|big )underset{w}{prec} lambda(|a|)*lambda(|b|)$ for all $a$ and $b$, where $P_u$ and $lambda(u)$ denote, respectively, the quadratic representation and the eigenvalue vector of an element $u$. We also describe inequalities of the form $lambda(|Abullet b|)underset{w}{prec} lambda({mathrm{diag}}(A))*lambda(|b|)$, where $A$ is a real symmetric positive semidefinite matrix and $A,bullet, b$ is the Schur product of $A$ and $b$. In the form of an application, we prove the generalized H{o}lder type inequality $||acirc b||_pleq ||a||_r,||b||_s$, where $||x||_p:=||lambda(x)||_p$ denotes the spectral $p$-norm of $x$ and $p,q,rin [1,infty]$ with $frac{1}{p}=frac{1}{r}+frac{1}{s}$. We also give precise values of the norms of the Lyapunov transformation $L_a$ and $P_a$ relative to two spectral $p$-norms.
We consider optimization problems on Riemannian manifolds with equality and inequality constraints, which we call Riemannian nonlinear optimization (RNLO) problems. Although they have numerous applications, the existing studies on them are limited especially in terms of algorithms. In this paper, we propose Riemannian sequential quadratic optimization (RSQO) that uses a line-search technique with an ell_1 penalty function as an extension of the standard SQO algorithm for constrained nonlinear optimization problems in Euclidean spaces to Riemannian manifolds. We prove its global convergence to a Karush-Kuhn-Tucker point of the RNLO problem by means of parallel transport and the exponential mapping. Furthermore, we establish its local quadratic convergence by analyzing the relationship between sequences generated by RSQO and the Riemannian Newton method. Ours is the first algorithm that has both global and local convergence properties for constrained nonlinear optimization on Riemannian manifolds. Empirical results show that RSQO finds solutions more stably and with higher accuracy compared with the existing Riemannian penalty and augmented Lagrangian methods.
Many problems in nonlinear analysis and optimization, among them variational inequalities and minimization of convex functions, can be reduced to finding zeros (namely, roots) of set-valued operators. Hence numerous algorithms have been devised in order to achieve this task. A lot of these algorithms are inexact in the sense that they allow perturbations to appear during the iterative process, and hence they enable one to better deal with noise and computational errors, as well as superiorization. For many years a certain fundamental question has remained open regarding many of these known inexact algorithmic schemes in various finite and infinite dimensional settings, namely whether there exist sequences satisfying these inexact schemes when errors appear. We provide a positive answer to this question. Our results also show that various theorems discussing the convergence of these inexact schemes have a genuine merit beyond the exact case. As a by-product we solve the standard and the strongly implicit inexact resolvent inclusion problems, introduce a promising class of functions (fully Legendre functions), establish continuous dependence (stability) properties of the solution of the inexact resolvent inclusion problem and continuity properties of the protoresolvent, and generalize the notion of strong monotonicity.
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