Symmetric Cone Research Articles

On a Box-Constrained Linear Symmetric Cone Optimization Problem

In this paper, an analytical expression of the optimal solution for a box-constrained linear symmetric cone optimization problem is proposed. The resulting theories are established based on the theory of the spectral decomposition of a symmetric cone. Moreover, we apply our results to develop algorithms for solving several symmetric cone optimization problems and conduct some preliminary numerical experiments to show the performance of the developed algorithms.

An Efficient Numerical Method for the Symmetric Positive Definite Second-Order Cone Linear Complementarity Problem

An efficient numerical method for solving a symmetric positive definite second-order cone linear complementarity problem (SOCLCP) is proposed. The method is shown to be more efficient than recently developed iterative methods for small-to-medium sized and dense SOCLCP. Therefore it can serve as an excellent core computational engine in solutions of large scale symmetric positive definite SOCLCP solved by subspace projection methods, solutions of general SOCLCP and the quadratic programming over a Cartesian product of multiple second-order cones, in which small-to-medium sized SOCLCPs have to be solved repeatedly, efficiently, and robustly.

Global Conic Shock Wave for the Steady Supersonic Flow Past a Curved Cone

In this paper, we establish the global existence of a steady symmetric conic shock wave when a supersonic incoming flow hits a symmetric cone for a steady potential flow. The speed of the incoming flow is close to the limit speed, and the generatrix of the cone is assumed to have an attack angle less than a critical value and to have a decay rate on its second order derivatives. Based on a background solution, which is introduced here for the first time, we apply the characteristic method to construct the global solution.

On some new sharp embedding theorems for multifunctional Herz-type and Bergman-type spaces in pseudoconvex domains

We introduce new multifunctional mixed norm analytic Herz-type spaces in strongly pseudoconvex domains and provide new sharp embedding theorems for them. Some results are new even in case of onefunctional holomorphic spaces. Some new related sharp results for new multifunctional Bergman-type spaces will be also provided under one condition on Bergman kernel. Similar results with similar proofs in unbounded tubular domains over symmetric cones and bounded symmetric domains will be also shortly mentioned.

Quadratic Capelli operators and Okounkov polynomials

Let $Z$ be the symmetric cone of $r \times r$ positive definite Hermitian matrices over a real division algebra $\mathbb F$. Then $Z$ admits a natural family of invariant differential operators -- the Capelli operators $C_\lambda$ -- indexed by partitions $\lambda$ of length at most $r$, whose eigenvalues are given by specialization of Knop--Sahi interpolation polynomials. In this paper we consider a double fibration $Y \longleftarrow X \longrightarrow Z$ where $Y$ is the Grassmanian of $r$-dimensional subspaces of $\mathbb F^n $ with $n \geq 2r$. Using this we construct a family of invariant differential operators $D_{\lambda,s}$ on $Y$ that we refer to as quadratic Capelli operators. Our main result shows that the eigenvalues of the $D_{\lambda,s}$ are given by specializations of Okounkov interpolation polynomials.

An Infeasible Interior-Point Algorithm for Stochastic Second-Order Cone Optimization

Alzalg (J Optim Theory Appl 163(1):148–164, 2014) derived a homogeneous self-dual algorithm for stochastic second-order cone programs with finite event space. In this paper, we derive an infeasible interior-point algorithm for the same stochastic optimization problem by utilizing the work of Rangarajan (SIAM J Optim 16(4), 1211–1229, 2006) for deterministic symmetric cone programs. We show that the infeasible interior-point algorithm developed in this paper has complexity less than that of the homogeneous self-dual algorithm mentioned above. We implement the proposed algorithm to show that they are efficient.

Optimality Conditions for Problems over Symmetric Cones and a Simple Augmented Lagrangian Method

In this work, we are interested in nonlinear symmetric cone problems (NSCPs), which contain as special cases nonlinear semidefinite programming, nonlinear second-order cone programming, and the classical nonlinear programming problems. We explore the possibility of reformulating NSCPs as common nonlinear programs (NLPs), with the aid of squared slack variables. Through this connection, we show how to obtain second-order optimality conditions for NSCPs in an easy manner, thus bypassing a number of difficulties associated to the usual variational analytical approach. We then discuss several aspects of this connection. In particular, we show a “sharp” criterion for membership in a symmetric cone that also encodes rank information. Also, we discuss the possibility of importing convergence results from nonlinear programming to NSCPs, which we illustrate by discussing a simple augmented Lagrangian method for nonlinear symmetric cones. We show that, employing the slack variable approach, we can use the results for NLPs to prove convergence results, thus extending a special case (i.e., the case with strict complementarity) of an earlier result by Sun et al. [Sun D, Sun J, Zhang L (2008) The rate of convergence of the augmented Lagrangian method for nonlinear semidefinite programming. Math. Programming 114(2):349–391] for nonlinear semidefinite programs.

A long-step feasible predictor–corrector interior-point algorithm for symmetric cone optimization

ABSTRACTIn this paper, we present a feasible predictor–corrector interior-point method for symmetric cone optimization problem in the large neighbourhood of the central path. The method is generalization of Ai-Zhang's predictor–corrector algorithm to the symmetric cone optimization problem. Starting with a feasible point in given large neighbourhood of the central path, the algorithm still terminates in at most iterations. This matches the best known iteration bound that is usually achieved by short-step methods, thereby, closing the complexity gap between long- and short-step interior-point methods for symmetric cone optimization. The preliminary numerical results on a selected set of NETLIB problems show advantage of the method in comparison with the version of the algorithm that is not based on the predictor–corrector scheme.

Large-Neighborhood Infeasible Predictor–Corrector Algorithm for Horizontal Linear Complementarity Problems over Cartesian Product of Symmetric Cones

We present an infeasible interior-point predictor–corrector algorithm, based on a large neighborhood of the central path, for horizontal linear complementarity problem over the Cartesian product of symmetric cones. Throughout the paper, we assume that a certain property holds for the above-mentioned problem. This condition is equivalent to the property of sufficiency for the particular case of horizontal linear complementarity problem. The polynomial convergence is shown for the commutative class of search directions. We specialize our algorithm further by prescribing some scaling elements and also consider the case of feasible starting points. We believe this to be the first interior-point method based on large neighborhoods for the problem in consideration.

Hilbert and Thompson isometries on cones in JB-algebras

Hilbert’s and Thompson’s metric spaces on the interior of cones in JB-algebras are important examples of symmetric Banach-Finsler spaces. In this paper we characterize the Hilbert’s metric isometries on the interiors of cones in JBW-algebras, and the Thompson’s metric isometries on the interiors of cones in JB-algebras. These characterizations generalize work by Bosché on the Hilbert’s and Thompson’s metric isometries on symmetric cones, and work by Hatori and Molnár on the Thompson’s metric isometries on the cone of positive self-adjoint elements in a unital C^*-algebra. To obtain the results we develop a variety of new geometric and Jordan algebraic techniques.

Some Carleson measures for the Hilbert–Hardy space of tube domains over symmetric cones

In the setting of a general tube domain over a symmetric cone, we obtain a full characterization of measures of the form $$\varphi (y)\,dxdy$$ which are Carleson measures for the Hilbert–Hardy space; for large derivatives, we also obtain a full characterization of general positive measures for which the corresponding embedding operator is continuous. Restricting to the case of the Lorentz cone of dimension three, we prove that by freezing one or two secondary variables, the problem of embedding derivatives of the Hilbert–Hardy space into Lebesgue spaces reduces to the characterization of Carleson measures for Hilbert–Bergman spaces of the upper-half plane or the product of two upper-half planes.

A long-step interior-point algorithm for symmetric cone Cartesian P*(κ)-HLCP

ABSTRACTIn this paper, we present a feasible interior-point algorithm for Cartesian horizontal linear complementarity problems in a new large neighbourhood of the central path. The new large neighbourhood is based on the infinity norm, and it is wider than the well-known neighbourhood based on negative infinity pseudonorm as well as the recently introduced large neighbourhood by Liu et al. [A new wide neighborhood primal-dual infeasible-interior-point method for symmetric cone programming. J Optim Theory Appl. 2013;158:796–815] which is based on Frobenius norm. The iterates are calculated by taking the largest possible step along the Nesterov–Todd search directions. Nevertheless, we show that the algorithm is globally convergent with the favourable polynomial iteration bound. Furthermore, the preliminary numerical results indicate that our method preforms quite well and outperforms the large-step Liu et al.'s method.

Schatten class Toeplitz operators on weighted Bergman spaces of tube domains over symmetric cones

We prove some characterizations of Schatten class of Toeplitz operators on Bergman spaces of tube domains over symmetric cones for small exponents. Our results are applied to obtain Schatten class characterization of some Cesàro-type operators.

Experimental and computational investigation of segregation during tumblers unloading

Dry mixing is a common operation in many industrial processes. The unloading of the mixer is an essential aspect of the process which actively contributes to the final quality of the mixture. Its effect however is often underestimate or even ignored. In the present work the consequences of the unloading process on mixture quality were investigated in three different types of tumbling mixers: a symmetric double cone, an asymmetric double cone and a conic mixer were considered. The symmetric double cone mixer is a standard geometry in the pharmaceutical industry, while the other two geometries are unconventional. The effects of mixer geometry on the mixture quality were studied and related to differences in the unloading flow patterns. A segregating polydispersed mixture was used during the experiments in order to emphasize the different performances of the three tumblers. Additional quasi-2D experiments and FEM simulation of the unloading process clearly showed the ability of the unconventional geometries to mitigate the effect of segregation improving the mixture quality evaluated at the outlet of the mixer.

Characterization of the Cubilche debris avalanche deposit, a controversial case from the northern Andes, Ecuador

In areas characterized by many volcanoes, identifying the source of a deposit may not be trivial. This becomes much more complicated when looking for the source of a debris avalanche deposit (DAD), the common products of catastrophic volcanic edifice collapses. To overcome this problem, in this work a methodology is proposed based on the integration of texture features and areal distribution of the deposit, comparison between the petrography of the coarser clasts within the DAD and of the proximal products, grain-size analysis, and the volumetric estimations of the deposit and the volume missing from the volcanic edifice. This methodology has been tested to a DAD occurred near the city of Ibarra (Imbabura Province; Northern Ecuador), having a controversial source. Two main volcanic edifice are located in proximity of the DAD, the Cubilche volcano (3826 m.a.s.l.), located immediately south of and east of the colossal dormant Imbabura volcano. The former displays a sharp horseshoe-shaped scar towards the north and inside this post-collapse edifice, that we name old Cubilche volcano (OCV), is located the young Cubilche volcano (YCV) that refilled a portion of the collapse scar and partially covered the southern flank of the OCV. Detailed knowledge of Cubilche volcano is critical because of its close proximity and interspersed activity with Imbabura volcano. In fact, Imbabura most recent edifice was built over the northwestern slope of the OCV and partially covered it. Recent studies linked the studied DAD to both Imbabura volcano as a product of its northern sector collapse, as well as neighboring Cubilche volcano. Our data points to Cubilche as the most likely source for this DAD. A perspective view of the shaded relief image of the present day OCV shows that the morphology of the volcano is well-preserved on its southern, eastern, and western flanks. This allows us to reconstruct the morphology of the OCV previous to the collapse through interpolation of elevation and altitude data of preserved flanks. A DEM of the present day topography was used for extrapolating the morphology. Using similar methodology, the post collapse base of the amphitheater was reconstructed by removing the relief of the present day YCV. The reconstructed topography of the OCV shows that it could have been a symmetric cone, reaching a maximum elevation of ~4100 m.a.s.l. with a lack volume of ~3.5 km3. Based on this scenario, the deposit originated from the OCV main collapse should have a volume >3–3.5 km3 in accordance to the volume calculated for the studied DAD.

Iterative complexities of a class of homogeneous algorithms for monotone nonlinear complementarity problems over symmetric cones

ABSTRACTThis paper provides an analysis of the iterative complexities of a class of homogeneous algorithms for monotone nonlinear complementarity problems over symmetric cones. The proof of the complexity bounds requires that the nonlinear transformation satisfies an SLC. To prove the complexity bounds of the homogeneous algorithm, this paper proposes an SLC which does not depend on the scaling parameter p and is very easy to be verified. More important, it has scaled invariance. Underlying SLC, the obtained complexity bounds of the short-step algorithm, the semi-long-step algorithm and the long-step algorithm with the sx-direction match that of the homogeneous algorithms proposed by Yoshise [Homogenous algorithms for monotone complementarity problems over symmetric cones. Pac J Optim. 2008; 5(2):313–337].

Beta Distributions and Sonine Integrals for Bessel Functions on Symmetric Cones

AbstractThere exist several multivariate extensions of the classical Sonine integral representation for Bessel functions of some index with respect to such functions of lower index μ. For Bessel functions on matrix cones, Sonine formulas involve beta densities on the cone and trace already back to Herz. The Sonine representations known so far on symmetric cones are restricted to continuous ranges , where the involved beta densities are probability measures and the limiting index depends on the rank of the cone. It is zero only in the one‐dimensional case, but larger than zero in all multivariate cases. In this paper, we study the extension of Sonine formulas for Bessel functions on symmetric cones to values of ν below the critical limit μ0. This is achieved by an analytic extension of the involved beta measures as tempered distributions. Following recent ideas by A. Sokal for Riesz distributions on symmetric cones, we analyze for which indices the obtained beta distributions are still measures. At the same time, we characterize the indices for which a Sonine formula between the related Bessel functions exists. As for Riesz distributions, there occur gaps in the admissible range of indices, which are determined by the so‐called Wallach set.

English

We obtain new sharp embedding theorems for mixed-norm Herz-type analytic spaces in tubular domains over symmetric cones. These results enlarge the list of recent sharp theorems in analytic spaces obtained by Nana and Sehba (2015).

The shape of low energy configurations of a thin elastic sheet with a single disclination

We consider a geometrically fully nonlinear variational model for thin elastic sheets that contain a single disclination. The free elastic energy contains the thickness $h$ as a small parameter. We give an improvement of a recently proved energy scaling law, removing the next-to leading order terms in the lower bound. Then we prove the convergence of (almost-)minimizers of the free elastic energy towards the shape of a radially symmetric cone, up to Euclidean motions, weakly in the spaces $W^{2,2}(B_1\setminus B_\rho;\mathbb{R}^3)$ for every $0<\rho<1$, as the thickness $h$ is sent to 0.

Singular Riesz measures on symmetric cones

A fondamental theorem due to Gindikin [Russian Math. Surveys, 29 (1964), 1-89] says that the‎ ‎generalized power $\Delta_{s}(-\theta^{-1})$ defined on a symmetric‎ ‎cone is the Laplace transform of a positive measure $R_{s}$ if and ‎only if $s$ is in a given subset $\Xi$ of $\mathbb{R}^{r}$‎, ‎where $r$‎ ‎is the rank of the cone‎. ‎When $s$ is in a well defined part of‎ ‎$\Xi$‎, ‎the measure $R_{s}$ is absolutely continuous with respect to‎ ‎Lebesgue measure and has a known expression‎. ‎For the other elements‎ ‎$s$ of $\Xi$‎, ‎the measure $R_{s}$ is concentrated on the boundary of‎ ‎the cone and it has never been explicitly determined‎. ‎The aim of the‎ ‎present paper is to give an explicit description of the measure‎ ‎$R_{s}$ for all $s$ in $\Xi$‎. ‎The work is motivated by the‎ ‎importance of these measures in probability theory and in statistics‎ ‎since they represent a generalization of the class of measures‎ ‎generating the famous Wishart probability distributions‎.