1. Motivation and basic concepts In this note, we consider two signomial functions whose convexity play important roles in some recent papers [4, 6, 7, 8, 9] dealing with geometric programming problems. However, the verifications therein contain some certain flaws and those incorrect arguments are repeatedly appeared and cited. From point of scientific researchs view, we hereby provide correct proofs for them.

We provide an affirmative answer to an question that the Fischer-Burmeister complementarity function associated with symmetric cones, named the FB SC complementarity function, is globally Lipschitz continuous and strongly semismooth everywhere for Hn and Qn. This is achieved with the help of embedding Hn and Qn into certain Sm.

Given a Hilbert space H, the infinite-dimensional Lorentz/second-order cone K is introduced. For any x H, a spectral decomposition is introduced, and for any function f: R R, we define a corresponding vector-valued function f H (x) on Hilbert space H by applying f to the spectral values of the spectral decomposition of x H with respect to K. We show that this vector-valued function inherits from f the properties of continuity, Lipschitz continuity, differentiability, smoothness, as well as s-semismoothness. These results can be helpful for designing and analyzing solution methods for solving infinite-dimensional second-order cone programs and complementarity problems.

Processing streaming data as they arrive is often necessary for high dimensional data analysis. In this paper, we analyze the convergence of a subspace online PCA iteration, as a followup of the recent work of Li, Wang, Liu, and Zhang [Math. Program., Ser. B, DOI 10.1007/s10107-017-1182-z] who considered the case for the most significant principal component only, i.e., a single vector. Under the sub-Gaussian assumption, we obtain a finite-sample error bound that closely matches the minimax information lower bound of Vu and Lei [Ann. Statist. 41:6 (2013), 2905-2947].

We introduce an entropy-like proximal algorithm for the problem of minimizing a closed proper convex function subject to symmetric cone constraints. The algorithm is based on a distance-like function that is an extension of the Kullback-Leiber relative entropy to the setting of symmetric cones. Like the proximal algorithms for convex programming with nonnegative orthant cone constraints, we show that, under some mild assumptions, the sequence generated by the proposed algorithm is bounded and every accumulation point is a solution of the considered problem. In addition, we also present a dual application of the proposed algorithm to the symmetric cone linear program, leading to a multiplier method which is shown to possess similar properties as the exponential multiplier method (Tseng and Bertsekas in Math. Program. 60:119, 1993) holds.