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In this paper, we consider a type of cone-constrained convex program in finite-dimensional space, and are interested in characterization of the solution set of this convex program with the help of the Lagrange multiplier. We establish necessary conditions for a feasible point being an optimal solution. Moreover, some necessary conditions and sufficient conditions are established which simplifies the corresponding results in Jeyakumar et al. (J Optim Theory Appl 123(1), 83103, 2004). In particular, when the cone reduces to three specific cones, that is, the p -order cone, p cone and circular cone, we show that the obtained results can be achieved by easier ways by exploiting the special structure of those three cones.

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].

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We make a unified analysis of interior proximal methods of solving convex second-order cone programming problems. These methods use a proximal distance with respect to second-order cones which can be produced with an appropriate closed proper univariate function in three ways. Under some mild conditions, the sequence generated is bounded with each limit point being a solution, and global rates of convergence estimates are obtained in terms of objective values. A class of regularized proximal distances is also constructed which can guarantee the global convergence of the sequence to an optimal solution. These results are illustrated with some examples. In addition, we also study the central paths associated with these distance-like functions, and for the linear SOCP we discuss their relations with the sequence generated by the interior proximal methods. From this, we obtain improved convergence

In this paper, we study the properties of the penalized Fischer-Burmeister (FB) second-order cone (SOC) complementarity function. We show that the function possesses similar desirable properties of the FB SOC complementarity function for local convergence; for example, with the function the second-order cone complementarity problem (SOCCP) can be reformulated as a (strongly) semismooth system of equations, and the corresponding nonsmooth Newton method has local quadratic convergence without strict complementarity of solutions. In addition, the penalized FB merit function has bounded level sets under a rather weak condition which can be satisfied by strictly feasible monotone SOCCPs or SOCCPs with the Cartesian <i>R</i> ₀₁-property, although it is not continuously differentiable. Numerical results are included to illustrate the theoretical considerations.

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.

In this paper we derive new fractional error bounds for polynomial systems with exponents explicitly determined by the dimension of the underlying space and the number/degree of the involved polynomials. Our major result extends the existing error bounds from the system involving only a single polynomial to a general polynomial system and do not require any regularity assumptions. In this way we resolve, in particular, some open questions posed in the literature. The developed techniques are largely based on variational analysis and generalized differentiation, which allow us to establish, e.g., a nonsmooth extension of the seminal Łojasiewicz’s gradient inequality to maxima of polynomials with explicitly determined exponents. Our major applications concern quantitative Hölderian stability of solution maps for parameterized polynomial optimization problems and nonlinear complementarity systems with polynomial data as well as high-order semismooth properties of the eigenvalues of symmetric tensors.

We investigate a one-parametric class of merit functions for the second-order cone complementarity problem (SOCCP) which is closely related to the popular FischerBurmeister (FB) merit function and natural residual merit function. In fact, it will reduce to the FB merit function if the involved parameter <i></i> equals 2, whereas as <i></i> tends to zero, its limit will become a multiple of the natural residual merit function. In this paper, we show that this class of merit functions enjoys several favorable properties as the FB merit function holds, for example, the smoothness. These properties play an important role in the reformulation method of an unconstrained minimization or a nonsmooth system of equations for the SOCCP. Numerical results are reported for some convex second-order cone programs (SOCPs) by solving the unconstrained minimization reformulation of the KKT optimality conditions, which indicate that the