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We propose a primal-dual interior-point method (IPM) with convergence to second-order stationary points (SOSPs) of nonlinear semidefinite optimization problems, abbreviated as NSDPs. As far as we know, the current algorithms for NSDPs only ensure convergence to first-order stationary points such as Karush-Kuhn-Tucker points. The proposed method generates a sequence approximating SOSPs while minimizing a primal-dual merit function for NSDPs by using scaled gradient directions and directions of negative curvature. Under some assumptions, the generated sequence accumulates at an SOSP with a worst-case iteration complexity. This result is also obtained for a primal IPM with slight modification. Finally, our numerical experiments show the benefits of using directions of negative curvature in the proposed method.
We provide a condition-based analysis of two interior-point methods for unconstrained geometric programs, a class of convex programs that arise naturally in applications including matrix scaling, matrix balancing, and entropy maximization. Our condit
We establish lower bounds on the complexity of finding $epsilon$-stationary points of smooth, non-convex high-dimensional functions using first-order methods. We prove that deterministic first-order methods, even applied to arbitrarily smooth functio
Finite-sum optimization problems are ubiquitous in machine learning, and are commonly solved using first-order methods which rely on gradient computations. Recently, there has been growing interest in emph{second-order} methods, which rely on both gr
We propose an algorithm for solving nonlinear convex programs defined in terms of a symmetric positive semidefinite matrix variable $X$. This algorithm rests on the factorization $X=Y Y^T$, where the number of columns of Y fixes the rank of $X$. It i
In this paper we study second-order optimality conditions for non-convex set-constrained optimization problems. For a convex set-constrained optimization problem, it is well-known that second-order optimality conditions involve the support function o