A New Homotopy Proximal Variable-Metric Framework for Composite Convex Minimization

This paper suggests two novel ideas to develop new proximal variable-metric methods for solving a class of composite convex optimization problems. The first idea is to utilize a new parameterization strategy of the optimality condition to design a class of homotopy proximal variable-metric algorithms that can achieve linear convergence and finite global iteration-complexity bounds. We identify at least three subclasses of convex problems in which our approach can apply to achieve linear convergence rates. The second idea is a new primal-dual-primal framework for implementing proximal Newton methods that has attractive computational features for a subclass of nonsmooth composite convex minimization problems. We specialize the proposed algorithm to solve a covariance estimation problem in order to demonstrate its computational advantages. Numerical experiments on the four concrete applications are given to illustrate the theoretical and computational advances of the new methods compared with other state-of-the-art algorithms.

Minimizing Uniformly Convex Functions by Cubic Regularization of Newton Method

In this paper, we study the iteration complexity of cubic regularization of Newton method for solving composite minimization problems with uniformly convex objective. We introduce the notion of second-order condition number of a certain degree and justify the linear rate of convergence in a nondegenerate case for the method with an adaptive estimate of the regularization parameter. The algorithm automatically achieves the best possible global complexity bound among different problem classes of uniformly convex objective functions with Hölder continuous Hessian of the smooth part of the objective. As a byproduct of our developments, we justify an intuitively plausible result that the global iteration complexity of the Newton method is always better than that of the gradient method on the class of strongly convex functions with uniformly bounded second derivative.

Complexities in Assessing Structural Health of Civil Infrastructures

The complexity in the health assessment of civil infrastructures, as it evolves over a long period of time, is briefly discussed. A simple problem can become very complex based on the current needs, sophistication required, and the technological advancements. To meet the current needs of locating defect spots and their severity accurately and efficiently, infrastructures are represented by finite elements. To increase the implementation potential, the stiffness parameters of all the elements are identified and tracked using only few noise-contaminated dynamic responses measured at small part of the infrastructure. To extract the required information, Kalman filter concept is integrated with other numerical schemes. An unscented Kalman filter (UKF) concept is developed for highly nonlinear dynamic systems. It is denoted as 3D UKF-UI-WGI. The basic UKF concept is improved in several ways. Instead of using one long duration time-history in one global iteration, very short duration time-histories and multiple global iterations with weight factors are used to locate the defect spot more accurately and efficiently. The capabilities of the procedure are demonstrated with the help of two informative examples. The proposed procedure is much superior to the extended Kalman filter-based procedures developed by the team earlier.

A Relax Inexact Accelerated Proximal Gradient Method for the Constrained Minimization Problem of Maximum Eigenvalue Functions

For constrained minimization problem of maximum eigenvalue functions, since the objective function is nonsmooth, we can use the approximate inexact accelerated proximal gradient (AIAPG) method (Wang et al., 2013) to solve its smooth approximation minimization problem. When we take the functiong(X)=δΩ(X) (Ω∶={X∈Sn:F(X)=b,X⪰0})in the problemmin{λmax(X)+g(X):X∈Sn}, whereλmax(X)is the maximum eigenvalue function,g(X)is a proper lower semicontinuous convex function (possibly nonsmooth) andδΩ(X)denotes the indicator function. But the approximate minimizer generated by AIAPG method must be contained inΩotherwise the method will be invalid. In this paper, we will consider the case where the approximate minimizer cannot be guaranteed inΩ. Thus we will propose two different strategies, respectively, constructing the feasible solution and designing a new method named relax inexact accelerated proximal gradient (RIAPG) method. It is worth mentioning that one advantage when compared to the former is that the latter strategy can overcome the drawback. The drawback is that the required conditions are too strict. Furthermore, the RIAPG method inherits the global iteration complexity and attractive computational advantage of AIAPG method.