Consider a semidefinite program (SDP) involving an $n\times n$ positive semidefinite matrix $X$. The Burer-Monteiro method consists in solving a nonconvex program in $Y$, where $Y$ is an $n\times p$ matrix such that $X = Y Y^T$. Despite nonconvexity, Boumal et al. showed that the method provably solves generic equality-constrained SDP’s when $p > \sqrt{2m}$, … Read more
Backtracking line-search is an old yet powerful strategy for finding better step size to be used in proximal gradient algorithms. The main principle is to locally find a simple convex upper bound of the objective function, which in turn controls the step size that is used. In case of inertial proximal gradient algorithms, the situation … Read more
In many cases in which one wishes to minimize a complicated or expensive function, it is convenient to employ cheap approximations, at least when the current approximation to the solution is poor. Adequate strategies for deciding the accuracy desired at each stage of optimization are crucial for the global convergence and overall efficiency of the … Read more
A new extragradient projection method is devised in this paper, which does not obviously require generalized monotonicity and assumes only that the so-called dual variational inequality has a solution in order to ensure its global convergence. In particular, it applies to quasimonotone variational inequality having a nontrivial solution. ArticleDownload View PDF
In this paper, based on a modified Gram-Schmidt (MGS) process, we propose a class of derivative-free conjugate gradient (CG) projection methods for nonsmooth equations with convex constraints. Two attractive features of the new class of methods are: (i) its generated direction contains a free vector, which can be set as any vector such that the … Read more
Deep learning has recently attracted a significant amount of attention due to its great empirical success. However, the effectiveness in training deep neural networks (DNNs) remains a mystery in the associated nonconvex optimizations. In this paper, we aim to provide some theoretical understanding on such optimization problems. In particular, the Kurdyka-{\L}ojasiewicz (KL) property is established … Read more
The Douglas-Rachford algorithm is a classical and a successful method for solving the feasibility problems. Here, we provide a region for global convergence of the algorithm for the feasibility problem involving a disc and a circle in the Euclidean space of dimension two. Citation1. Borwein, J.M., Sims, B.: The Douglas-Rachford algorithm in the absence of … Read more
To improve the performance of the limited-memory variable metric L-BFGS method for large scale unconstrained optimization, repeating of some BFGS updates was proposed in [1, 2]. But the suitable extra updates need to be selected carefully, since the repeating process can be time consuming. We show that for the limited-memory variable metric BNS method, matrix … Read more
In this paper, we extend a class of globally convergent evolution strategies to handle general constrained optimization problems. The proposed framework handles relaxable constraints using a merit function approach combined with a specific restoration procedure. The unrelaxable constraints in our framework, when present, are treated either by using the extreme barrier function or through a … Read more
Concise complexity analyses are presented for simple trust region algorithms for solving unconstrained optimization problems. In contrast to a traditional trust region algorithm, the algorithms considered in this paper require certain control over the choice of trust region radius after any successful iteration. The analyses highlight the essential algorithm components required to obtain certain complexity … Read more