Given two arbitrary closed sets in Euclidean space, a simple transversality condition guarantees that the method of alternating projections converges locally, at linear rate, to a point in the intersection. Exact projection onto nonconvex sets is typically intractable, but we show that computationally-cheap inexact projections may suffice instead. In particular, if one set is defined … Read more
We consider optimization problems over the Stiefel manifold whose objective function is the summation of a smooth function and a nonsmooth function. Existing methods for solving this kind of problems can be classified into three classes. Algorithms in the first class rely on information of the subgradients of the objective function and thus tend to … Read more
The sparse portfolio selection problem is one of the most famous and frequently-studied problems in the optimization and financial economics literatures. In a universe of risky assets, the goal is to construct a portfolio with maximal expected return and minimum variance, subject to an upper bound on the number of positions, linear inequalities and minimum … Read more
An optimization approach is proposed to construct sparse portfolios with mean-reverting price behaviors. Our objectives are threefold: (i) design a multi-asset long-short portfolio that best fits an Ornstein-Uhlenbeck process in terms of maximum likelihood, (ii) select portfolios with desirable characteristics of high mean reversion and low variance though penalization, and (iii) select a parsimonious portfolio … Read more
In this article, we present a framework for the numerical solution of optimal control problems, constrained by ordinary differential equations which can run in (finitely many) different modes, where a change of modes leads to additional switching cost in the cost function, and whenever the system changes its mode, jumps in the differential states are … Read more
The main focus in this paper is exact linesearch methods for minimizing a quadratic function whose Hessian is positive definite. We give two classes of limited-memory quasi-Newton Hessian approximations that generate search directions parallel to those of the method of preconditioned conjugate gradients, and hence give finite termination on quadratic optimization problems. The Hessian approximations … Read more
We study the class of nonsmooth nonconvex problems in which the objective is to minimize the difference between a continuously differentiable function (possibly nonconvex) and a convex (possibly nonsmooth) function over a convex polytope. This general class contains many types of problems, including difference of convex functions (DC) problems, and as such, can be used … Read more
Gradient-based optimization algorithms can be studied from the perspective of limiting or- dinary differential equations (ODEs). Motivated by the fact that existing ODEs do not distin- guish between two fundamentally different algorithms—Nesterov’s accelerated gradient method for strongly convex functions (NAG-SC) and Polyak’s heavy-ball method—we study an alter- native limiting process that yields high-resolution ODEs. We … Read more
We propose a novel framework for analyzing convergence rates of stochastic optimization algorithms with adaptive step sizes. This framework is based on analysing properties of an underlying generic stochastic process, in particular by deriving a bound on the expected stopping time of this process. We utilise this framework to analyse the bounds on expected global … Read more
In this work we consider the problem of minimizing a differentiable functional restricted to the set of $n\times p$ matrices with orthonormal columns. This problem appears in several fields such as statistics, signal processing, global positioning system, machine learning, physics, chemistry and others. We present an algorithm based on a recent non-monotone variation of the … Read more