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
We propose a new stepsize for the gradient method. It is shown that this new stepsize will converge to the reciprocal of the largest eigenvalue of the Hessian, when Dai-Yang’s asymptotic optimal gradient method (Computational Optimization and Applications, 2006, 33(1): 73-88) is applied for minimizing quadratic objective functions. Based on this spectral property, we develop … Read more
A regularization algorithm using inexact function values and inexact derivatives is proposed and its evaluation complexity analyzed. This algorithm is applicable to unconstrained problems and to problems with inexpensive constraints (that is constraints whose evaluation and enforcement has negligible cost) under the assumption that the derivative of highest degree is beta-H\”{o}lder continuous. It features a … Read more
Regularized regression problems are ubiquitous in statistical modeling, signal processing, and machine learning. Sparse regression in particular has been instrumental in scientific model discovery, including compressed sensing applications, vari- able selection, and high-dimensional analysis. We propose a broad framework for sparse relaxed regularized regression, called SR3. The key idea is to solve a relaxation of … Read more
We address the problem of deterministically determining upper bounds in continuous non-convex global minimization of box-constrained problems with equality constraints. These upper bounds are important for the termination of spatial branch-and-bound algorithms. Our method is based on the theorem of Miranda which helps to ensure the existence of feasible points in certain boxes. Then, the … Read more
We provide sharp worst-case evaluation complexity bounds for nonconvex minimization problems with general inexpensive constraints, i.e.\ problems where the cost of evaluating/enforcing of the (possibly nonconvex or even disconnected) constraints, if any, is negligible compared to that of evaluating the objective function. These bounds unify, extend or improve all known upper and lower complexity bounds … Read more
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