A second-derivative trust-region SQP method with a "trust-region-free" predictor step

In (NAR 08/18 and 08/21, Oxford University Computing Laboratory, 2008) we introduced a second-derivative SQP method (S2QP) for solving nonlinear nonconvex optimization problems. We proved that the method is globally convergent and locally superlinearly convergent under standard assumptions. A critical component of the algorithm is the so-called predictor step, which is computed from a strictly convex quadratic program with a trust-region constraint. This step is essential for proving global convergence, but its propensity to identify the optimal active set is Paramount for recovering fast local convergence. Thus the global and local efficiency of the method is intimately coupled with the quality of the predictor step.\ud \ud In this paper we study the effects of removing the trust-region constraint from the computation of the predictor step; this is reasonable since the resulting problem is still strictly convex and thus well-defined. Although this is an interesting theoretical question, our motivation is based on practicality. Our preliminary numerical experience with S2QP indicates that the trust-region constraint occasionally degrades the quality of the predictor step and diminishes its ability to correctly identify the optimal active set. Moreover, removal of the trust-region constraint allows for re-use of the predictor step over a sequence of failed iterations thus reducing computation. We show that the modified algorithm remains globally convergent and preserves local superlinear convergence provided a nonmonotone strategy is incorporated

A second derivative SQP method: theoretical issues

Sequential quadratic programming (SQP) methods form a class of highly efficient algorithms for solving nonlinearly constrained optimization problems. Although second derivative information may often be calculated, there is little practical theory that justifies exact-Hessian SQP methods. In particular, the resulting quadratic programming (QP) subproblems are often nonconvex, and thus finding their global solutions may be computationally nonviable. This paper presents a second-derivative SQP method based on quadratic subproblems that are either convex, and thus may be solved efficiently, or need not be solved globally. Additionally, an explicit descent-constraint is imposed on certain QP subproblems, which “guides” the iterates through areas in which nonconvexity is a concern. Global convergence of the resulting algorithm is established

Forward-backward truncated Newton methods for convex composite optimization

This paper proposes two proximal Newton-CG methods for convex nonsmooth optimization problems in composite form. The algorithms are based on a a reformulation of the original nonsmooth problem as the unconstrained minimization of a continuously differentiable function, namely the forward-backward envelope (FBE). The first algorithm is based on a standard line search strategy, whereas the second one combines the global efficiency estimates of the corresponding first-order methods, while achieving fast asymptotic convergence rates. Furthermore, they are computationally attractive since each Newton iteration requires the approximate solution of a linear system of usually small dimension

MAGMA: Multi-level accelerated gradient mirror descent algorithm for large-scale convex composite minimization

Composite convex optimization models arise in several applications, and are especially prevalent in inverse problems with a sparsity inducing norm and in general convex optimization with simple constraints. The most widely used algorithms for convex composite models are accelerated first order methods, however they can take a large number of iterations to compute an acceptable solution for large-scale problems. In this paper we propose to speed up first order methods by taking advantage of the structure present in many applications and in image processing in particular. Our method is based on multi-level optimization methods and exploits the fact that many applications that give rise to large scale models can be modelled using varying degrees of fidelity. We use Nesterov's acceleration techniques together with the multi-level approach to achieve $\mathcal{O}(1/\sqrt{\epsilon})$ convergence rate, where $\epsilon$ denotes the desired accuracy. The proposed method has a better convergence rate than any other existing multi-level method for convex problems, and in addition has the same rate as accelerated methods, which is known to be optimal for first-order methods. Moreover, as our numerical experiments show, on large-scale face recognition problems our algorithm is several times faster than the state of the art

DC Proximal Newton for Non-Convex Optimization Problems

We introduce a novel algorithm for solving learning problems where both the loss function and the regularizer are non-convex but belong to the class of difference of convex (DC) functions. Our contribution is a new general purpose proximal Newton algorithm that is able to deal with such a situation. The algorithm consists in obtaining a descent direction from an approximation of the loss function and then in performing a line search to ensure sufficient descent. A theoretical analysis is provided showing that the iterates of the proposed algorithm {admit} as limit points stationary points of the DC objective function. Numerical experiments show that our approach is more efficient than current state of the art for a problem with a convex loss functions and non-convex regularizer. We have also illustrated the benefit of our algorithm in high-dimensional transductive learning problem where both loss function and regularizers are non-convex

Nonmonotone Barzilai-Borwein Gradient Algorithm for ℓ1\ell_1-Regularized Nonsmooth Minimization in Compressive Sensing

This paper is devoted to minimizing the sum of a smooth function and a nonsmooth $\ell_1$-regularized term. This problem as a special cases includes the $\ell_1$-regularized convex minimization problem in signal processing, compressive sensing, machine learning, data mining, etc. However, the non-differentiability of the $\ell_1$-norm causes more challenging especially in large problems encountered in many practical applications. This paper proposes, analyzes, and tests a Barzilai-Borwein gradient algorithm. At each iteration, the generated search direction enjoys descent property and can be easily derived by minimizing a local approximal quadratic model and simultaneously taking the favorable structure of the $\ell_1$-norm. Moreover, a nonmonotone line search technique is incorporated to find a suitable stepsize along this direction. The algorithm is easily performed, where the values of the objective function and the gradient of the smooth term are required at per-iteration. Under some conditions, the proposed algorithm is shown to be globally convergent. The limited experiments by using some nonconvex unconstrained problems from CUTEr library with additive $\ell_1$-regularization illustrate that the proposed algorithm performs quite well. Extensive experiments for $\ell_1$-regularized least squares problems in compressive sensing verify that our algorithm compares favorably with several state-of-the-art algorithms which are specifically designed in recent years.Comment: 20 page