Successive quadratic approximations, or second-order proximal methods, are useful for minimizing functions that are a sum of a smooth part and a convex, possibly nonsmooth part that promotes a regularized solution. Most analyses of iteration complexity focus on the special case of proximal gradient method, or accelerated variants thereof. There have been only a few studies of methods that use a second-order approximation to the smooth part, due in part to the difficulty of obtaining closed-form solutions to the subproblems at each iteration. In practice, iterative algorithms need to be used to find inexact solutions to the subproblems. In this work, we present global analysis of the iteration complexity of inexact successive quadratic approximation methods, showing that it is sufficient to obtain an inexact solution of the subproblem to fixed multiplicative precision in order to guarantee the same order of convergence rate as the exact version, with complexity related proportionally to the degree of inexactness. Our result allows flexible choices of the second-order terms, including Newton and quasi-Newton choices, and does not necessarily require more time to be spent on the subproblem solves on later iterations. For problems exhibiting a property related to strong convexity, the algorithm con- verges at a global linear rate. For general convex problems, the convergence rate is linear in early stages, while the overall rate is @math . For nonconvex problems, a first-order optimality criterion converges to zero at a rate of @math .
When @math is the indicator function of a convex constraint set, our approach includes an inexact variant of a constrained Newton or quasi-Newton method. There are a number of papers on this approach, but their convergence results generally have a different flavor from ours. They typically show only asymptotic convergence rates, together with global convergence results without rates, under weaker smoothness and convexity assumptions on @math than we make here. For example, when @math is the indicator function of a box'' defined by bound constraints, applies a trust-region framework to solve approximately, and shows asymptotic convergence. The paper @cite_25 uses a line-search approach, with @math defined by an L-BFGS update, and omits convergence results. For constraint sets defined by linear inequalities, or general convex constraints, @cite_2 shows global convergence of a trust region method using the Cauchy point. A similar approach using the exact Hessian as @math is considered in @cite_29 , proving local superlinear or quadratic convergence in the case of linear constraints.
@cite_29 @cite_25 @cite_2
2151568819 2035741312 2034171981 2768546550
We investigate the behavior of quasi-Newton algorithms applied to minimize a nonsmooth function f, not necessarily convex. We introduce an inexact line search that generates a sequence of nested intervals containing a set of points of nonzero measure that satisfy the Armijo and Wolfe conditions if f is absolutely continuous along the line. Furthermore, the line search is guaranteed to terminate if f is semi-algebraic. It seems quite difficult to establish a convergence theorem for quasi-Newton methods applied to such general classes of functions, so we give a careful analysis of a special but illuminating case, the Euclidean norm, in one variable using the inexact line search and in two variables assuming that the line search is exact. In practice, we find that when f is locally Lipschitz and semi-algebraic with bounded sublevel sets, the BFGS (Broyden–Fletcher–Goldfarb–Shanno) method with the inexact line search almost always generates sequences whose cluster points are Clarke stationary and with function values converging R-linearly to a Clarke stationary value. We give references documenting the successful use of BFGS in a variety of nonsmooth applications, particularly the design of low-order controllers for linear dynamical systems. We conclude with a challenging open question. We develop a convergence theory for convex and linearly constrained trust region methods which only requires that the step between iterates produce a sufficient reduction in the trust region subproblem. Global convergence is established for general convex constraints while the local analysis is for linearly constrained problems. The main local result establishes that if the sequence converges to a nondegenerate stationary point then the active constraints at the solution are identified in a finite number of iterations. As a consequence of the identification properties, we develop rate of convergence results by assuming that the step is a truncated Newton method. Our development is mainly geometrical; this approach allows the development of a convergence theory without any linear independence assumptions. This paper considers local convergence and rate of convergence results for algorithms for minimizing the composite functionF(x)=f(x)+h(c(x)) wheref andc are smooth buth(c) may be nonsmooth. Local convergence at a second order rate is established for the generalized Gauss—Newton method whenh is convex and globally Lipschitz and the minimizer is strongly unique. Local convergence at a second order rate is established for a generalized Newton method when the minimizer satisfies nondegeneracy, strict complementarity and second order sufficiency conditions. Assuming the minimizer satisfies these conditions, necessary and sufficient conditions for a superlinear rate of convergence for curvature approximating methods are established. Necessary and sufficient conditions for a two-step superlinear rate of convergence are also established when only reduced curvature information is available. All these local convergence and rate of convergence results are directly applicable to nonlinearing programming problems. First-order methods have been popularly used for solving large-scale problems. However, many existing works only consider unconstrained problems or those with simple constraint. In this paper, we develop two first-order methods for constrained convex programs, for which the constraint set is represented by affine equations and smooth nonlinear inequalities. Both methods are based on the classic augmented Lagrangian function. They update the multipliers in the same way as the augmented Lagrangian method (ALM) but employ different primal variable updates. The first method, at each iteration, performs a single proximal gradient step to the primal variable, and the second method is a block update version of the first one. For the first method, we establish its global iterate convergence as well as global sublinear and local linear convergence, and for the second method, we show a global sublinear convergence result in expectation. Numerical experiments are carried out on the basis pursuit denoising and a convex quadratically constrained quadratic program to show the empirical performance of the proposed methods. Their numerical behaviors closely match the established theoretical results.