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 .
A version of Algorithm without line search but requiring @math to overestimate the Hessian, as follows: is considered in @cite_19 . Asymptotic convergence is proved, but no rates are given. Convergence of an inexact proximal-gradient method (for which @math for all @math ) is discussed in @cite_5 . With this choice of @math , always holds with @math . They also discuss its accelerated version for convex and strongly convex problems. Instead of our multiplicative inexactness criterion, they assume an additive inexactness criterion in the subproblem, of the form Their analysis also allows for an error @math in the gradient term in . The paper shows that for general convex problems, the objective value converges at a @math rate provided that @math and @math converge. For strongly convex problems, they proved R-linear convergence of @math , provided that the sequence @math and @math both decrease linearly to zero. When our approaches are specialized to proximal gradient ( @math ), our analysis shows a Q-linear rate (rather than R-linear) for the strongly convex case, and applies to the convergence of the objective value rather than the iterates. Additionally, our results shows convergence for nonconvex problems.
@cite_19 @cite_5
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In this paper, we generalize the well-known Nesterov's accelerated gradient (AG) method, originally designed for convex smooth optimization, to solve nonconvex and possibly stochastic optimization problems. We demonstrate that by properly specifying the stepsize policy, the AG method exhibits the best known rate of convergence for solving general nonconvex smooth optimization problems by using first-order information, similarly to the gradient descent method. We then consider an important class of composite optimization problems and show that the AG method can solve them uniformly, i.e., by using the same aggressive stepsize policy as in the convex case, even if the problem turns out to be nonconvex. We demonstrate that the AG method exhibits an optimal rate of convergence if the composite problem is convex, and improves the best known rate of convergence if the problem is nonconvex. Based on the AG method, we also present new nonconvex stochastic approximation methods and show that they can improve a few existing rates of convergence for nonconvex stochastic optimization. To the best of our knowledge, this is the first time that the convergence of the AG method has been established for solving nonconvex nonlinear programming in the literature. In this paper, we generalize the well-known Nesterov's accelerated gradient (AG) method, originally designed for convex smooth optimization, to solve nonconvex and possibly stochastic optimization problems. We demonstrate that by properly specifying the stepsize policy, the AG method exhibits the best known rate of convergence for solving general nonconvex smooth optimization problems by using first-order information, similarly to the gradient descent method. We then consider an important class of composite optimization problems and show that the AG method can solve them uniformly, i.e., by using the same aggressive stepsize policy as in the convex case, even if the problem turns out to be nonconvex. We demonstrate that the AG method exhibits an optimal rate of convergence if the composite problem is convex, and improves the best known rate of convergence if the problem is nonconvex. Based on the AG method, we also present new nonconvex stochastic approximation methods and show that they can improve a few existing rates of convergence for nonconvex stochastic optimization. To the best of our knowledge, this is the first time that the convergence of the AG method has been established for solving nonconvex nonlinear programming in the literature. We consider the problem of minimizing the sum of a smooth function @math with a bounded Hessian and a nonsmooth function. We assume that the latter function is a composition of a proper closed function @math and a surjective linear map @math , with the proximal mappings of @math , @math , simple to compute. This problem is nonconvex in general and encompasses many important applications in engineering and machine learning. In this paper, we examined two types of splitting methods for solving this nonconvex optimization problem: the alternating direction method of multipliers and the proximal gradient algorithm. For the direct adaptation of the alternating direction method of multipliers, we show that if the penalty parameter is chosen sufficiently large and the sequence generated has a cluster point, then it gives a stationary point of the nonconvex problem. We also establish convergence of the whole sequence under an additional assumption that the functions @math and @math are semialgebraic. Further... This paper analyzes the iteration-complexity of a quadratic penalty accelerated inexact proximal point method for solving linearly constrained nonconvex composite programs. More specifically, the objective function is of the form @math where @math is a differentiable function whose gradient is Lipschitz continuous and @math is a closed convex function with bounded domain. The method, basically, consists of applying an accelerated inexact proximal point method for solving approximately a sequence of quadratic penalized subproblems associated to the linearly constrained problem. Each subproblem of the proximal point method is in turn approximately solved by an accelerated composite gradient method. It is shown that the proposed scheme generates a @math -approximate stationary point in at most @math . Finally, numerical results showing the efficiency of the proposed method are also given.