Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userInexact Proximal-Point Penalty Methods for Constrained Non-Convex Optimization
99
0
0.0
(
0
)
Added by
Yangyang Xu
Publication date
2019
fields
Informatics Engineering
and research's language is
English
Ask ChatGPT about the research
No Arabic abstract
In this paper, an inexact proximal-point penalty method is studied for constrained optimization problems, where the objective function is non-convex, and the constraint functions can also be non-convex. The proposed method approximately solves a sequence of subproblems, each of which is formed by adding to the original objective function a proximal term and quadratic penalty terms associated to the constraint functions. Under a weak-convexity assumption, each subproblem is made strongly convex and can be solved effectively to a required accuracy by an optimal gradient-based method. The computational complexity of the proposed method is analyzed separately for the cases of convex constraint and non-convex constraint. For both cases, the complexity results are established in terms of the number of proximal gradient steps needed to find an $varepsilon$-stationary point. When the constraint functions are convex, we show a complexity result of $tilde O(varepsilon^{-5/2})$ to produce an $varepsilon$-stationary point under the Slaters condition. When the constraint functions are non-convex, the complexity becomes $tilde O(varepsilon^{-3})$ if a non-singularity condition holds on constraints and otherwise $tilde O(varepsilon^{-4})$ if a feasible initial solution is available.
rate research
Read More
We consider the problem of optimizing the sum of a smooth convex function and a non-smooth convex function using proximal-gradient methods, where an error is present in the calculation of the gradient of the smooth term or in the proximity operator with respect to the non-smooth term. We show that both the basic proximal-gradient method and the accelerated proximal-gradient method achieve the same convergence rate as in the error-free case, provided that the errors decrease at appropriate rates.Using these rates, we perform as well as or better than a carefully chosen fixed error level on a set of structured sparsity problems.
For solving large-scale non-convex problems, we propose inexact variants of trust region and adaptive cubic regularization methods, which, to increase efficiency, incorporate various approximations. In particular, in addition to approximate sub-problem solves, both the Hessian and the gradient are suitably approximated. Using rather mild conditions on such approximations, we show that our proposed inexact methods achieve similar optimal worst-case iteration complexities as the exact counterparts. Our proposed algorithms, and their respective theoretical analysis, do not require knowledge of any unknowable problem-related quantities, and hence are easily implementable in practice. In the context of finite-sum problems, we then explore randomized sub-sampling methods as ways to construct the gradient and Hessian approximations and examine the empirical performance of our algorithms on some real datasets.
In this paper, we develop a parameterized proximal point algorithm (P-PPA) for solving a class of separable convex programming problems subject to linear and convex constraints. The proposed algorithm is provable to be globally convergent with a worst-case O(1/t) convergence rate, wheret denotes the iteration number. By properly choosing the algorithm parameters, numerical experiments on solving a sparse optimization problem arising from statistical learning show that our P-PPA could perform significantly better than other state-of-the-art methods, such as the alternating direction method of multipliers and the relaxed proximal point algorithm.
Nonsmooth sparsity constrained optimization captures a broad spectrum of applications in machine learning and computer vision. However, this problem is NP-hard in general. Existing solutions to this problem suffer from one or more of the following limitations: they fail to solve general nonsmooth problems; they lack convergence analysis; they lead to weaker optimality conditions. This paper revisits the Penalty Alternating Direction Method (PADM) for nonsmooth sparsity constrained optimization problems. We consider two variants of the PADM, i.e., PADM based on Iterative Hard Thresholding (PADM-IHT) and PADM based on Block Coordinate Decomposition (PADM-BCD). We show that the PADM-BCD algorithm finds stronger stationary points of the optimization problem than previous methods. We also develop novel theories to analyze the convergence rate for both the PADM-IHT and the PADM-BCD algorithms. Our theoretical bounds can exploit the inherent sparsity of the optimization problem. Finally, numerical results demonstrate the superiority of PADM-BCD to existing sparse optimization algorithms. Keywords: Sparsity Recovery, Nonsmooth Optimization, Non-Convex Optimization, Block Coordinate Decomposition, Iterative Hard Thresholding, Convergence Analysis
We propose a framework to use Nesterovs accelerated method for constrained convex optimization problems. Our approach consists of first reformulating the original problem as an unconstrained optimization problem using a continuously differentiable exact penalty function. This reformulation is based on replacing the Lagrange multipliers in the augmented Lagrangian of the original problem by Lagrange multiplier functions. The expressions of these Lagrange multiplier functions, which depend upon the gradients of the objective function and the constraints, can make the unconstrained penalty function non-convex in general even if the original problem is convex. We establish sufficient conditions on the objective function and the constraints of the original problem under which the unconstrained penalty function is convex. This enables us to use Nesterovs accelerated gradient method for unconstrained convex optimization and achieve a guaranteed rate of convergence which is better than the state-of-the-art first-order algorithms for constrained convex optimization. Simulations illustrate our results.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
