Pdf parallel computing subgradient method for nonsmooth. More especially as the resolution of a nonconvex optimization problem requires in general, at each step, the resolution of a convex optimization problem. A parallel line search subspace correction method for. Dec 15, 2015 convergence rate analysis of the two methods under certain situations is provided to illustrate the two methods efficiency. Distributed subgradient projection algorithm for convex optimization s. Nonsmooth convex optimization, incremental subgradient method, parallel subgradient method.
Pdf we study subgradient methods for convex optimization that use projections. Convergence analysis of iterative methods for nonsmooth. Rdis a closed convex set, and then receives a convex loss function f t. Subgradient methods simple constrained convex problems min fx s. We study a distributed computation model for optimizing a sum of convex objective functions corresponding to multiple agents. Network optimization lies in the middle of the great divide that separates the two major types of optimization problems, continuous and discrete. In this paper, we present two parallel optimization methods for solving the problem. Although asynchronous blockmethods have a long history see, e. Convex problems are often solved by the same methods as lps. In the first part of this paper we studied subgradient methods for convex optimization that use projections onto successive approximations of level sets of the objective corresponding to estimates of the optimal value.
The only structure assumed is that a strictly feasible point is known. Local minimax complexity of stochastic convex optimization, sabyasachi chatterjee. U r is a realvalued convex function defined on a convex open set in the euclidean space r n, a vector in that space is called a subgradient at a point x 0 in u if for any x in u one has. Adaptive subgradient methods for online learning and stochastic optimization. Extensions of convex optimization include the optimization of biconvex, pseudo convex, and quasiconvex functions. Post 1990s lps are often best solved by nonsimplex convex methods. We study subgradient methods for convex optimization that use projections onto successive approximations of level sets of the objective corresponding to estimates of the optimal value.
Radial subgradient method siam journal on optimization. Convergence rate analysis of the two methods under certain situations is provided to illustrate the two methods efficiency. Pdf the efficiency of subgradient projection methods for convex. Distributed subgradient methods for multiagent optimization. Distributed asynchronous incremental subgradient methods. Primaldual subgradient methods 225 note that the value f. The ties between linear programming and combinatorial optimization can be traced to the representation of the constraint polyhedron as the convex hull of its extreme points. Sep 15, 2015 randomized block subgradient methods for convex nonsmooth and stochastic optimization article pdf available september 2015 with 44 reads how we measure reads. Randomized block subgradient methods for convex nonsmooth and stochastic optimization article pdf available september 2015 with 44 reads how we measure reads. We consider a distributed multiagent network system where the goal is to minimize a sum of convex objective functions of the agents subject to a common convex constraint set.
For simplicity, consider the basic online convex optimization setting. Ie 521 convex optimization niao he recap dual subgradient method augmented lagrangian method alternating direction method of multipliers admm summary and outlook recap. C x p cx jjx p cxjj 2 where p cx is the projection of xonto c. Mirrordescentandnonlinearprojectedsubgradientmethods. On improving relaxation methods by modifying gradient techniques. Pdf primal convergence from dual subgradient methods for. As a simple case, prove that if cis closed and midpoint convex, then cis convex. We present a method for nonsmooth convex minimization which is based on subgradient directions and stringaveraging techniques. On the other hand, in convex optimization there is only one way to get a lower bound for the optimal solution of a minimization problem. The proposed algorithm does not use any proximity operators, in contrast to conventional parallel algorithms for nonsmooth convex optimization. Optimal subgradient algorithms for largescale convex optimization. The subgradient method is far slower than newtons method, but is much simpler and can be applied to a far wider variety of problems. Abstract pdf 462 kb 2015 a subgradient method based on gradient sampling for solving convex optimization problems. Distributed subgradient methods for convex optimization.
Subgradient algorithm with nonlinear projections sanp. C is convex, so f ix is convex, and the max of a set of convex functions is still convex. The method involves every agent minimizing hisher own objective function while exchanging information locally with other agents in. In another paper we discuss possible implementations of such methods. In a convergent variant of this method 6, 14 we need to choose in advance a sequence of steps.
Incremental gradient, subgradient, and proximal methods for. Then, each agent combines weighted averages of the received iterates with its own iterate, and adjusts the iterate by. Ee364b convex optimization ii stanford engineering everywhere. May 16, 2015 we propose a parallel subgradient algorithm for solving the problem by using the operators attribution such that it can communicate with all users. The ties between linear programming and combinatorial optimization can be traced to the representation of the constraint. A set cis midpoint convex if whenever two points a. We propose a parallel subgradient algorithm for solving the problem by using the operators attribution such that it can communicate with all users. An iterative method to solve the convex feasibility problem for a finite family of convex sets is presented. The objective of this paper is to accelerate the existing incremental and parallel subgradient methods for constrained nonsmooth convex optimization to solve the problem of minimizing the sum of nonsmooth, convex functionals over a constraint set in a hilbert space. The efficiency of subgradient projection methods for convex. Distributed subgradient methods for convex optimization over. Parallel subgradient methods for convex optimization.
A line of analysis that provides a new framework for the convergence analysis. In 21, the authors solve a multiagent unconstrained convex optimization problem through a novel combination of average consensus algorithms with subgradient methods. Inherently parallel algorithms in feasibility and optimization and their applications. In these algorithms, we typically have a subroutine that receives as input a value x, and has output. Subgradient methods in network resource allocation.
Subgradients of the lagrangian dual can be obtained by solving nscenariowise subproblems. The subgradient method was originally developed by shor and others in the soviet union in the 1960s and 1970s. Convex optimization with sparsityinducing norm this chapter is on convex optimization of the form where f is convex differentiable function and. For solving this not necessarily smooth optimization problem, we consider a. A unitary distributed subgradient method for multiagent. Although asynchronous block methods have a long history see, e. The proposed algorithm does not use any proximity operators, in contrast to conventional parallel algorithms for.
In this approach, the set of available data is split into sequences strings and a given iterate is processed independently along each string, possibly in parallel, by an incremental subgradient method ism. Extensions of convex optimization include the optimization of biconvex, pseudoconvex, and quasiconvex functions. Stochastic gradient methods for distributionally robust optimization with fdivergences, hongseok namkoong, john duchi. For general convex programs in the form of, where the objective functionfx is convex but not necessarily strongly convex, the convergence time of the driftpluspenalty algorithm is shown to be o1 2 in 12. We study subgradient methods for minimizing a sum of convex functions over a closed convex set. The driftpluspenalty method is similar to the dual subgradient method, but takes a time average of the primal variables. To generate a search direction, each iteration employs subgradients of a subset of the objectives evaluated at the current iterate, as well as past subgradients of the remaining objectives. Pdf randomized block subgradient methods for convex. Incremental subgradient methods for nondifferentiable. Historically, a subgradient method with constant step was the. Unlike the ordinary gradient method, the subgradient method is notadescentmethod.
Abstractwe study diffusion and consensus based optimization of a sum of unknown convex objective functions over distributed networks. Parallel computing subgradient method for nonsmooth convex optimization over the intersection of fixed point sets of nonexpansive mappings. Decentralized convex optimization via primal and dual decomposition. Primal convergence from dual subgradient methods for convex optimization article pdf available in mathematical programming 1502 may 2014 with 312 reads how we measure reads. Stochastic subgradient algorithms for strongly convex optimization over distributed networks muhammed o. Random minibatch subgradient algorithms for convex problems. The variables are first divided into a few blocks based on certain rules. Post 1990s lps are often best solved by nonsimplexconvex methods. In any case, subgradient methods are well worth knowing about. Distributed subgradient projection algorithm for convex. Adaptive subgradient methods for online learning and. At each iteration, the algorithms solve a suitable subproblem on each block simultaneously, construct a search direction by combining their solutions on all blocks, then identify a new point along this direction. In this paper we are interested in studying parallel asynchronous stochastic subgradient descent for general nonconvex nonsmooth objectives, as arising in the training of deep neural network architectures. Asynchronous stochastic subgradient methods for general.
Pdf the efficiency of subgradient projection methods for. Parallelizing subgradient methods for the lagrangian dual in. The numerical results are evaluated with a multicore computer and show that our parallel method reduces the running time and iterations needed to nd an optimal solution compared with other ones. A lot of literature focuses on the convergence time of dual subgradient methods to an approximiate solution. We present several variants and show that they enjoy almost optimal efficiency estimates. Random minibatch projection algorithms for convex problems with. It can be proved that under mild conditions midpoint convexity implies convexity. Subgradient cutting plane interior point subgradient polyhedral approximation lps are solved by simplex method nlps are solved by gradientnewton methods. These subproblems, which generalize the problem of projecting a point onto a convex set, often admit closed. In contrast to the classical approach, where the constraints are usually represented as intersection of simple sets, which are easy to project onto, in this paper we consider that each constraint set is given as the level set of a convex but not necessarily differentiable. Distributed subgradient methods for convex optimization over random networks ilan lobelyand asuman ozdaglarz december 4, 2009 abstract we consider the problem of cooperatively minimizing the sum of convex functions, where the functions represent local objective functions of the agents. Development of a distributed subgradient method for multiagent optimization nedic, ozdaglar 08 convergence analysis and performance bounds for timevarying topologies under general connectivity assumptions e. We presented several variants and showed that they enjoy almost optimal efficiency estimates.
We introduce a unified algorithmic framework for a variety of such methods, some involving gradient and subgradient iterations, which are known, and some involving combinations of subgradient and proximal methods, which are new and offer. We present a subgradient method for minimizing nonsmooth, nonlipschitz convex optimization problems. A parallel subgradient projections method for the convex. Subgradient methods and consensus algorithms for solving. Stringaveraging incremental subgradients for constrained.
Subgradient optimization, generalized and nonconvex duality. Dual subgradient methods are subgradient methods applied to a dual problem. This paper also discusses nonsmooth convex optimization over sublevel sets of convex functions and provides numerical comparisons that demonstrate the effectiveness of the proposed methods. Distributed optimization techniques offer high quality solutions to various engineering problems, such as resource. Moreover, incremental proximal point algorithms 8 have been proposed for nonsmooth convex optimization.
Each agent maintains an iterate sequence and communicates the iterates to its neighbors. In this paper we consider nonsmooth convex optimization problems with possibly infinite intersection of constraints. Convex optimization has been shown to provide efficient algorithms for computing. Neural information processing systems neurips 2016. The concepts of subderivative and subdifferential can be generalized to functions of several variables. For solving this not necessarily smooth optimization problem, we consider a subgradient method that is distributed among the agents. Stochastic subgradient algorithms for strongly convex. In this paper, we study generating approximate primal optimal solutions for general convex constrained optimization problems using. Distributed subgradient methods for saddlepoint problems. Our methods consist of iterations applied to single components, and have proved very effective in practice.
607 1048 3 1289 1503 1352 1364 1463 1135 1499 85 1298 509 88 668 856 493 514 1441 504 297 1191 716 496 288 892 1238 1200 238 1226 928 1010 217 406 367 1054 605