Efficiency of Proximal Bundle Methods

We give efficiency estimates for proximal bundle methods for finding f*≔minXf, where f and X are convex. We show that, for any accuracy ∈<0, these methods find a point xk∈X such that f(xk)−f*≤∈ after at most k=O(1/∈3) objective and subgradient evaluations.

[1]  P. Wolfe.  Note on a method of conjugate subgradients for minimizing nondifferentiable functions , 1974 .

[2]  John Darzentas,et al.  Problem Complexity and Method Efficiency in Optimization , 1983 .

[3]  K. Kiwiel.  Methods of Descent for Nondifferentiable Optimization , 1985 .

[4]  Krzysztof C. Kiwiel,et al.  Proximity control in bundle methods for convex nondifferentiable minimization , 1990, Math. Program..

[5]  Jochem Zowe,et al.  A Version of the Bundle Idea for Minimizing a Nonsmooth Function: Conceptual Idea, Convergence Analysis, Numerical Results , 1992, SIAM J. Optim..

[6]  J. Hiriart-Urruty,et al.  Convex analysis and minimization algorithms , 1993 .

[7]  Claude Lemaréchal,et al.  Convergence of some algorithms for convex minimization , 1993, Math. Program..

[8]  K. Kiwiel.  A Cholesky dual method for proximal piecewise linear programming , 1994 .

[9]  Yurii Nesterov,et al.  Complexity estimates of some cutting plane methods based on the analytic barrier , 1995, Math. Program..

[10]  Yurii Nesterov,et al.  New variants of bundle methods , 1995, Math. Program..

[11]  K. Kiwiel.  Approximations in proximal bundle methods and decomposition of convex programs , 1995 .

[12]  Krzysztof C. Kiwiel,et al.  Proximal level bundle methods for convex nondifferentiable optimization, saddle-point problems and variational inequalities , 1995, Math. Program..

[13]  Krzysztof C. Kiwiel,et al.  Restricted Step and Levenberg-Marquardt Techniques in Proximal Bundle Methods for Nonconvex Nondifferentiable Optimization , 1996, SIAM J. Optim..

[14]  K. Kiwiel.  The efficiency of subgradient projection methods for convex optimization, part I: general level methods , 1996 .

[15]  Krzysztof C. Kiwiel,et al.  Efficiency of the Analytic Center Cutting Plane Method for Convex Minimization , 1997, SIAM J. Optim..

[16]  Torbjörn Larsson,et al.  The Efficiency of Ballstep Subgradient Level Methods for Convex Optimization , 1999, Math. Oper. Res..

[17]  Andrzej Ruszczynski,et al.  Proximal Decomposition Via Alternating Linearization , 1999, SIAM J. Optim..