Approximations in proximal bundle methods and decomposition of convex programs

A proximal bundle method is presented for minimizing a nonsmooth convex functionf. At each iteration, it requires only one approximate evaluation off and its ε-subgradient, and it finds a search direction via quadratic programming. When applied to the Lagrangian decomposition of convex programs, it allows for inexact solutions of decomposed subproblems; yet, increasing their required accuracy automatically, it asymptotically finds both the primal and dual solutions. It is an implementable approximate version of the proximal point algorithm. Some encouraging numerical experience is reported.

[1]  George B. Dantzig,et al.  Decomposition Principle for Linear Programs , 1960 .

[2]  Leon S. Lasdon,et al.  Optimization Theory of Large Systems , 1970 .

[3]  R. Rockafellar.  Monotone Operators and the Proximal Point Algorithm , 1976 .

[4]  Dimitri P. Bertsekas,et al.  Constrained Optimization and Lagrange Multiplier Methods , 1982 .

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

[6]  V. F. Dem'yanov,et al.  Nondifferentiable Optimization , 1985 .

[7]  Jonathan E. Spingarn,et al.  Applications of the method of partial inverses to convex programming: Decomposition , 1985, Math. Program..

[8]  K. Kiwiel.  An algorithm for nonsmooth convex minimization with errors , 1985 .

[9]  Stephen M. Robinson.  Bundle-based decomposition: Description and preliminary results , 1986 .

[10]  K. Kiwiel.  A Method for Solving Certain Quadratic Programming Problems Arising in Nonsmooth Optimization , 1986 .

[11]  Hanif D. Sherali,et al.  A class of convergent primal-dual subgradient algorithms for decomposable convex programs , 1986, Math. Program..

[12]  K. Kiwiel.  A Dual Method for Certain Positive Semidefinite Quadratic Programming Problems , 1989 .

[13]  John N. Tsitsiklis,et al.  Parallel and distributed computation , 1989 .

[14]  Stephen M. Robinson.  Bundle-Based Decomposition: Conditions for Convergence , 1989 .

[15]  H. Sherali,et al.  A primal-dual conjugate subgradient algorithm for specially structured linear and convex programming problems , 1989 .

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

[17]  Paul Tseng,et al.  Further applications of a splitting algorithm to decomposition in variational inequalities and convex programming , 1990, Math. Program..

[18]  Krzysztof C. Kiwiel.  Approximations in Decomposition of Large-Scale Convex Programs via a Nondifferentiable Optimization Method , 1990 .

[19]  Krzysztof C. Kiwiel,et al.  Exact penalty functions in proximal bundle methods for constrained convex nondifferentiable minimization , 1991, Math. Program..

[20]  P. Tseng.  Applications of splitting algorithm to decomposition in convex programming and variational inequalities , 1991 .

[21]  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..

[22]  Dimitri P. Bertsekas,et al.  On the Douglas—Rachford splitting method and the proximal point algorithm for maximal monotone operators , 1992, Math. Program..

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

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

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