The progressive hedging algorithm of Rockafellar and Wets for multistage stochastic programming problems could be viewed as a two-block alternating direction method of multipliers. This correspondence brings in some useful results. In particular, it provides a new proof for the convergence of the progressive hedging algorithm with a flexibility in the selection of primal and dual step lengths and it helps to develop a new progressive hedging algorithm for solving risk averse stochastic optimization problems with cross constraints.
Citation: |
[1] |
M. Ang, J. Sun and Q. Yao, On the dual representation of coherent risk measures, Ann. Oper. Res., 262 (2018), 29-46.
doi: 10.1007/s10479-017-2441-3.![]() ![]() ![]() |
[2] |
M. Fazel, T. K. Pong, D. Sun and P. Tseng, Hankel matrix rank minimization with applications to system identification and realization, SIAM J. Matrix Anal. Appl., 34 (2013), 946-977.
doi: 10.1137/110853996.![]() ![]() ![]() |
[3] |
D. Gabay and B. Mercier, A dual algorithm for the solution of nonlinear variational problems via finite element approximations, Comput. Math. Appl., 2 (1976), 17-40.
doi: 10.1016/0898-1221(76)90003-1.![]() ![]() |
[4] |
J. J. Moreau, Proximité et dualité dans un espace Hilbertien, Bull. Soc. Math. de France, 93 (1975), 273-299.
![]() ![]() |
[5] |
R. T. Rockafellar, Monotone operators and the proximal point algorithm, SIAM J. Control Optim., 14 (1976), 877-898.
doi: 10.1137/0314056.![]() ![]() ![]() |
[6] |
R. T. Rockafellar, Solving stochastic programming problems with risk measures by progressive hedging, Set-Valued Var. Anal., 26 (2018), 759-768.
doi: 10.1007/s11228-017-0437-4.![]() ![]() ![]() |
[7] |
R. T. Rockafellar, Progressive decoupling of linkages in monotone variational inequalities and convex optimization, in Proceedings of the Conference on Nonlinear Analysis and Convex Analysis, Chitose, Japan, (2017).
![]() |
[8] |
R. T. Rockafellar and J. Sun, Solving monotone stochastic variational inequalities and complementarity problems by progressive hedging, Math. Program., (2018).
doi: 10.1007/s10107-018-1251-y.![]() ![]() |
[9] |
R. T. Rockafellar and R. J. B. Wets, Scenario and policy aggregation in optimization under uncertainty, Math. Oper. Res., 16 (1991), 119-147.
doi: 10.1287/moor.16.1.119.![]() ![]() ![]() |
[10] |
R. T. Rockafellar and R. J. B. Wets, Stochastic variational inequalities: Single-stage to multistage, Math. Program., 135 (2016), 331-360.
doi: 10.1007/s10107-016-0995-5.![]() ![]() ![]() |
[11] |
J. E. Spingarn, Applications of the method of partial inverses to convex programming: decomposition, Math. Program., 32 (1985), 199-223.
doi: 10.1007/BF01586091.![]() ![]() ![]() |
[12] |
J. Sun, X. M. Yang, Q. Yao and M. Zhang, Risk minimization, regret minimization and progressive hedging algorithms, work in process.
![]() |