A Scalable Bounding Method for Multi-Stage Stochastic Integer Programs

2014 ◽  
Author(s):  
Burhaneddin Sandikci ◽  
Osman Y. zalttn
Keyword(s):  
Integer Programs ◽  
Multi Stage ◽  
Stochastic Integer Programs

scholarly journals About the complexity of two-stage stochastic IPs

2021 ◽  
Author(s):  
Kim-Manuel Klein
Keyword(s):  
Lower Bound ◽  
Vector Space ◽  
Exponential Function ◽  
Commutative Algebra ◽  
Integer Program ◽  
Constraint Matrix ◽  
Exponential Bound ◽  
Integer Programs ◽  
Multi Stage ◽  
Stochastic Integer Programs

AbstractWe consider so called 2-stage stochastic integer programs (IPs) and their generalized form, so called multi-stage stochastic IPs. A 2-stage stochastic IP is an integer program of the form $$\max \{ c^T x \mid {\mathcal {A}} x = b, \,l \le x \le u,\, x \in {\mathbb {Z}}^{s + nt} \}$$ max { c T x ∣ A x = b , l ≤ x ≤ u , x ∈ Z s + n t } where the constraint matrix $${\mathcal {A}} \in {\mathbb {Z}}^{r n \times s +nt}$$ A ∈ Z r n × s + n t consists roughly of n repetitions of a matrix $$A \in {\mathbb {Z}}^{r \times s}$$ A ∈ Z r × s on the vertical line and n repetitions of a matrix $$B \in {\mathbb {Z}}^{r \times t}$$ B ∈ Z r × t on the diagonal. In this paper we improve upon an algorithmic result by Hemmecke and Schultz from 2003 [Hemmecke and Schultz, Math. Prog. 2003] to solve 2-stage stochastic IPs. The algorithm is based on the Graver augmentation framework where our main contribution is to give an explicit doubly exponential bound on the size of the augmenting steps. The previous bound for the size of the augmenting steps relied on non-constructive finiteness arguments from commutative algebra and therefore only an implicit bound was known that depends on parameters r, s, t and $$\Delta $$ Δ , where $$\Delta $$ Δ is the largest entry of the constraint matrix. Our new improved bound however is obtained by a novel theorem which argues about intersections of paths in a vector space. As a result of our new bound we obtain an algorithm to solve 2-stage stochastic IPs in time $$f(r,s,\Delta ) \cdot \mathrm {poly}(n,t)$$ f ( r , s , Δ ) · poly ( n , t ) , where f is a doubly exponential function. To complement our result, we also prove a doubly exponential lower bound for the size of the augmenting steps.

scholarly journals Integration of progressive hedging and dual decomposition in stochastic integer programs

2015 ◽  
Vol 43 (3) ◽  
pp. 311-316 ◽  
Author(s):  
Ge Guo ◽  
Gabriel Hackebeil ◽  
Sarah M. Ryan ◽  
Jean-Paul Watson ◽  
David L. Woodruff
Keyword(s):  
Dual Decomposition ◽  
Integer Programs ◽  
Progressive Hedging ◽  
Stochastic Integer Programs

scholarly journals Cutting Planes for Multistage Stochastic Integer Programs

2009 ◽  
Vol 57 (2) ◽  
pp. 287-298 ◽  
Author(s):  
Yongpei Guan ◽  
Shabbir Ahmed ◽  
George L. Nemhauser
Keyword(s):  
Cutting Planes ◽  
Integer Programs ◽  
Stochastic Integer Programs

scholarly journals A finite branch-and-bound algorithm for two-stage stochastic integer programs

2004 ◽  
Vol 100 (2) ◽  
pp. 355-377 ◽  
Author(s):  
Shabbir Ahmed ◽  
Mohit Tawarmalani ◽  
Nikolaos V. Sahinidis
Keyword(s):  
Branch And Bound ◽  
Branch And Bound Algorithm ◽  
Two Stage ◽  
Integer Programs ◽  
Stochastic Integer Programs

scholarly journals An approximation framework for two-stage ambiguous stochastic integer programs under mean-MAD information

2019 ◽  
Vol 274 (2) ◽  
pp. 432-444 ◽  
Author(s):  
Krzysztof Postek ◽  
Ward Romeijnders ◽  
Dick den Hertog ◽  
Maarten H. van der Vlerk
Keyword(s):  
Two Stage ◽  
Integer Programs ◽  
Stochastic Integer Programs
Sign in / Sign up

Export Citation Format

Share Document