7,048 research outputs found
Lagrangean decomposition for large-scale two-stage stochastic mixed 0-1 problems
In this paper we study solution methods for solving the dual problem corresponding to the Lagrangean Decomposition of two stage stochastic mixed 0-1 models. We represent the two stage stochastic mixed 0-1 problem by a splitting variable representation of the deterministic equivalent model, where 0-1 and continuous variables appear at any stage. Lagrangean Decomposition is proposed for satisfying both the integrality constraints for the 0-1 variables and the non-anticipativity constraints. We compare the performance of four iterative algorithms based on dual Lagrangean Decomposition schemes, as the Subgradient method, the Volume algorithm, the Progressive Hedging algorithm and the Dynamic Constrained Cutting Plane scheme. We test the conditions and properties of convergence for medium and large-scale dimension stochastic problems. Computational results are reported.Progressive Hedging algorithm, volume algorithm, Lagrangean decomposition, subgradient method
Phase Transitions of the Typical Algorithmic Complexity of the Random Satisfiability Problem Studied with Linear Programming
Here we study the NP-complete -SAT problem. Although the worst-case
complexity of NP-complete problems is conjectured to be exponential, there
exist parametrized random ensembles of problems where solutions can typically
be found in polynomial time for suitable ranges of the parameter. In fact,
random -SAT, with as control parameter, can be solved quickly
for small enough values of . It shows a phase transition between a
satisfiable phase and an unsatisfiable phase. For branch and bound algorithms,
which operate in the space of feasible Boolean configurations, the empirically
hardest problems are located only close to this phase transition. Here we study
-SAT () and the related optimization problem MAX-SAT by a linear
programming approach, which is widely used for practical problems and allows
for polynomial run time. In contrast to branch and bound it operates outside
the space of feasible configurations. On the other hand, finding a solution
within polynomial time is not guaranteed. We investigated several variants like
including artificial objective functions, so called cutting-plane approaches,
and a mapping to the NP-complete vertex-cover problem. We observed several
easy-hard transitions, from where the problems are typically solvable (in
polynomial time) using the given algorithms, respectively, to where they are
not solvable in polynomial time. For the related vertex-cover problem on random
graphs these easy-hard transitions can be identified with structural properties
of the graphs, like percolation transitions. For the present random -SAT
problem we have investigated numerous structural properties also exhibiting
clear transitions, but they appear not be correlated to the here observed
easy-hard transitions. This renders the behaviour of random -SAT more
complex than, e.g., the vertex-cover problem.Comment: 11 pages, 5 figure
Stochastic Cutting Planes for Data-Driven Optimization
We introduce a stochastic version of the cutting-plane method for a large
class of data-driven Mixed-Integer Nonlinear Optimization (MINLO) problems. We
show that under very weak assumptions the stochastic algorithm is able to
converge to an -optimal solution with high probability. Numerical
experiments on several problems show that stochastic cutting planes is able to
deliver a multiple order-of-magnitude speedup compared to the standard
cutting-plane method. We further experimentally explore the lower limits of
sampling for stochastic cutting planes and show that for many problems, a
sampling size of appears to be sufficient for high quality
solutions
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