81 research outputs found
Efficient solution of two-stage stochastic linear programs using interior point methods
Solving deterministic equivalent formulations of two-stage stochastic linear programs using interior point methods may be computationally difficult due to the need to factorize quite dense search direction matrices (e.g., AA T ). Several methods for improving the algorithmic efficiency of interior point algorithms by reducing the density of these matrices have been proposed in the literature. Reformulating the program decreases the effort required to find a search direction, but at the expense of increased problem size. Using transpose product formulations (e.g., A T A ) works well but is highly problem dependent. Schur complements may require solutions with potentially near singular matrices. Explicit factorizations of the search direction matrices eliminate these problems while only requiring the solution to several small, independent linear systems. These systems may be distributed across multiple processors. Computational experience with these methods suggests that substantial performance improvements are possible with each method and that, generally, explicit factorizations require the least computational effort.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/44758/1/10589_2004_Article_BF00249637.pd
An interior-point and decomposition approach to multiple stage stochastic programming
There is no abstract of this report
Minimization of Collateral Damage in Airdrops and Airstrikes
Collateral damage presents a significant risk during air drops and airstrikes, risking citizens\u27 lives and property, straining the relationship between the United States Air Force and host nations. This dissertation presents a methodology to determine the optimal location for making supply airdrops in order to minimize collateral damage while maintaining a high likelihood of successful recovery. A series of non-linear optimization algorithms is presented along with their relative success in finding the optimal location in the airdrop problem. Additionally, we present a quick algorithm for accurately creating the Pareto frontier in the multi-objective airstrike problem. We demonstrate the effect of differing guidelines, damage functions, and weapon employment selection which significantly alter the location of the optimal aimpoint in this targeting problem. Finally, we have provided a framework for making policy decisions in fast-moving troops-in-contact situations where observers are unsure of the nature of possible enemy forces in both finite horizon and infinite horizon problems. Through the recursive technique of solving this Markov decision process we have demonstrated the effect of improved intelligence and differing weights for waiting and incorrect decisions in the face of uncertain situations
Volumetric center method for stochastic convex programs using sampling
We develop an algorithm for solving the stochastic convex program (SCP) by combining Vaidya's volumetric center interior point method (VCM) for solving non-smooth convex programming problems with the Monte-Carlo sampling technique to compute a subgradient. A near-central cut variant of VCM is developed, and for this method an approach to perform bulk cut translation, and adding multiple cuts is given. We show that by using near-central VCM the SCP can be solved to a desirable accuracy with any given probability. For the two-stage SCP the solution time is independent of the number of scenarios
Sublinear upper bounds for stochastic programs with recourse
Separable sublinear functions are used to provide upper bounds on the recourse function of a stochastic program. The resulting problem's objective involves the inf-convolution of convex functions. A dual of this problem is formulated to obtain an implementable procedure to calculate the bound. Function evaluations for the resulting convex program only require a small number of single integrations in contrast with previous upper bounds that require a number of function evaluations that grows exponentially in the number of random variables. The sublinear bound can often be used when other suggested upper bounds are intractable. Computational results indicate that the sublinear approximation provides good, efficient bounds on the stochastic program objective value.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/47918/1/10107_2005_Article_BF01582286.pd
An interior-point and decomposition approach to multiple stage stochastic programming
There is no abstract of this repor
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Numerical methods on some structured matrix algebra problems
This proposal concerned the design, analysis, and implementation of serial and parallel algorithms for certain structured matrix algebra problems. It emphasized large order problems and so focused on methods that can be implemented efficiently on distributed-memory MIMD multiprocessors. Such machines supply the computing power and extensive memory demanded by the large order problems. We proposed to examine three classes of matrix algebra problems: the symmetric and nonsymmetric eigenvalue problems (especially the tridiagonal cases) and the solution of linear systems with specially structured coefficient matrices. As all of these are of practical interest, a major goal of this work was to translate our research in linear algebra into useful tools for use by the computational scientists interested in these and related applications. Thus, in addition to software specific to the linear algebra problems, we proposed to produce a programming paradigm and library to aid in the design and implementation of programs for distributed-memory MIMD computers. We now report on our progress on each of the problems and on the programming tools
Improving an interior-point approach for large block-angular problems by hybrid preconditioners
The computational time required by interior-point methods
is often domi-
nated by the solution of linear systems of equations. An efficient spec
ialized
interior-point algorithm for primal block-angular proble
ms has been used to
solve these systems by combining Cholesky factorizations for the
block con-
straints and a conjugate gradient based on a power series precon
ditioner for
the linking constraints. In some problems this power series prec
onditioner re-
sulted to be inefficient on the last interior-point iterations, wh
en the systems
became ill-conditioned. In this work this approach is combi
ned with a split-
ting preconditioner based on LU factorization, which is main
ly appropriate
for the last interior-point iterations. Computational result
s are provided for
three classes of problems: multicommodity flows (oriented and no
noriented),
minimum-distance controlled tabular adjustment for statistic
al data protec-
tion, and the minimum congestion problem. The results show that
, in most
cases, the hybrid preconditioner improves the performance an
d robustness of
the interior-point solver. In particular, for some block-ang
ular problems the
solution time is reduced by a factor of 10.Peer ReviewedPreprin
On the Convergence of L-shaped Algorithms for Two-Stage Stochastic Programming
In this paper, we design, analyze, and implement a variant of the two-loop
L-shaped algorithms for solving two-stage stochastic programming problems that
arise from important application areas including revenue management and power
systems. We consider the setting in which it is intractable to compute exact
objective function and (sub)gradient information, and instead, only estimates
of objective function and (sub)gradient values are available. Under common
assumptions including fixed recourse and bounded (sub)gradients, the algorithm
generates a sequence of iterates that converge to a neighborhood of optimality,
where the radius of the convergence neighborhood depends on the level of the
inexactness of objective function estimates. The number of outer and inner
iterations needed to find an approximate optimal iterate is provided. Finally,
we show a sample complexity result for the algorithm with a Polyak-type
step-size policy that can be extended to analyze other situations. We also
present a numerical study that verifies our theoretical results and
demonstrates the superior empirical performance of our proposed algorithms over
classic solvers.Comment: 39 pages, 2 figure
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