A Reach and Bound algorithm for acyclic dynamic-programming networks

Node pruning is a commonly used technique for solution acceleration in a dynamic-programming network. In pruning, nodes are adaptively removed from the dynamic programming network when they are determined not to lie on an optimal path. We introduce an [epsiv]-pruning condition that extends pruning to include a possible error in the pruning step. This results in a greater reduction of the computation time; however, as a result of the inclusion of this error, the solution can be suboptimal or possibly infeasible. This condition requires the ability to compare the costs of an optimal path from a node to a terminal node. Therefore, we focus on the class of acyclic dynamic programming networks with monotonically decreasing optimal costs-to-go. We provide an easily implementable algorithm, Reach and Bound, which maintains feasibility and bounds the solution's error. We conclude by illustrating the applicability of Reach and Bound on a problem of single location capacity expansion. © 2007 Wiley Periodicals, Inc. NETWORKS, 2008Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/60450/1/20219_ftp.pd

Convergence of selections with applications in optimization

We consider the problem of finding an easily implemented tie-breaking rule for a convergent set-valued algorithm, i.e., a sequence of compact, non-empty subsets of a metric space converging in the Hausdorff metric. Our tie-breaking rule is determined by nearest-point selections defined by "uniqueness" points in the space, i.e., points having a unique best approximation in the limit set of the convergent algorithm. Convergence of the algorithm is shown to be equivalent to convergence of all such nearest-point selections. Under reasonable additional hypotheses, all points in the metric space have the uniqueness property. Consequently, all points yield convergent nearest-point selections, i.e., tie-breaking rules, for a convergent algorithm.We then show how to apply these results to approximate solutions for the following types of problems: infinite systems of inequalities, semi-infinite mathematical programming, non-convex optimization, and infinite horizon optimization.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/29485/1/0000571.pd

Conditions for the discovery of solution horizons

We present necessary and sufficient conditions for discrete infinite horizon optimization problems with unique solutions to be solvable. These problems can be equivalently viewed as the task of finding a shortest path in an infinite directed network. We provide general forward algorithms with stopping rules for their solution. The key condition required is that of weak reachability, which roughly requires that for any sequence of nodes or states, it must be possible from optimal states to reach states close in cost to states along this sequence. Moreover the costs to reach these states must converge to zero. Applications are considered in optimal search, undiscounted Markov decision processes, and deterministic infinite horizon optimization.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/47927/1/10107_2005_Article_BF01581244.pd

An adaptive group theoretic algorithm for integer programming problems

At head of title: Preliminary draft

Análise comparativa de alguns algoritmos na soluçao do problema da mochila

Dissertação (mestrado) - Universidade Federal de Santa Catarina, Centro Tecnológico. Programa de Pós-Graduação em Engenharia de Produção

Optimisation of an integrated transport and distribution system

Imperial Users onl

Application of mathematical programming techniques to power system optimization

Imperial Users onl

Solvability in Discrete, Nonstationary, Infinite Horizon Optimization

For several time-staged operations management problems, the optimal immediate decision is dependent on the choice of problem horizon. When that horizon is very long or indefinite, an appropriate modeling technique is infinite horizon optimization. For problems that have stationary data over time, optimizing system performance over an infinite horizon is generally no more difficult than optimizing over a finite horizon. However, restricting problem data to be stationary can render the models unrealistic, failing to include nonstationary aspects of the real world. The primary difficulty in nonstationary, infinite horizon optimization is that the problem to solve can never be known in its entirety. Thus, solution techniques must rely upon increasingly longer finite horizon problems. Ideally, the optimal immediate decisions to these finite horizon problems converge to an infinite horizon optimum. When finite detection of that optimal decision is possible, we call the underlying infinite horizon problem well-posed. The literature on nonstationary, infinite horizon optimization has generally relied upon either uniqueness of the optimal immediate decision or monotonicity of that decision as a function of horizon length. In this thesis, we require neither of these, instead developing a more general structural condition called coalescence that is equivalent to well-posedness. Chapters 2-4 study infinite horizon variants of three deterministic optimization applications: concave cost production planning, single machine replacement, and capacitated inventory planning. For each problem, we show that coalescence is equivalent to well-posedness. We also give a solution procedure for each application that will uncover an infinite horizon optimal immediate decision for any well-posed problem. In Chapter 5, we generalize the results of these applications to a generic classes of optimization problems expressible as dynamic programs. Under two different sets of assumptions concerning the finiteness of and reachability between states, we show that coalescence and well-posedness are equivalent. We also give solution procedures that solve any well-posed problem under each set of assumptions. Finally, in Chapter 6, we introduce a stochastic application: the infinite horizon asset selling problem, and again show that coalescence and well-posedness are equivalent and give a solution procedure to solve any such well-posed problem.Ph.D.Industrial & Operations EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/60810/1/tlortz_1.pd