Abstract. We present a modification of Karmarkar's linear programming algorithm. Our algorithm uses a recentered projected gradient approach thereby obviating a priori knowledge of the optimal objective function value. Assuming primal and dual nondegeneracy, we prove that our algorithm converges. We present computational comparisons between our algorithm and the revised simplex method. For small, dense constraint matrices we saw little difference between the two methods.Key Words. Linear programming, Karmarkar's algorithm, Projected gradient methods, Least squares.1. Introduction. This paper proposes a modification to Karmarkar's original algorithm [6] for solving linear programs. Our algorithm is formulated in the positive orthant instead of the simplex. This makes it easier to conceptualize and leads to computational simplicity. Karmarkar's sliding-objective-function method is replaced by a projected gradient search for the optimum. Empirically, this leads to a decrease in the number of iterations the algorithm requires to solve a problem.In describing our algorithm, we show how to start it, when to stop it, and how to identify infeasibility and unboundedness easily. Assuming primal and dual nondegeneracy, we prove convergence to the optimal solution. (In practice, the algorithm works equally well on problems not satisfying these assumptions.) We also show that duality plays an important role.Finally, we present results comparing the performance of our algorithm with the revised simplex method for problems with fewer than 200 variables and randomly generated, dense constraint matrices. For this class of problems we saw little difference between the two methods. However, our results may not be indicative of the behavior for large sparse problems.
We wish to estimate the variance of the sample mean from a continuous-time stationary stochastic process. This article expands on the results of a technical note (Goldsman and Schruben 1990) by using the theory of standardized time series to investigate weighted generalizations of Schruben's area variance estimator. We find a simple expression for the bias of the weighted area variance estimator, and we give weights which yield variance estimators with lower asymptotic bias than certain other popular estimators. We use the weighted area variance estimators to derive asymptotically valid confidence interval estimators (CIEs) for the mean of a stationary stochastic process. Although the weighted area CIEs have the same asymptotic expected value and variance of the length as Schruben's area CIE, we show that the new CIEs sometimes yield coverages which are closer to the nominal value.simulation output analysis, confidence intervals, standardized time series
North American railways have traditionally practiced tonnage-based dispatching, running trains only when they have enough freight. As a result, their customer service and their use of crews, fixed assets, locomotives, and railcars are poor. Canadian Pacific Railway is using new decision-support tools developed in-house and by MultiModal Applied Systems to create a scheduled railway. These tools use operations research approaches, such as an optimal block-sequencing algorithm, a heuristic algorithm for block design, (very fast) simulation, and time-space network algorithms for planning locomotive use and distributing empty cars. This implementation has saved $300 million Canadian (US$170 million) from mid-1999 through autumn 2000. We estimate it has saved at least an additional $210 million Canadian during 2001 and 2002 in fuel and labor costs alone. Labor productivity, locomotive productivity, fuel consumption, and railcar velocity have improved by 40, 35, 17, and 41 percent, respectively. Furthermore, Canadian Pacific Railway now provides its customers with reliable delivery times and has received many customer and shipping association awards for its improvement in service.
The special structure of regenerative processes is exploited to derive a new point estimate with very low bias for steady state quantities of regenerative simulations. If the simulation run length is t units of tune, the bias of the new estimate is of order 1/t 2 as opposed to the bias of order 1/t associated with more standard estimates. The bias reduction is achieved by continuing the simulation until the first regeneration after time t and then forming the ratio estimate which involves the random number of regenerative cycles observed during the simulation. Empirical results for several queueing models demonstrate that the bias reduction can be substantial for small values of t.simulation, regenerative, bias reduction, renewal theory, cumulative process, ratio estimator
This paper describes the AT&T KORBX system, which implements certain variants of the Karmarkar algorithm on a computer that has multiple processors, each capable of performing vector arithmetic. We provide an overview of the KORBX system that covers its optimization algorithms, the hardware architecture, and software implementation techniques. Performance is characterized for a variety of applications found in industry, government, and academia.
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