Computational study of the step size parameter of the subgradient optimization method

The subgradient optimization method is a simple and flexible linear programming iterative algorithm. It is much simpler than Newton's method and can be applied to a wider variety of problems. It also converges when the objective function is non-differentiable. Since an efficient algorithm will not only produce a good solution but also take less computing time, we always prefer a simpler algorithm with high quality. In this study a series of step size parameters in the subgradient equation is studied. The performance is compared for a general piecewise function and a specific p-median problem. We examine how the quality of solution changes by setting five forms of step size parameter.

On statistical bounds of heuristic solutions to location problems

Solutions to combinatorial optimization problems, such as problems of locating facilities, frequently rely on heuristics to minimize the objective function. The optimum is sought iteratively and a criterion is needed to decide when the procedure (almost) attains it. Pre-setting the number of iterations dominates in OR applications, which implies that the quality of the solution cannot be ascertained. A small, almost dormant, branch of the literature suggests using statistical principles to estimate the minimum and its bounds as a tool to decide upon stopping and evaluating the quality of the solution. In this paper we examine the functioning of statistical bounds obtained from four different estimators by using simulated annealing on p-median test problems taken from Beasley’s OR-library. We find the Weibull estimator and the 2nd order Jackknife estimator preferable and the requirement of sample size to be about 10 being much less than the current recommendation. However, reliable statistical bounds are found to depend critically on a sample of heuristic solutions of high quality and we give a simple statistic useful for checking the quality. We end the paper with an illustration on using statistical bounds in a problem of locating some 70 distribution centers of the Swedish Post in one Swedish region. 

A stopping rule while searching for optimal solution of facility-location

Solutions to combinatorial optimization, such as p-median problems of locating facilities, frequently rely on heuristics to minimize the objective function. The minimum is sought iteratively and a criterion is needed to decide when the procedure (almost) attains it. However, pre-setting the number of iterations dominates in OR applications, which implies that the quality of the solution cannot be ascertained. A small branch of the literature suggests using statistical principles to estimate the minimum and use the estimate for either stopping or evaluating the quality of the solution. In this paper we use test-problems taken from Baesley's OR-library and apply Simulated Annealing on these p-median problems. We do this for the purpose of comparing suggested methods of minimum estimation and, eventually, provide a recommendation for practioners. An illustration ends the paper being a problem of locating some 70 distribution centers of the Swedish Post in a region.

Confidence in heuristic solutions?

Solutions to combinatorial optimization problems frequently rely on heuristics to minimize an objective function. The optimum is sought iteratively and pre-setting the number of iterations dominates in operations research applications, which implies that the quality of the solution cannot be ascertained. Deterministic bounds offer a mean of ascertaining the quality, but such bounds are available for only a limited number of heuristics and the length of the interval may be difficult to control in an application. A small, almost dormant, branch of the literature suggests using statistical principles to derive statistical bounds for the optimum. We discuss alternative approaches to derive statistical bounds. We also assess their performance by testing them on 40 test p-median problems on facility location, taken from Beasley’s OR-library, for which the optimum is known. We consider three popular heuristics for solving such location problems; simulated annealing, vertex substitution, and Lagrangian relaxation where only the last offers deterministic bounds. Moreover, we illustrate statistical bounds in the location of 71 regional delivery points of the Swedish Post. We find statistical bounds reliable and much more efficient than deterministic bounds provided that the heuristic solutions are sampled close to the optimum. Statistical bounds are also found computationally affordable.

A pharmacokinetic-pharmacodynamic model for duodenal levodopa infusion

Objective Levodopa in presence of decarboxylase inhibitors is following two-compartment kinetics and its effect is typically modelled using sigmoid Emax models. Pharmacokinetic modelling of the absorption phase of oral distributions is problematic because of irregular gastric emptying. The purpose of this work was to identify and estimate a population pharmacokinetic- pharmacodynamic model for duodenal infusion of levodopa/carbidopa (Duodopa®) that can be used for in numero simulation of treatment strategies. Methods The modelling involved pooling data from two studies and fixing some parameters to values found in literature (Chan et al. J Pharmacokinet Pharmacodyn. 2005 Aug;32(3-4):307-31). The first study involved 12 patients on 3 occasions and is described in Nyholm et al. Clinical Neuropharmacology 2003:26:156-63. The second study, PEDAL, involved 3 patients on 2 occasions. A bolus dose (normal morning dose plus 50%) was given after a washout during night. Plasma samples and motor ratings (clinical assessment of motor function from video recordings on a treatment response scale between -3 and 3, where -3 represents severe parkinsonism and 3 represents severe dyskinesia.) were repeatedly collected until the clinical effect was back at baseline. At this point, the usual infusion rate was started and sampling continued for another two hours. Different structural absorption models and effect models were evaluated using the value of the objective function in the NONMEM package. Population mean parameter values, standard error of estimates (SE) and if possible, interindividual/interoccasion variability (IIV/IOV) were estimated. Results Our results indicate that Duodopa absorption can be modelled with an absorption compartment with an added bioavailability fraction and a lag time. The most successful effect model was of sigmoid Emax type with a steep Hill coefficient and an effect compartment delay. Estimated parameter values are presented in the table. Conclusions The absorption and effect models were reasonably successful in fitting observed data and can be used in simulation experiments.

An empirical test of the gravity p-median model

A customer is presumed to gravitate to a facility by the distance to it and the attractiveness of it. However regarding the location of the facility, the presumption is that the customer opts for the shortest route to the nearest facility.This paradox was recently solved by the introduction of the gravity p-median model. The model is yet to be implemented and tested empirically. We implemented the model in an empirical problem of locating locksmiths, vehicle inspections, and retail stores ofv ehicle spare-parts, and we compared the solutions with those of the p-median model. We found the gravity p-median model to be of limited use for the problem of locating facilities as it either gives solutions similar to the p-median model, or it gives unstable solutions due to a non-concave objective function.

Testing the gravity p-median model empirically

Regarding the location of a facility, the presumption in the widely used p-median model is that the customer opts for the shortest route to the nearest facility. However, this assumption is problematic on free markets since the customer is presumed to gravitate to a facility by the distance to and the attractiveness of it. The recently introduced gravity p-median model offers an extension to the p-median model that account for this. The model is therefore potentially interesting, although it has not yet been implemented and tested empirically. In this paper, we have implemented the model in an empirical problem of locating vehicle inspections, locksmiths, and retail stores of vehicle spare-parts for the purpose of investigating its superiority to the p-median model. We found, however, the gravity p-median model to be of limited use for the problem of locating facilities as it either gives solutions similar to the p-median model, or it gives unstable solutions due to a non-concave objective function.

Digital spiral analysis for objective assessment of fine motor timing variability in Parkinson's disease

OBJECTIVES: To develop a method for objective assessment of fine motor timing variability in Parkinson’s disease (PD) patients, using digital spiral data gathered by a touch screen device. BACKGROUND: A retrospective analysis was conducted on data from 105 subjects including65 patients with advanced PD (group A), 15 intermediate patients experiencing motor fluctuations (group I), 15 early stage patients (group S), and 10 healthy elderly subjects (HE) were examined. The subjects were asked to perform repeated upper limb motor tasks by tracing a pre-drawn Archimedes spiral as shown on the screen of the device. The spiral tracing test was performed using an ergonomic pen stylus, using dominant hand. The test was repeated three times per test occasion and the subjects were instructed to complete it within 10 seconds. Digital spiral data including stylus position (x-ycoordinates) and timestamps (milliseconds) were collected and used in subsequent analysis. The total number of observations with the test battery were as follows: Swedish group (n=10079), Italian I group (n=822), Italian S group (n = 811), and HE (n=299). METHODS: The raw spiral data were processed with three data processing methods. To quantify motor timing variability during spiral drawing tasks Approximate Entropy (APEN) method was applied on digitized spiral data. APEN is designed to capture the amount of irregularity or complexity in time series. APEN requires determination of two parameters, namely, the window size and similarity measure. In our work and after experimentation, window size was set to 4 and similarity measure to 0.2 (20% of the standard deviation of the time series). The final score obtained by APEN was normalized by total drawing completion time and used in subsequent analysis. The score generated by this method is hence on denoted APEN. In addition, two more methods were applied on digital spiral data and their scores were used in subsequent analysis. The first method was based on Digital Wavelet Transform and Principal Component Analysis and generated a score representing spiral drawing impairment. The score generated by this method is hence on denoted WAV. The second method was based on standard deviation of frequency filtered drawing velocity. The score generated by this method is hence on denoted SDDV. Linear mixed-effects (LME) models were used to evaluate mean differences of the spiral scores of the three methods across the four subject groups. Test-retest reliability of the three scores was assessed after taking mean of the three possible correlations (Spearman’s rank coefficients) between the three test trials. Internal consistency of the methods was assessed by calculating correlations between their scores. RESULTS: When comparing mean spiral scores between the four subject groups, the APEN scores were different between HE subjects and three patient groups (P=0.626 for S group with 9.9% mean value difference, P=0.089 for I group with 30.2%, and P=0.0019 for A group with 44.1%). However, there were no significant differences in mean scores of the other two methods, except for the WAV between the HE and A groups (P<0.001). WAV and SDDV were highly and significantly correlated to each other with a coefficient of 0.69. However, APEN was not correlated to neither WAV nor SDDV with coefficients of 0.11 and 0.12, respectively. Test-retest reliability coefficients of the three scores were as follows: APEN (0.9), WAV(0.83) and SD-DV (0.55). CONCLUSIONS: The results show that the digital spiral analysis-based objective APEN measure is able to significantly differentiate the healthy subjects from patients at advanced level. In contrast to the other two methods (WAV and SDDV) that are designed to quantify dyskinesias (over-medications), this method can be useful for characterizing Off symptoms in PD. The APEN was not correlated to none of the other two methods indicating that it measures a different construct of upper limb motor function in PD patients than WAV and SDDV. The APEN also had a better test-retest reliability indicating that it is more stable and consistent over time than WAV and SDDV.