### Table 2: Non-convex quarticly constrained optimization problem for hierarchy and policy discovery in bounded stochastic recursive controllers.

in Abstract

"... In PAGE 5: ... 3.3 Algorithms Since the problem in Table2 has non-convex (quartic) constraints in Eq. 5 and 6, it is difficult to solve.... In PAGE 5: ... 5 and 6, it is difficult to solve. We consider three approaches inspired from the techniques for non-hierarchical controllers: Non-convex optimization: Use a general non-linear solver, such as SNOPT, to directly tackle the optimization problem in Table2 . This is the most convenient approach, however a globally optimal solution may not be found due to the non-convex nature of the problem.... In PAGE 7: ... 4 Experiments We report on some preliminary experiments with three toy problems (paint, shuttle and maze) from the POMDP repository3. We used the SNOPT package to directly solve the non-convex optimization problem in Table2 and bounded hierarchical policy iteration (BHPI) to solve it iteratively. Table 3 reports the running time and the value of the hierarchical policies found.... ..."

### Table 2: Non-convex quarticly constrained optimization problem for hierarchy and policy discovery in bounded stochastic recursive controllers.

in Abstract

"... In PAGE 5: ... 3.3 Algorithms Since the problem in Table2 has non-convex (quartic) constraints in Eq. 5 and 6, it is difficult to solve.... In PAGE 5: ... 5 and 6, it is difficult to solve. We consider three approaches inspired from the techniques for non-hierarchical controllers: Non-convex optimization: Use a general non-linear solver, such as SNOPT, to directly tackle the optimization problem in Table2 . This is the most convenient approach, however a globally optimal solution may not be found due to the non-convex nature of the problem.... In PAGE 7: ... 4 Experiments We report on some preliminary experiments with three toy problems (paint, shuttle and maze) from the POMDP repository3. We used the SNOPT package to directly solve the non-convex optimization problem in Table2 and bounded hierarchical policy iteration (BHPI) to solve it iteratively. Table 3 reports the running time and the value of the hierarchical policies found.... ..."

### Table 2: Non-convex quarticly constrained optimization problem for hierarchy and policy discovery in bounded stochastic recursive controllers.

"... In PAGE 5: ... 3.3 Algorithms Since the problem in Table2 has non-convex (quartic) constraints in Eq. 5 and 6, it is difficult to solve.... In PAGE 5: ... 5 and 6, it is difficult to solve. We consider three approaches inspired from the techniques for non-hierarchical controllers: Non-convex optimization: Use a general non-linear solver, such as SNOPT, to directly tackle the optimization problem in Table2 . This is the most convenient approach, however a globally optimal solution may not be found due to the non-convex nature of the problem.... In PAGE 7: ... 4 Experiments We report on some preliminary experiments with three toy problems (paint, shuttle and maze) from the POMDP repository3. We used the SNOPT package to directly solve the non-convexoptimization problem in Table2 and bounded hierarchical policy iteration (BHPI) to solve it iteratively. Table 3 reports the running time and the value of the hierarchical policies found.... ..."

### Table 3b. Solution Statistics for Model 2 (Minimization)

1999

"... In PAGE 4: ...6 Table 2. Problem Statistics Model 1 Model 2 Pt Rows Cols 0/1 Vars Rows Cols 0/1 Vars 1 4398 4568 4568 4398 4568 170 2 4546 4738 4738 4546 4738 192 3 3030 3128 3128 3030 3128 98 4 2774 2921 2921 2774 2921 147 5 5732 5957 5957 5732 5957 225 6 5728 5978 5978 5728 5978 250 7 2538 2658 2658 2538 2658 120 8 3506 3695 3695 3506 3695 189 9 2616 2777 2777 2616 2777 161 10 1680 1758 1758 1680 1758 78 11 5628 5848 5848 5628 5848 220 12 3484 3644 3644 3484 3644 160 13 3700 3833 3833 3700 3833 133 14 4220 4436 4436 4220 4436 216 15 2234 2330 2330 2234 2330 96 16 3823 3949 3949 3823 3949 126 17 4222 4362 4362 4222 4362 140 18 2612 2747 2747 2612 2747 135 19 2400 2484 2484 2400 2484 84 20 2298 2406 2406 2298 2406 108 Table3 a. Solution Statistics for Model 1 (Maximization) Pt Initial First Heuristic Best Best LP Obj.... In PAGE 5: ...) list the elapsed time when the heuristic procedure is first called and the objective value corresponding to the feasible integer solution returned by the heuristic. For Table3 a, the columns Best LP Obj. and Best IP Obj.... In PAGE 5: ... report, respectively, the LP objective bound corresponding to the best node in the remaining branch-and-bound tree and the incumbent objective value corresponding to the best integer feasible solution upon termination of the solution process (10,000 CPU seconds). In Table3 b, the columns Optimal IP Obj., bb nodes, and Elapsed Time report, respectively, the optimal IP objective value, the total number of branch-and-bound tree nodes solved, and the total elapsed time for the solution process.... ..."

### Table 1 ABC ABCABC ABC ABC Max regret

2006

"... In PAGE 8: ... 3. The problem admits five feasible states, and the pairwise max regret R(x,xprime,U) for each pair of states (with x along columns and xprime along the rows) is shown in Table1 . The max regret of each feasible state is shown in the final column, from which we see that state ABC is the minimax optimal decision.... ..."

### Table 5 Gradually Mixed Multilevel Optimization of a 20D bump

1998

Cited by 2

### TABLE I UPPER BOUND ON MAXIMUM REGRET OF CES RULES UNDER SIMPLE RANDOM SAMPLING (q = periodori5)

2004

Cited by 3

### Table 5-7: Summary for Offsets From Various offline Methods for State Route 26, Lafayette, Indiana (Cycle length = 120 seconds)

"... In PAGE 8: ...Figure Page 5-8 Comparison of offline offset design methods, Eastbound State Route 26 (Graph created with values from Table5 -7 through Table 5-11) .... In PAGE 8: ...Graph created with values from Table 5-7 through Table 5-11) ................90 5-9 Comparison of offline offset design methods, Eastbound State Route 26 (Graph created with values from Table5 -7 through Table 5-11) .... In PAGE 8: ...Graph created with values from Table 5-7 through Table 5-11) ................90 5-10 Comparison of offline offset design methods, Westbound State Route 26 (Graph created with values from Table5 -7 through Table 5-11).... In PAGE 8: ...Graph created with values from Table 5-7 through Table 5-11)..............91 5-11 Comparison of offline offset design methods, Westbound State Route 26 (Graph created with values from Table5 -7 through Table 5-11).... In PAGE 74: ... Subjective values are required for determining the arterial classification, as well as for the vehicle progression classifications provided in HCM Table 11-5. Table5 -1a and Table 5-1b show the HCM tables in which subjective values are required. Table 5-2 provides the HCM definitions for the progression adjustment factors (PF) for different arrival types.... In PAGE 74: ... Table 5-1a and Table 5-1b show the HCM tables in which subjective values are required. Table5 -2 provides the HCM definitions for the progression adjustment factors (PF) for different arrival types. These subjective requirements are explicitly acknowledged in the HCM, and the manual states that if knowledge of the intended signal timings and quality of progression are not available, no meaningful estimation of arterial level of service is possible, even on a planning level [NRC, 1997].... In PAGE 76: ... Table5 -1: Highway Capacity Manual Tables Requiring Subjective Values HCM TABLE 11-3. ARTERIAL CLASSIFICATION ACCORDING TO FUNCTIONAL AND DESIGN CATEGORIES FUNCTIONAL CATEGORY DESIGN CATEGORY PRINCIPAL ARTERIAL MINOR ARTERIAL High Speed design And control I Not Applicable Typical suburban Design and Control II II Intermediate Design II III or IV Typical urban Design III or IV IV (a) Arterial classification subjective values HCM TABLE 11-5.... In PAGE 77: ... Table5 -2: Highway Capacity Manual Definitions for Progression Adjustment Factors HIGHWAY CAPACITY MANUAL DESCRIPTIONS FOR PROGRESSION ADJUSTMENT FACTORS ARRIVAL TYPE DESCRIPTION FOR PROGRESSION ADJUSTMENT FACTOR 1 Dense platoon containing more than 80 percent of the lane group volume and arriving at the start of the red phase. This arrival type is representative of arterials that experience very poor progression quality as a result of conditions such as lack of overall network signal optimization.... In PAGE 78: ... However with advancements in traffic controller hardware functions to include actuated controls, the delay equation does not account for how variations in green splits affect the start of green times in modern coordinated-actuated controllers. Such variations in the start of green times directly impact the quality of progression (HCM PF Factors shown in Table5 -1) and the amount of delay experienced. However, the HCM has no procedure for estimating which PF Factors shall be used with design volumes or for the design of coordinated- actuated controller timings.... In PAGE 79: ...Furthermore, determining the quality of progression factor for the PF term of the HCM average intersection delay equation is a difficult task, even if observed in the field by an engineer. For existing arterial conditions, one analyst may conclude that current arterial signal timings are not facilitating progression and assign an arrival of type-2 ( Table5 -2). But, another analyst may conclude that because of platoon dispersion, vehicle progression for the exact same arterial resembles random arrivals and assign an arrival of type-3 (Table 5-2).... In PAGE 79: ... For existing arterial conditions, one analyst may conclude that current arterial signal timings are not facilitating progression and assign an arrival of type-2 (Table 5-2). But, another analyst may conclude that because of platoon dispersion, vehicle progression for the exact same arterial resembles random arrivals and assign an arrival of type-3 ( Table5 -2). Both of these arrival types are subjective values and are difficult to distinguish between one another in the field by technicians typically employed to do so.... In PAGE 80: ... An example of how slight discrepancies with both of the discussed issues impact an arterial level of service is provided. Table5 -3 and Table 5-4 show the quantitative calculations used to compute an arterial level of service for a hypothetical 0.2 mile suburban arterial section with slightly different values assigned to the g/c ratios and the quality of progression factors.... In PAGE 81: ...developed for accurately modeling the multitude of coordinated-actuated control parameters now in use on most modern traffic signal systems. Table5 -3: HCM Calculations for Average Control Delay per Vehicle on Arterial Approach SUMMARY OF ARTERIAL INTERSECTION DELAY ESTIMATES Arterial Description: 4 lane suburban arterial, 0.2 mile section, volume = 1500 vph Adjusted Saturation flow rate = 3000 vph, Unit extension of 2.... In PAGE 81: ...or X = 0.833 and X = 0.714 respectively. Table5 -4: HCM Calculations for Arterial Level of Service COMPUTATION OF ARTERIAL LOS WORKSHEET Arterial Description: 4 lane suburban arterial, 0.2 mile section, Volume = 1500 vph, Adjusted Saturation flow rate = 3000 vph, Unit extension of 2.... In PAGE 84: ...then tabulated to calculate the arterial cumulative delay and travel times with respect to intersection locations. See Table5 -5 for an example of arterial cumulative value calculations. Plots of these cumulative delay and travel times can provide insight on the performance of the system.... In PAGE 85: ... Additionally, although the graphic procedure discussed is limited to cumulative delay and travel times, similar graphic procedures can be expanded to include the number of stops or emission estimates for HC, CO, and NOX. Table5 -5: Cumulative Delay and Travel Times used to Construct Figure 5-4 and Figure 5-5 MID-DAY TRAFFIC S.R.... In PAGE 87: ... STEP 2: Data Calculations Individual link values are then used to compute the cumulative values for link lengths, travel times, and delay times at each of the node locations on the arterial. After these cumulative data are compiled, the averages ( Table5 -5, cols. 7 amp; 8), and standard deviations (Table 5-5, cols.... In PAGE 87: ... After these cumulative data are compiled, the averages (Table 5-5, cols. 7 amp; 8), and standard deviations ( Table5 -5, cols. 9 amp; 10) for the cumulative arterial measures of effectiveness (MOEs) are computed.... In PAGE 88: ... In contrast to current methods, this proposed performance evaluation procedure can provide a graphic comparison of different system plans to validate new proposed signal timings. As shown in Figure 5-6, Figure 5-7, and Table5 -6 ,a proposed arterial timing plan can be compared with an existing arterial timing plan through graphical and tabular analysis procedures to quantitatively illustrate that the proposed timing plan accomplishes the design objective of reducing delay and travel times. This comparison of alternate signal timing plans provides the designer a tool that illustrates the improvement of the proposed arterial signal timing plan over the existing arterial signal timing plan.... In PAGE 90: ... Table5 -6: Lane Group Comparison of Delay per Vehicle at Individual Intersection; State Route 26, Node 1 Lane Group Movements Existing Delay (sec/veh) Proposed Delay (sec/veh) Existing Level of Service Proposed Level of Service 41.4 38.... In PAGE 94: ... All offset setting methodologies were used with the intent to replicate how practicing engineers typically specify offsets for coordinated-actuated systems. Table5 -7 summarizes the offsets determined with each method and what offset setting technique was used within that package. Table 5-7: Summary for Offsets From Various offline Methods for State Route 26, Lafayette, Indiana (Cycle length = 120 seconds)... In PAGE 95: ...Cumulative results for measures of effectiveness consisting of travel time (sec/veh) and delay (per-min) for the arterial through movement for each timing strategy are provided in Table5 -8 through Table 5-12. Graphical performance summaries comparing the alternate offset timing strategies are shown in Figure 5-8 through Figure 5-11.... In PAGE 96: ... Table5 -8: Existing Offset Results LINK START END CUMM TRAVEL TIME (sec) STDEV CUMM TRAVEL TIME (sec) CUMM DELAY TIME (per-min) STDEV CUMM DELAY TIME (per-min) 101 1 32.6 1.... In PAGE 97: ... Table5 -9: Fine-Tuned Offset Results LINK START END CUMM TRAVEL TIME (sec) STDEV CUMM TRAVEL TIME (sec) CUMM DELAY TIME (per-min) STDEV CUMM DELAY TIME (per-min) 101 1 32.2 1.... In PAGE 98: ... Table5 -10: PASSER II-90 Offset Results LINK START END CUMM TRAVEL TIME (sec) STDEV CUMM TRAVEL TIME (sec) CUMM DELAY TIME (per-min) STDEV CUMM DELAY TIME (per-min) 101 1 32.1 0.... In PAGE 99: ... Table5 -11: SYNCHRO Offset Results LINK START END CUMM TRAVEL TIME (sec) STDEV CUMM TRAVEL TIME (sec) CUMM DELAY TIME (per-min) STDEV CUMM DELAY TIME (per-min) 101 1 32.1 1.... In PAGE 100: ... Table5 -12: TRANSYT-7F Offset Results LINK START END CUMM TRAVEL TIME (sec) STDEV CUMM TRAVEL TIME (sec) CUMM DELAY TIME (per-min) STDEV CUMM DELAY TIME (per-min) 101 1 31.4 0.... In PAGE 101: ...Cumulative Travel Time (State Route 26 (Eastbound)) (Earl Avenue to Creasy Lane) 0 50 100 150 200 250 0 1000 2000 3000 4000 5000 6000 7000 Linear Distance Along Corridor (ft) Time (s) Signal Location Fine-tuned Existing PASSER2 Offsets SYNCHRO TRANSYT 7F Posted Speed Limit Figure 5-8: Comparison of offline offset design methods, Eastbound State Route 26 (Graph created with values from Table5 -8 through Table 5-12) Cumulative Delay (State Route 26 (Eastbound)) (Earl Avenue to Creasy Lane) 0 100 200 300 400 500 600 700 800 900 1000 0 1000 2000 3000 4000 5000 6000 7000 Linear Distance Along Corridor (ft) Delay (person-min) Signal Location Fine-tuned Existing PASSER2 SYNCHRO TRANSYT 7F Figure 5-9: Comparison of offline offset design methods, Eastbound State Route... In PAGE 101: ...Cumulative Travel Time (State Route 26 (Eastbound)) (Earl Avenue to Creasy Lane) 0 50 100 150 200 250 0 1000 2000 3000 4000 5000 6000 7000 Linear Distance Along Corridor (ft) Time (s) Signal Location Fine-tuned Existing PASSER2 Offsets SYNCHRO TRANSYT 7F Posted Speed Limit Figure 5-8: Comparison of offline offset design methods, Eastbound State Route 26 (Graph created with values from Table 5-8 through Table 5-12) Cumulative Delay (State Route 26 (Eastbound)) (Earl Avenue to Creasy Lane) 0 100 200 300 400 500 600 700 800 900 1000 0 1000 2000 3000 4000 5000 6000 7000 Linear Distance Along Corridor (ft) Delay (person-min) Signal Location Fine-tuned Existing PASSER2 SYNCHRO TRANSYT 7F Figure 5-9: Comparison of offline offset design methods, Eastbound State Route 26 (Graph created with values from Table5... In PAGE 102: ...Cumulative Travel Time (State Route 26 (Westbound)) (Creasy Lane to Earl Avenue) 0 50 100 150 200 250 300 0 1000 2000 3000 4000 5000 6000 7000 Linear Distance Along Corridor (ft) Time (s) Signal Location Fine-tuned Existing PASSER2 SYNCHRO TRANSYT 7F Posted Speed Limit Figure 5-10: Comparison of offline offset design methods, Westbound State Route 26 (Graph created with values from Table5 -8 through Table 5-12) Cumulative Delay (State Route 26 (Westbound)) (Creasy Lane to Earl Avenue) 0 200 400 600 800 1000 1200 1400 1600 0 1000 2000 3000 4000 5000 6000 7000 Linear Distance Along Corridor (ft) Delay (person-min) Signal Location Fine-tuned Existing PASSER2 SYNCHRO TRANSYT 7F Figure 5-11: Comparison of offline offset design methods, Westbound State... In PAGE 102: ...Cumulative Travel Time (State Route 26 (Westbound)) (Creasy Lane to Earl Avenue) 0 50 100 150 200 250 300 0 1000 2000 3000 4000 5000 6000 7000 Linear Distance Along Corridor (ft) Time (s) Signal Location Fine-tuned Existing PASSER2 SYNCHRO TRANSYT 7F Posted Speed Limit Figure 5-10: Comparison of offline offset design methods, Westbound State Route 26 (Graph created with values from Table 5-8 through Table 5-12) Cumulative Delay (State Route 26 (Westbound)) (Creasy Lane to Earl Avenue) 0 200 400 600 800 1000 1200 1400 1600 0 1000 2000 3000 4000 5000 6000 7000 Linear Distance Along Corridor (ft) Delay (person-min) Signal Location Fine-tuned Existing PASSER2 SYNCHRO TRANSYT 7F Figure 5-11: Comparison of offline offset design methods, Westbound State Route 26 (Graph created with values from Table5... In PAGE 103: ... To accommodate left turning vehicles from upstream intersections, an online algorithm should attempt to keep the average start of green as low as possible in relation to downstream intersections while also accounting for downstream queues that may impede progression. Table5 -13 through Table 5-15 provide statistical significance summaries comparing the fine-tuning offset... In PAGE 104: ... Table5 -13: State Route 26 statistical significance summary; Fine-tuned offsets versus PASSER-II 90 offsets Measure of Effectiveness FINE TUNED PASSER II-90 Percent Reduction Calculated t-statistic Test statistic for 95% C.I.... In PAGE 104: ...078 -1.688 (#) Standard deviation; n1 = n2 = 20 replications Table5 -14: State Route 26 statistical significance summary; Fine-tuned offsets versus SYNCHRO offsets Measure of Effectiveness FINE TUNED SYNCHRO Percent Reduction Calculated t-statistic Test statistic for 95% C.I.... In PAGE 105: ... Table5 -15: State Route 26 statistical significance summary; Fine-tuned offsets versus TRANSYT-7F Measure of Effectiveness FINE TUNED PASSER II-90 Percent Reduction Calculated t-statistic Test statistic for 95% C.I.... ..."

### Table 1: Lower bounds for difierent versions of the static on-line range searching problem using O(m) storage.

2007

"... In PAGE 4: ...n a relatively easy way. These models are better suited for describing algorithms and upper bounds. In the following, we will examine the models of computation that have been used to obtain nontrivial lower bounds in range searching. Table1 summarizes the best known lower bounds for difierent variations of non-orthogonal range searching problems. Many of the following results for the non-orthogonal case have their orthogonal counterparts [53, 23, 24] as well.... ..."