This section discusses the results obtained from the LR approach for the Predecessor-Jump model from section 4.1.4. The results herein are also compared to the values presented in [Gru06]6.
As mentioned above the combination of Relax-and-Cut and Subgradient Optimization proved to be very sensitive to even minor changes in its param-eters when applied to the Lagrangian relaxation for the Predecessor-Jump model. After extensive testing I decided to set SG_baseAgility_Termination-Level to 0.05 and SG_baseAgility_ReductionAfterNoImprove to 10, since these values seemed to be a reasonable compromise between quality of the lower bound and total running time. More relaxed values improve the lower bounds slightly on average, but increase the total running time signicantly.
5More relaxed values in this context means smaller values for SG_baseAgility_-TerminationLevel and/or bigger values for SG_baseAgility_ReductionAfterNoImprove.
6Note that the values produced by [Gru06] are based on the Predecessor-Depth model and not on the Predecessor-Jump model.
CHAPTER 6. COMPUTATIONAL EXPERIMENTS 47 The three dierent combinations of dynamic constraint separation strate-gies7turned out to dier in their success to improve the lower bound. There-fore, I decided to use dierent settings for SG_repetitionsNoImproveLimit for each of the three combinations. When both strategies are applied, the lower bounds can be improved early in the Subgradient Optimization and further improvements are achieved with some regularity. In opposition to this, the strategies[V1]and[Vdepth], applied for their own, lead to a faster convergence of the Subgradient Optimization. This happens because fewer improvements can be achieved. The consequence is that, compared to the conguration where both separation strategies are applied, SG terminates after fewer it-erations and after a shorter computation time. To compensate this and to allow for a fairer comparison between the dierent combinations of separation strategies, I decided to setSG_repetitionsNoImproveLimitto6for the combined strategy, to 8when only [V1]was applied, and to 10 when only [Vdepth]was applied. These increased values improved the found lower bounds to some degree, but clearly any of the strategies[V1]or [Vdepth]applied for their own produces inferior lower bounds compared to the combined strategy.
Results for the Predecessor-Jump Approach with [V1]
Table 6.2 shows the results of the Relax-and-Cut algorithm applied to the LR approach of the Predecessor-Jump model, together with jump separation according to the strategy[V1].
The following observations can be made:
1. The lower bounds are better than the values of the minimum spanning trees for all considered instances (G≤0.759).
2. In almost all cases, the lower bounds are better than the values of the LP relaxation with cuts from [Gru06].
3. Feasible solutions are found for three instances.
4. Heuristic solutions are found for all but one instance. In general, they are very close to the optimum. Actually, the optimum is reached for most of the instances.
5. Two instances can be solved to proven optimality.
Results for the Predecessor-Jump Approach with [Vdepth] Table 6.3 presents the results of the Relax-and-Cut algorithm applied to the LR approach of the Predecessor-Jump model, together with jump separation according to the strategy[Vdepth].
The following observations can be made:
7Only[V1]or only[Vdepth]or both.
CHAPTER 6. COMPUTATIONAL EXPERIMENTS 48
InstVEDMLPLPCOptLBFHUCAitreptgG TE201004292313.43315.76369321.53-370‡79411021468746364.90.1290.616 TE201006292286.67303.00322305.25-3323548932485648498.80.0520.558 TE201008292279.50298.00308301.96-30833810140490446738.60.0200.378 TE302004396444.12447.31599456.51--599215309001892152.20.2380.702 TE302006396390.39412.18482420.34-491‡1256141435153551358.40.1280.717 TE302008396364.07404.67437409.73-437482206588270724448.40.0620.665 TR201004145180.27183.55233192.79-233‡97111820689269563.80.1730.457 TR201006145141.76153.00178164.77-1782221442810101761626.90.0740.401 TR201008145136.00151.00154152.001541541855537692161654.40.0130.222 TR302004132166.22169.20234170.99-234331172275355551074.50.2690.618 TR302006132126.83137.67157139.63-15720911440441943963.80.1110.695 TR302008132115.00135.00135∗134.31-1351703652291921.30.0050.231 c201904271267.21284.83349300.39-349349137386336741642.30.1390.623 c201906261234.15267.00298271.53-29932013655128071234617.80.0890.715 c2019010313239.38317.50324319.20-33235966864057451685.90.0150.436 c253004366376.58394.65500411.97-500521185489619884737.70.1760.657 c253006342315.23349.25378354.60-378424153708351795693.60.0620.650 c2530010366333.30368.50379369.48-379439961113026342351.10.0250.732 g20504332393.88395.69442389.03-446‡75115514870384406.50.1200.482 g20506300237.79304.50329307.00-3293667918689656316.80.0670.759 g205010357240.89359.00359§358.303593593777249640.80.0020.352 g401004500602.85623.67755595.54-755‡159723120725578768.80.2110.625 g401006570417.77581.92599584.67-5996557219513347209.50.0240.494 g4010010570392.35571.50574571.50-5746297537501955367.70.0040.625 Table6.2:ComputationalresultsforthePredecessor-Jumpapproachwithseparationstrategy[V1].SGsettings: SG_repetitionsNoImproveLimit8 ∗ ThecomputationstoppedbecausetheLBwasgreaterthanH−1,i.e.Histheoptimum. § ThecomputationstoppedbecausetheLBwasgreaterthanF−1,i.e.Fistheoptimum. ‡Theupperboundisthesumofthe|V|−1mostexpensiveedges.
CHAPTER 6. COMPUTATIONAL EXPERIMENTS 49
InstVEDMLPLPCOptLBFHUCAitreptgG TE201004292313.43315.76369292.00--‡794232717531372.20.2091.000 TE201006292286.67303.00322292.00--3543141167913122.80.0931.000 TE201008292279.50298.00308292.00-320338135019461299.40.0521.000 TE302004396444.12447.31599396.00--59989435637301072.20.3391.000 TE302006396390.39412.18482396.00-482‡12563855286114522.10.1781.000 TE302008396364.07404.67437396.00760437482186806415426187.40.0941.000 TR201004145180.27183.55233145.00-316‡9713428310019192.50.3781.000 TR201006145141.76153.00178145.003801792223638241916175.40.1851.000 TR201008145136.00151.00154145.002891711851949214812141.00.0581.000 TR302004132166.22169.20234133.00-2343311884116404836929.40.4320.990 TR302006132126.83137.67157132.00457161209167615017303734.80.1591.000 TR302008132115.00135.00135132.00255135170106763443203669.90.0221.000 c201904271267.21284.83349271.7562934934915238139581003094.60.2210.990 c201906261234.15267.00298261.0059129932066674523291164.40.1241.000 c2019010313239.38317.50324313.50344327359101117126418717186.60.0320.955 c253004366376.58394.65500367.001130500521173488309603554.40.2660.993 c253006342315.23349.25378342.0012413874243079175311448.40.0951.000 c2530010366333.30368.50379366.0074939343931142249612742.30.0341.000 g20504332393.88395.69442335.00-446‡7511172313066661294.90.2420.973 g20506300237.79304.50329300.00-3293665230363627190.70.0881.000 g205010357240.89359.00359357.1235935937712538156311032568.80.0050.938 g401004500602.85623.67755500.00-757‡15973341286815268.20.3381.000 g401006570417.77581.92599570.001443619655100524255221352.00.0481.000 g4010010570392.35571.50574570.007905806296578312519943.00.0071.000 Table6.3:ComputationalresultsforthePredecessor-Jumpapproachwithseparationstrategy[Vdepth].SGsettings: SG_repetitionsNoImproveLimit10 ‡Theupperboundisthesumofthe|V|−1mostexpensiveedges.
CHAPTER 6. COMPUTATIONAL EXPERIMENTS 50 1. The lower bounds are not better than the values of the corresponding minimum spanning trees for most of the considered instances (G ≥ 0.938).
2. More feasible solutions are found compared to the experiments with the separation strategy[V1].
3. Heuristic solutions are found for all but two instances. In general, they are very close to the optimum. The optimum is reached for some of the instances.
Results for the Predecessor-Jump Approach with [V1] and
[Vdepth]
Table 6.4 shows the results of the Relax-and-Cut algorithm applied to the LR approach of the Predecessor-Jump model, together with jump separation according to the strategy [V1] and [Vdepth]. In this table the rst value in column C gives the number of constraints that were separated according to [V1], and the second value gives the number of separated constraints accord-ing to[Vdepth].
The following observations can be made:
1. The lower bounds are better than the values of the corresponding min-imum spanning trees for all of the considered instances.
2. The lower bound is worse than the value of the corresponding LP relaxation with cuts for only one instance. For all other instances it is equally or even better.
3. Compared to the experiments with the separation strategy [Vdepth], even more feasible solutions are found.
4. Heuristic solutions are found for all but one instance. In general, they are very close to the optimum. Actually the optimum is reached for most of the instances.
5. Two instances can be solved to proven optimality.
To summarize, it can be said that combining the two separation strate-gies actually combines the strength's of both. The lower bounds are, with one exception, always equal or better than the values of the LP relaxation with cuts, and for most of the instances feasible solutions can be found.
Unfortunately, the running times to compute the lower bounds are much longer than the ones of the LP relaxation with cuts. It seems that while the Lagrangian relaxation presented in section 4.1.2 does produce good lower bounds, the Relax-and-Cut approach based on Subgradient Optimization requires too many iterations. It might be promising to substitute the Sub-gradient Optimization with another scheme for solving Lagrangian duals.
CHAPTER 6. COMPUTATIONAL EXPERIMENTS 51
InstVEDMLPLPCOptLBFHUCAitreptgG TE201004292313.43315.76369325.49-369‡794137/374285146521546.50.1180.565 TE201006292286.67303.00322307.30-328354113/277393203351066.00.0460.490 TE201008292279.50298.00308301.9946630833863/13050196022443.20.0200.375 TE302004396444.12447.31599449.37--599156/326404554432123.00.2500.737 TE302006396390.39412.18482426.95-489‡1256168/562644138363637.00.1140.640 TE302008396364.07404.67437409.83723437482165/611883748374249.60.0620.663 TR201004145180.27183.55233200.80-233‡971149/572265506412292.90.1380.366 TR201006145141.76153.00178164.76360178222154/388389255944982.20.0740.401 TR201008145136.00151.00154152.0015415418549/11751383142813.30.0130.222 TR302004132166.22169.20234176.75-234331255/1773391166810019067.60.2450.561 TR302006132126.83137.67157143.63489157209137/8496780157516796.80.0850.535 TR302008132115.00135.00135∗134.6628313517070/696867211131.80.0030.113 c201904271267.21284.83349301.241138349349138/366386761713685.20.1370.612 c201906261234.15267.00298272.5965029932081/215654043432555.40.0850.687 c2019010313239.38317.50324319.3649932735966/1261064047433603.60.0140.422 c253004366376.58394.65500418.26941500521172/390485982584991.10.1630.610 c253006342315.23349.25378354.67905378424110/392864738477312.70.0620.648 c2530010366333.30368.50379369.33118637943985/1701465320548746.60.0260.744 g20504332393.88395.69442399.63693442‡751163/40122117871132484.30.0960.385 g20506300237.79304.50329306.5047432936669/9824346335290.50.0680.776 g205010357240.89359.00359∗358.734693593774/4331230.10.0010.135 g401004500602.85623.67755590.40-757‡1597179/602467333794730.90.2180.645 g401006570417.77581.92599584.66145661665543/30168233825851.40.0240.494 g4010010570392.35571.50574571.50-57462969/16368334635911.90.0040.625 Table6.4:ComputationalresultsforthePredecessor-Jumpapproachwithseparationstrategies[V1]and[Vdepth].SGsettings: SG_repetitionsNoImproveLimit6 ∗ ThecomputationstoppedbecausetheLBwasgreaterthanH−1,i.e.Histheoptimum. ‡ Theupperboundisthesumofthe|V|−1mostexpensiveedges.
CHAPTER 6. COMPUTATIONAL EXPERIMENTS 52