Column generationalgorithmsfornonlinearoptimization,II: Numericalinvestigations

Garcı´a etal. [1] presentaclassofcolumngeneration(CG)algorithmsfornonlinearprograms.Itsmain motivationfromatheoreticalviewpointisthatundersomecircumstances,finiteconvergencecanbe achieved,inmuchthesamewayasfortheclassicsimplicialdecompositionmethod;themainpractical motivationisthatwithintheclasstherearecertainnonlinearcolumngenerationproblemsthatcan acceleratetheconvergenceofasolutionapproachwhichgeneratesasequenceoffeasiblepoints.This algorithmcan,forexample,acceleratesimplicialdecompositionschemesbymakingthesubproblems nonlinear.Thispapercomplementsthetheoreticalstudyontheasymptoticandfiniteconvergenceof these methodsgivenin [1] with anexperimentalstudyfocusedontheircomputationalefficiency. Three typesofnumericalexperimentsareconducted.Thefirstgroupoftestproblemshasbeen designedtostudytheparametersinvolvedinthesemethods.Thesecondgrouphasbeendesignedto investigatetheroleandthecomputationoftheprolongationofthegeneratedcolumnstotherelative boundary.Thelastonehasbeendesignedtocarryoutamorecompleteinvestigationofthedifference in computationalefficiencybetweenlinearandnonlinearcolumngenerationapproaches. In ordertocarryoutthisinvestigation,weconsidertwotypesoftestproblems:thefirstoneisthe nonlinear,capacitatedsingle-commoditynetworkflowproblemofwhichseverallarge-scaleinstances with varieddegreesofnonlinearityandtotalcapacityareconstructedandinvestigated,andthesecond one isacombinedtrafficassignmentmodel.