Smoothing Techniques and Augmented Lagrangian Method for Recourse Problem of Two-Stage Stochastic Linear Programming

Table 1

Comparative between smooth augmented Lagrangian Newton method (SALN) and CPLEX solver.

N. P	Recourse problem; 𝑛2×𝑚2×𝑑	Solver	‖𝑊𝑧−𝑞‖∞	|𝑄(𝜉)−𝜉^𝑇𝑧|	‖𝑧‖	Time (second)
P1	50×50×0.68	SALN	9.9135𝑒−011	7.2760𝑒−012	41.0362	0.3292
		CPLEX	1.0717𝑒−007	5.2589𝑒−007	41.0362	0.3182
P2	100×105×0.4	SALN	1.8622𝑒−010	4.3074𝑒−009	57.1793	0.1275
		CPLEX	6.5591𝑒−008	1.6105𝑒−006	57.1826	0.1585
P3	150×150×0.5	SALN	3.1559𝑒−010	5.6361𝑒−009	74.9098	0.6593
		CPLEX	5.9572𝑒−010	1.2014𝑒−009	74.9098	0.1605
P4	200×200×0.5	SALN	5.3819𝑒−010	1.0506𝑒−008	85.6646	0.2530
		CPLEX	1.1734𝑒−007	1.3964𝑒−006	85.6646	0.1820
P5	300×300×0.5	SALN	9.4178𝑒−010	2.7951𝑒−008	102.4325	2.1356
		CPLEX	4.1638𝑒−010	4.4456𝑒−009	102.4325	0.1830
P6	350×350×0.5	SALN	1.2787𝑒−009	2.6226𝑒−008	110.4189	3.2102
		CPLEX	7.7398𝑒−010	3.4452𝑒−009	110.4189	0.2116
P7	450×500×0.05	SALN	1.6564𝑒−010	4.7094𝑒−009	124.1204	0.7807
		CPLEX	9.0949𝑒−013	1.6371𝑒−011	124.1205	0.2606
P8	500×550×0.04	SALN	1.0425𝑒−010	1.0241𝑒−009	134.3999	0.7567
		CPLEX	6.1618𝑒−011	1.9081𝑒−009	134.3999	0.2660
P9	700×800×0.6	SALN	4.6139𝑒−009	2.3908𝑒−007	153.9782	7.1364
		CPLEX	9.1022𝑒−009	1.3768𝑒−007	153.9906	0.8435
P10	900×1100×0.4	SALN	5.3396𝑒−009	1.5207𝑒−007	178.1151	7.9383
		CPLEX	4.5020𝑒−011	2.0373𝑒−010	178.1163	1.3643
P11	1500×2000×0.1	SALN	1.1289𝑒−008	2.0696𝑒−007	231.5284	13.2493
		CPLEX	7.5886𝑒−011	1.1059𝑒−009	231.5286	2.1343
P12	1000×2000×0.01	SALN	3.6398𝑒−010	7.9162𝑒−009	198.4905	5.2620
		CPLEX	6.8212𝑒−013	1.1642𝑒−010	198.4922	0.6752
P13	1000×5000×0.001	SALN	2.7853𝑒−010	1.9281𝑒−010	190.0141	0.2709
		CPLEX	4.8203𝑒−010	1.5425𝑒−009	190.0142	0.2162
P14	1000×10000×0.001	SALN	1.1221𝑒−010	1.4988𝑒−009	212.2416	0.4867
		CPLEX	7.9094𝑒−009	2.6691𝑒−007	212.3453	0.3353
P15	1000×1𝑒5×0.001	SALN	9.7702𝑒−010	2.3283𝑒−009	231.7930	3.7511
		CPLEX	2.7285𝑒−012	1.8044𝑒−009	231.8763	1.2472
P16	1000×1𝑒6×0.0002	SALN	9.9432𝑒−013	2.0464𝑒−012	121.3937	9.0948
		CPLEX	9.9098𝑒−006	1.1089𝑒−004	121.3940	3.7848
P17	100×1𝑒6×0.001	SALN	3.9563𝑒−010	5.6607𝑒−009	73.8493	6.8537
		CPLEX	2.0082𝑒−003	1.1777𝑒−002	73.9412	2.5582
P18	10×1𝑒4×0.001	SALN	2.2737𝑒−013	7.9581𝑒−013	19.5735	0.0166
		CPLEX	2.2737𝑒−013	1.1369𝑒−013	19.5739	0.1386
P19	10×1𝑒6×0.001	SALN	1.2478𝑒−009	1.0710𝑒−008	18.8192	5.6399
		CPLEX	5.9615𝑒−004	6.4811𝑒−005	18.8863	2.5623
P20	10×5𝑒6×0.01	SALN	2.0425𝑒−008	1.1816𝑒−008	20.8470	28.7339
		CPLEX	4.9966𝑒+004	3.3500𝑒+005	0.0000	1.9482
P21	100×1𝑒6×0.01	SALN	3.8654𝑒−012	7.7875𝑒−012	42.0698	7.8895
		CPLEX	1.7994𝑒−004	6.2712𝑒−005	42.0931	8.8324
P22	100×1𝑒4×0.1	SALN	1.1084𝑒−012	2.1600𝑒−012	39.4563	0.1440
		CPLEX	1.2518𝑒−005	1.3174𝑒−005	39.4563	0.3099
P23	100×1𝑒5×0.05	SALN	1.7053𝑒−012	6.3121𝑒−012	43.5944	1.1379
		CPLEX	3.9827𝑒−006	2.8309𝑒−006	43.5944	1.5729
P24	1000×5𝑒4×0.1	SALN	1.3074𝑒−012	2.4102𝑒−011	117.4693	15.1065
		CPLEX	6.7455𝑒−007	6.7379𝑒−006	117.4699	20.3967
P25	1000×1𝑒5×0.08	SALN	2.5011𝑒−012	1.0516𝑒−011	116.6964	23.2540
		CPLEX	9.4298𝑒−006	5.5861𝑒−004	116.6964	33.4319