A Comparison Among Simple Algorithms for Linear Programming

SILVA, J.; GHIDINI, C.T.L.S.; OLIVEIRA, A.R.L.; FONTOVA, M.I.V.

doi:10.5540/tema.2018.019.02.0305

J. SILVA ^**Corresponding author: Jair Silva - E-mail: jairsilva@ufpr.br

Campus Avançado da Universidade Federal do Paraná em Jandaia do Sul, Rua Doutor João Maximiano, 426, Vila Operária, 86900-000, Jandaia do Sul, PR, Brazil. E-mail: jairsilva@ufpr.br

C.T.L.S. GHIDINI

Faculdade de Ciências Aplicadas, Universidade de Campinas, Rua Pedro Zaccaria, 1300, 13484-350, Limeira, SP, Brazil. E-mail: carla.ghidini@fca.unicamp.br

A.R.L. OLIVEIRA

Instituto de Matemática, Estatística e Computação Científica, Universidade de Campinas, Rua Sérgio Buarque de Holanda, 651, 13083-859, Campinas, SP, Brazil. E-mail: aurelio@ime.unicamp.br

M.I.V. FONTOVA

Centro Universitário de Campo Limpo Paulista, Rua Guatemala, 167, 13231-230, Jd.América, Campo Limpo Paulista, SP, Brazil. E-mail: martavelazco@cc.faccamp.br

*Corresponding author: Jair Silva - E-mail: jairsilva@ufpr.br

Problem	Line	Column	Time t5	Problem	Line	Column	Time t5
25fv47	769	1821	0.060000	ship04l	292	1905	0.030000
80bau3b	1965	10701	0.240000	ship04s	216	1281	0.090000
adlittle	53	134	0.000001	ship08l	470	3121	0.120000
afiro	25	48	0.000001	ship08s	276	1604	0.100000
agg	319	404	0.000001	ship12l	610	4171	0.040000
agg2	455	689	0.010000	ship12s	340	1943	0.150000
agg3	455	689	0.020000	sierra	1129	2618	0.080000
bandm	211	366	0.000001	stair	356	531	0.060000
beaconfd	73	148	0.020000	standata	292	582	0.050000
blend	66	101	0.020000	standgub	292	582	0.050000
bnl1	558	1439	0.020000	standmps	388	1146	0.040000
bnl2	1848	3800	0.080000	stocfor1	94	142	0.010000
boeing1	294	660	0.040000	stocfor2	1968	2856	0.030000
boeing2	125	264	0.000001	stocfor3	15336	22202	0.190000
bore3d	64	90	0.010000	truss	1000	8806	0.050000
brandy	116	216	0.020000	tuff	246	553	0.120000
capri	235	421	0.010000	vtp_base	46	82	0.000001
cycle	1400	2749	0.020000	wood1p	171	1718	0.180000
czprob	661	2705	0.040000	woodw	708	5364	0.090000
d2q06c	2012	5561	0.130000	cre-a	2994	6692	0.040000
d6cube	403	5443	0.480000	cre-b	5336	36382	0.230000
degen2	444	757	0.050000	cre-c	2375	5412	0.030000
degen3	1503	2604	0.260000	cre-d	4102	28601	0.190000
dfl001	5907	12065	1.440000	ken-07	1427	2603	0.030000
e226	161	392	0.020000	ken-11	10061	16709	0.540000
etamacro	331	666	0.020000	ken-13	22519	36546	2.060000
fffff800	313	817	0.040000	ken-18	78823	128395	21.310000
finnis	359	775	0.010000	osa-07	1047	24911	0.160000
fit1d	24	1047	0.010000	osa-14	2266	54535	0.380000
fit1p	678	1706	0.010000	osa-30	4279	103978	0.000001
fit2d	25	10387	0.280000	osa-60	10209	242411	1.840000
fit2p	3170	13695	0.240000	pds-02	2603	7333	0.070000
forplan	104	411	0.090000	pds-06	9119	28435	0.490000
ganges	840	1197	0.030000	pds-10	15587	48719	1.240000
gfrd-pnc	590	1134	0.020000	pds-20	32287	106080	5.440000
greenbea	1872	4081	0.070000	BL	5468	12038	0.830000
greenbeb	1865	4065	0.090000	BL2	5480	12063	0.840000
grow15	300	645	0.010000	CO5	4471	10318	0.240000
grow22	440	946	0.020000	CO9	8510	19276	0.470000
grow7	140	301	0.000001	CQ9	7073	17806	0.300000
israel	166	307	0.010000	GE	8361	14096	0.200000
kb2	43	68	0.000001	NL	6478	14393	0.560000
lotfi	117	329	0.000001	a1	42	73	0.000001
maros	626	1365	0.030000	fort45	1037	1467	0.060000
maros-r7	2152	6578	0.090000	fort46	1037	1467	0.060000
modszk1	658	1405	0.010000	fort47	1037	1467	0.050000
nesm	646	2850	0.080000	fort48	1037	1467	0.060000
perold	580	1412	0.020000	scagr25	344	543	0.020000
pilot	1350	4506	0.090000	fort49	1037	1467	0.050000
pilot4	389	1069	0.030000	fort51	1042	1473	0.060000
pilot87	1968	6367	0.220000	fort52	1041	1471	0.060000
pilot ja	795	1834	0.040000	fort53	1041	1471	0.050000
pilot we	691	2621	0.050000	fort54	1041	1471	0.050000
pilotnov	830	2089	0.050000	fort55	1041	1471	0.060000
recipe	61	120	0.000001	fort56	1041	1471	0.040000
sc105	104	162	0.000001	fort57	1041	1471	0.060000
sc205	203	315	0.000001	fort58	1041	1471	0.050000
sc50a	49	77	0.000001	fort59	1041	1471	0.050000
sc50b	48	76	0.000001	fort60	1041	1471	0.060000
scagr25	344	543	0.010000	fort61	1041	1471	0.050000
scagr7	92	147	0.000001	x1	983	1413	0.050000
scfxm1	268	526	0.020000	x2	983	1413	0.050000
scfxm2	536	1052	0.030000	pata01	122	1241	0.010000
scfxm3	804	1578	0.040000	pata02	122	1241	0.020000
scorpion	180	239	0.060000	patb01	57	143	0.000001
scrs8	418	1183	0.080000	patb02	57	143	0.000001
scsd1	77	760	0.260000	vschna02	122	1363	0.010000
scsd6	148	1350	0.430000	vschnb01	57	144	0.000001
scsd8	397	2750	0.070000	vschnb02	58	202	0.000001
sctap1	269	608	0.010000	willett	184	588	0.030000
sctap2	977	2303	0.010000	ex01	234	1555	0.070000
sctap3	1346	3113	0.020000	ex02	226	1547	0.070000
seba	2	9	0.130000	ex05	831	7747	0.090000
share1b	107	243	0.040000	ex06	824	7778	0.090000
share2b	92	158	0.010000	ex09	1821	18184	0.340000
shell	487	1450	0.020000

Algorithm	t1	t2	t3	t4	t5
Algorithm with p=2	7.28%	9.93%	19.20%	4.63%	11.92 %
Algorithm with p=4	23.84%	28.47%	23.17%	19.20%	19.20 %
Algorithm with p=10	17.88%	17.21%	15.89%	19.21%	13.25 %
Algorithm with p=20	23.85%	6.63%	8.63%	27.81%	3.98%
Algorithm with p=40	4.64%	4.64%	1.33%	2.66%	9.27%
Algorithm with p=100	22.51%	33.12%	31.78%	26.49%	42.38%

	H		NH			H		NH
Problem	p	it	p	it	Problem	p	it	p	it
25fv47	4	100	9	100	ship08s	4	100	6	100
80bau3b	8	47	3	17	ship12l	4	100	6	100
adlittle	4	100	5	100	ship12s	4	100	6	100
afiro	4	100	3	100	sierra	4	100	2	100
agg	4	100	5	49	stair	4	100	9	100
agg2	4	6	8	63	standata	4	100	3	100
agg3	4	6	8	8	standgub	4	100	3	100
bandm	4	100	7	100	standmps	4	100	4	100
beaconfd	4	100	11	100	stocfor1	4	100	4	100
blend	4	100	6	100	stocfor2	4	100	4	100
bnl1	4	100	6	100	stocfor3	20	100	4	100
bnl2	4	100	6	100	truss	4	100	10	100
boeing1	4	100	5	100	tuff	4	46	11	100
boeing2	4	100	4	100	wood1p	4	100	83	3
bore3d	4	100	5	100	woodw	4	100	11	100
brandy	4	100	11	100	pds-10	20	100	3	100
capri	4	100	4	100	pds-20	20	100	2	100
cycle	4	100	8	100	scagr25	4	100	4	100
czprob	4	100	5	100	gfrdpnc	4	100	3	100
d2q06c	4	100	10	100	pilotja	4	100	8	100
d6cube	4	100	24	100	pilotwe	4	82	6	100
degen2	4	100	8	100	vtpbase	4	63	3	100
degen3	4	100	13	100	cre-a	4	100	4	100
dfl001	8	100	5	3	cre-b	20	100	9	100
e226	4	100	9	100	cre-c	4	100	4	100
etamacro	4	13	4	13	cre-d	20	100	9	100
fffff800	4	100	11	100	ken-07	4	100	2	100
finnis	4	100	4	100	ken11	20	100	2	100
fit1d	4	33	10	53	ken13	20	100	2	100
fit1p	4	100	7	100	ken18	20	100	2	100
fit2d	8	5	9	5	osa-07	20	100	18	45
fit2p	8	100	4	100	osa-14	20	65	18	71
forplan	4	6	18	96	osa-30	20	73	18	67
ganges	4	2	4	2	osa-60	20	100	2	9
greenbea	4	54	8	40	pds-02	4	100	3	100
greenbeb	4	100	8	68	pds-06	20	100	3	100
grow15	4	100	6	100	BL	8	100	4	100
grow22	4	100	6	100	BL2	8	100	4	100
grow7	4	100	6	100	CO5	8	100	7	100
israel	4	100	11	100	CO9	20	100	7	100
kb2	4	100	5	100	CQ9	20	100	7	100
lotfi	4	55	5	100	GE	20	6	4	6
maros	4	6	7	72	NL	20	100	4	100
maros-r7	4	28	26	100	a1	4	100	3	100
modszk1	4	25	3	34	fort45	4	62	3	84
nesm	4	100	5	100	fort46	4	100	3	100
perold	4	100	6	100	fort47	4	83	3	100
pilot	4	100	12	54	fort48	4	92	3	100
pilot4	4	100	8	100	fort49	4	34	3	54
pilot87	4	100	14	100	fort51	4	42	4	42
pilotnov	4	100	8	52	fort52	4	100	3	100
recipe	4	100	4	100	fort53	4	100	3	100
sc105	4	100	3	100	fort54	4	36	3	100
sc205	4	100	3	100	fort55	4	34	3	100
sc50a	4	100	3	100	fort56	4	100	3	100
sc50b	4	100	3	100	fort57	4	100	3	100
scagr25	4	100	4	100	fort58	4	100	3	100
scagr7	4	100	3	100	fort59	4	100	3	100
scfxm1	4	100	6	100	fort60	4	100	3	100
scfxm2	4	100	6	100	fort61	4	100	3	100
scfxm3	4	100	6	100	x1	4	4	3	6
scorpion	4	100	4	100	x2	4	100	3	76
scrs8	4	100	5	100	pata01	4	100	7	100
scsd1	4	100	10	100	pata02	4	100	7	100
scsd6	4	100	10	100	patb01	4	100	4	100
scsd8	4	100	9	100	patb02	4	100	4	100
sctap1	4	100	5	100	vschna02	4	100	7	100
sctap2	4	100	5	100	vschnb01	4	100	4	100
sctap3	4	100	5	100	vschnb02	4	100	4	100
seba	4	100	4	100	willett	4	100	8	100
share1b	4	88	7	100	ex01	4	100	5	100
share2b	4	100	7	100	ex02	4	100	5	100
shell	4	68	3	100	ex05	4	100	5	100
ship04l	4	100	6	100	ex06	4	100	5	100
ship04s	4	100	6	100	ex09	20	100	5	100
ship08l	4	100	6	100

\begin{array}{l} λ_{0} (1 - x_{S}^{k - 1}) + λ_{1} = 1 \Leftrightarrow λ_{1} = 1 - λ_{0} (1 - x_{S}^{k - 1}) \geq 0 \\ \bar{b} = λ_{0} (b^{k - 1} - x_{S}^{k - 1} P_{S}) + λ_{1} P_{S} \\ \begin{matrix}  \end{matrix} = λ_{0} (b^{k - 1} - x_{S}^{k - 1} P_{S}) + (1 - λ_{0} (1 - x_{S}^{k - 1})) P_{S} \\ \begin{matrix}  \end{matrix} = λ_{0} b^{k - 1} + (1 - λ_{0}) P_{S} . \end{array}

x_{j}^{k} = {\begin{matrix} λ x_{j}^{k - 1}, \begin{matrix}  \end{matrix} j \neq s \\ 1 - λ (1 - x_{S}^{k - 1}), \begin{matrix}  \end{matrix} j = s . \end{matrix}

\begin{array}{l} minimize \begin{matrix}  \end{matrix} {‖ \bar{b} ‖}^{2} \\ s .t . \begin{matrix}  \end{matrix} λ_{0} (1 - x_{S^{+}}^{k - 1} - x_{S^{-}}^{k - 1}) + λ_{1} + λ_{2} = 1, \\ \begin{matrix}  \end{matrix} λ_{i} \geq 0, for i = 0,1,2. \end{array}

\begin{array}{l} \bar{b} = λ_{0} (b^{k - 1} - x_{S^{+}}^{k - 1} P_{S^{+}} - x_{S^{-}}^{k - 1} P_{S^{-}}) + λ_{1} P_{S^{+}} + λ_{2} P_{S^{-}} \\ \begin{matrix}  \end{matrix} = λ_{0} b^{k - 1} + (λ_{1} - λ_{0} x_{S^{+}}^{k - 1}) P_{S^{+}} + (λ_{2} - λ_{0} x_{S^{-}}^{k - 1}) P_{S^{-}} \end{array}

\begin{array}{l} \bar{b} = λ_{0} (b^{k - 1} - \sum_{i = 1}^{s_{1}} x_{η_{i}^{+}}^{k - 1} P_{η_{i}^{+}} - \sum_{j = 1}^{s_{2}} x_{η_{j}^{-}}^{k - 1} P_{η_{j}^{-}}) + \sum_{i = 1}^{s_{1}} λ_{η_{i}^{+}} + P_{η_{i}^{+}} + \sum_{j = 1}^{s_{2}} λ_{η_{j}^{-}} P_{η_{j}^{-}} \\ \begin{matrix}  \end{matrix} = λ_{0} b^{k - 1} + \sum_{i = 1}^{s_{1}} (λ_{η_{i}^{+}} - λ_{0} x_{η_{i}^{+}}^{k - 1}) P_{η_{i}^{+}} + \sum_{j = 1}^{s_{2}} (λ_{η_{j}^{-}} - λ_{0} x_{η_{j}^{-}}^{k - 1}) P_{η_{j}^{-}} \end{array}

P_{ν} = \frac{1}{1 - \sum_{i = 1}^{s_{1}} x_{η_{i}^{+}}^{k - 1} - \sum_{j = 1}^{s_{2}} x_{η_{j}^{-}}^{k - 1}} (b^{k - 1} - \sum_{i = 1}^{s_{1}} x_{η_{i}^{+}}^{k - 1} P_{η_{i}^{+}} - \sum_{j = 1}^{s_{2}} x_{η_{j}^{-}}^{k - 1} P_{η_{j}^{-}}) .

[\begin{matrix} A^{t} A & c & - I d \\ U & 0 & Λ \\ c^{t} & 0 & 0 \end{matrix}] [\begin{matrix} d λ \\ d τ \\ d μ \end{matrix}] = [\begin{matrix} r_{1} \\ r_{2} \\ r_{3} \end{matrix}]

\begin{array}{l} d μ = Λ^{- 1} r_{2} - Λ^{- 1} U d λ, \\ d λ = {(A^{t} A + Λ^{- 1} U)}^{- 1} r_{4} - {(A^{t} A + Λ^{- 1} U)}^{- 1} c d τ, \\ c^{t} {(A^{t} A + Λ^{- 1} U)}^{- 1} c d τ = c^{t} {(A^{t} A + Λ^{- 1} U)}^{- 1} r_{4} - r_{3}, \end{array}

b_{p_{2}}^{k} = {\bar{λ}}_{1} (b^{k - 1} - \sum_{i = 1}^{s_{1}} x_{η_{i}^{+}}^{k - 1} P_{η_{i}^{+}} - \sum_{j = 1}^{s_{2}} x_{η_{j}^{-}}^{k - 1} P_{η_{j}^{-}}) + \sum_{i = 1}^{s_{1}} {\bar{λ}}_{η_{i}^{+}} P_{η_{i}^{+}} + \sum_{j = 1}^{s_{2}} {\bar{λ}}_{η_{j}^{-}} P_{η_{j}^{-}},

\begin{array}{l} ‖ {\tilde{λ}}_{1} (b^{k - 1} - \sum_{i = 1}^{s_{3}} x_{η_{i}^{+}}^{k - 1} P_{η_{i}^{+}} - \sum_{j = 1}^{s_{4}} x_{η_{j}^{-}}^{k - 1} P_{η_{j}^{-}}) + \sum_{i = 1}^{s_{3}} {\tilde{λ}}_{η_{i}^{+}} P_{η_{i}^{+}} + \sum_{j = 1}^{s_{4}} {\tilde{λ}}_{η_{j}^{-}} P_{η_{j}^{-}} ‖ = \\ = ‖ b_{p_{1}}^{k} ‖ \geq ‖ b_{p_{2}}^{k} ‖, \end{array}

[1] *Corresponding author: Jair Silva - E-mail: jairsilva@ufpr.br

0	<	(m + n)	<	10000	→	p = 4;
10000	<	(m + n)	<	20000	→	p = 8;
20000	<	(m + n)	<	400000	→	p = 20;
400000	<	(m + n)	<	600000	→	p = 40;
600000	<	(m + n)	<		→	p = 80

Brasil

Brasil

A Comparison Among Simple Algorithms for Linear Programming

ABSTRACT