A survey of gradient methods for solving nonlinear optimization

Predrag S. Stanimirović; Branislav Ivanov; Haifeng Ma; Dijana Mosić

doi:10.3934/era.2020115

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December 2020, 28(4): 1573-1624. doi: 10.3934/era.2020115

A survey of gradient methods for solving nonlinear optimization

Predrag S. Stanimirović ^1,, , Branislav Ivanov ^2, , Haifeng Ma ^3, and Dijana Mosić ^1,

1.	University of Niš, Faculty of Sciences and Mathematics, Višegradska 33, 18000 Niš, Serbia
2.	Technical Faculty in Bor, University of Belgrade, Vojske Jugoslavije 12, 19210 Bor, Serbia
3.	School of Mathematical Science, Harbin Normal University, Harbin 150025, China

^* Corresponding author: Predrag S. Stanimirović

Received October 2020 Revised November 2020 Published November 2020

The paper surveys, classifies and investigates theoretically and numerically main classes of line search methods for unconstrained optimization. Quasi-Newton (QN) and conjugate gradient (CG) methods are considered as representative classes of effective numerical methods for solving large-scale unconstrained optimization problems. In this paper, we investigate, classify and compare main QN and CG methods to present a global overview of scientific advances in this field. Some of the most recent trends in this field are presented. A number of numerical experiments is performed with the aim to give an experimental and natural answer regarding the numerical one another comparison of different QN and CG methods.

Keywords: Gradient methods, conjugate gradient methods, unconstrained optimization, nonlinear programming, sufficient descent condition, global convergence, line search.

Mathematics Subject Classification: Primary: 65K05, 90C30; Secondary: 90C06, 90C26.

Citation: Predrag S. Stanimirović, Branislav Ivanov, Haifeng Ma, Dijana Mosić. A survey of gradient methods for solving nonlinear optimization. Electronic Research Archive, 2020, 28 (4) : 1573-1624. doi: 10.3934/era.2020115

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Figure 1. IT performance profile for AGD, MSM and SM methods

Quasi-Newton Eqs.	$ \tilde{\mathbf{y}}_{k-1} $	Ref.
$ B_{k}{\mathbf s}_{k-1}\!=\!\tilde{\mathbf{y}}_{k-1} $	$ \tilde{\mathbf{y}}_{k-1} \!=\! \varphi_{k-1} \mathbf{y}_{k-1} + (1-\varphi_{k-1})B_{k-1}\mathbf{s}_{k-1} $	[104]
$ B_{k}{\mathbf s}_{k-1}=\tilde{\mathbf{y}}_{k-1} $	$ \tilde{\mathbf{y}}_{k-1} \!=\! \mathbf{y}_{k-1} + t_{k-1}\mathbf{s}_{k-1}, t_{k-1}\leq10^{-6} $	[72]
$ B_{k}{\mathbf s}_{k-1}=\tilde{\mathbf{y}}_{k-1} $	$ \tilde{\mathbf{y}}_{k-1} \!=\! \mathbf{y}_{k-1} + \frac{2(f_{k-1} - f_k)+(\mathbf{g}_k + \mathbf{g}_{k-1})^ \mathrm{T} \mathbf{s}_{k-1}}{\\|\mathbf{s}_{k-1}\\|^2}\mathbf{s}_{k-1} $	[123]
$ B_{k}{\mathbf s}_{k-1}=\tilde{\mathbf{y}}_{k-1} $	$ \tilde{\mathbf{y}}_{k-1} \!=\! \mathbf{y}_{k-1} + \frac{\max (0, 2(f_{k-1} - f_k)+(\mathbf{g}_k + \mathbf{g}_{k-1})^ \mathrm{T} \mathbf{s}_{k-1})}{\\|\mathbf{s}_{k-1}\\|^2}\mathbf{s}_{k-1} $	[133]
$ B_{k}{\mathbf s}_{k-1}=\tilde{\mathbf{y}}_{k-1} $	$ \tilde{\mathbf{y}}_{k-1} \!=\! \mathbf{y}_{k-1} + \frac{\max (0, 6(f_{k-1} - f_k)+3(\mathbf{g}_k + \mathbf{g}_{k-1})^ \mathrm{T} \mathbf{s}_{k-1})}{\\|\mathbf{s}_{k-1}\\|^2}\mathbf{s}_{k-1} $	[134]
$ B_{k}{\mathbf s}_{k-1}=\tilde{\mathbf{y}}_{k-1} $	$ \tilde{\mathbf{y}}_{k-1} \!=\! \frac{1}{2}\mathbf{y}_{k-1} + \frac{(f_{k-1} - f_k) - \frac{1}{2} \mathbf{g}_k^ \mathrm{T} \mathbf{s}_{k-1}}{\mathbf{s}_{k-1}^ \mathrm{T} \mathbf{y}_{k-1}}\mathbf{y}_{k-1} $	[59]

$ \beta_k $	Title	Year	Reference
$ \beta_k^{\mathrm{HS}}\!=\!\dfrac{{\mathbf y}_{k-1}^ \mathrm{T} {\mathbf g}_k}{{\mathbf y}_{k-1}^ \mathrm{T} {\mathbf d}_{k-1}} $	$ \rm Hestenses–Stiefel $	1952	[60]
$ \beta_k^{\mathrm{FR}}\!=\!\dfrac{\\|{\mathbf g}_k\\|^2}{\\|{\mathbf g}_{k-1}\\|^2} $	$ \rm Fletcher–Reeves $	1964	[48]
$ \beta_k^{\mathrm{D}}\!=\!\dfrac{{\mathbf g}_{k}^ \mathrm{T} {\mathbf G}_{k-1} {\mathbf d}_{k-1}}{{\mathbf d}_{k-1}^ \mathrm{T} {\mathbf G}_{k-1}{\mathbf d}_{k-1}} $		1967	[38]
$ \beta_k^{\mathrm{PRP}}\!=\!\dfrac{{\mathbf y}_{k-1}^ \mathrm{T} {\mathbf g}_k}{\\|{\mathbf g}_{k-1}\\|^2} $	$ \rm Polak–Ribiere–Polyak $	1969	[102,103]
$ \beta_k^{\mathrm{CD}}\!=\!-\dfrac{\\|{\mathbf g}_k\\|^2}{{\mathbf g}_{k-1}^ \mathrm{T} {\mathbf d}_{k-1}} $	$ \rm Conjugate Descent $	1987	[47]
$ \beta_k^{\mathrm{LS}}\!=\!-\dfrac{{\mathbf y}_{k-1}^ \mathrm{T} {\mathbf g}_k}{{\mathbf g}_{k-1}^ \mathrm{T} {\mathbf d}_{k-1}} $	$ \rm Liu–Storey $	1991	[79]
$ \beta_k^{\mathrm{DY}}\!=\!\dfrac{\\|{\mathbf g}_k\\|^2}{{\mathbf y}_{k-1}^ \mathrm{T} {\mathbf d}_{k-1}} $	$ \rm Dai–Yuan $	1999	[30]

	Denominator
Numerator	$\\|\mathbf{g}_{k-1}\\|^2$	$\mathbf{y}_{k-1}^{\rm T} \mathbf{d}_{k-1}$	$-\mathbf{g}_{k-1}^{\rm T} \mathbf{d}_{k-1}$
$\\|\mathbf{g}_{k}\\|^2$	FR	DY	CD
$\mathbf{y}_{k-1}^{\rm T} \mathbf{g}_{k}$	PRP	HS	LS

	IT profile			FE profile			CPU time
Test function	AGD	MSM	SM	AGD	MSM	SM	AGD	MSM	SM
Perturbed Quadratic	353897	34828	59908	13916515	200106	337910	6756.047	116.281	185.641
Raydan 1	22620	26046	14918	431804	311260	81412	158.359	31.906	36.078
Diagonal 3	120416	7030	12827	4264718	38158	69906	5527.844	52.609	102.875
Generalized Tridiagonal 1	670	346	325	9334	1191	1094	11.344	1.469	1.203
Extended Tridiagonal 1	3564	1370	4206	14292	10989	35621	55.891	29.047	90.281
Extended TET	443	156	156	3794	528	528	3.219	0.516	0.594
Diagonal 4	120	96	96	1332	636	636	0.781	0.203	0.141
Extended Himmelblau	396	260	196	6897	976	668	1.953	0.297	0.188
Perturbed quadratic diagonal	2542050	37454	44903	94921578	341299	460028	44978.750	139.625	185.266
Quadratic QF1	366183	36169	62927	13310016	208286	352975	12602.563	81.531	138.172
Extended quadratic penalty QP1	210	369	271	2613	2196	2326	1.266	1.000	0.797
Extended quadratic penalty QP2	395887	1674	3489	9852040	11491	25905	3558.734	3.516	6.547
Quadratic QF2	100286	32727	64076	3989239	183142	353935	1582.766	73.438	132.703
Extended quadratic exponential EP1	48	100	73	990	894	661	0.750	0.688	0.438
Extended Tridiagonal 2	1657	659	543	8166	2866	2728	3.719	1.047	1.031
ARWHEAD (CUTE)	5667	430	270	214284	5322	3919	95.641	1.969	1.359
Almost Perturbed Quadratic	356094	33652	60789	14003318	194876	338797	13337.125	73.047	133.516
LIARWHD (CUTE)	1054019	3029	18691	47476667	27974	180457	27221.516	9.250	82.016
ENGVAL1 (CUTE)	743	461	375	6882	2285	2702	3.906	1.047	1.188
QUARTC (CUTE)	171	217	290	402	494	640	2.469	1.844	2.313
Generalized Quartic	187	181	189	849	493	507	0.797	0.281	0.188
Diagonal 7	72	147	108	333	504	335	0.625	0.547	0.375
Diagonal 8	60	120	118	304	383	711	0.438	0.469	0.797
Full Hessian FH3	45	63	63	1352	566	631	1.438	0.391	0.391
Diagonal 9	329768	10540	13619	13144711	68189	89287	6353.172	43.609	38.672

Test function	HCG1	HCG2	HCG3	HCG4	HCG5	HCG6	HCG7	HCG8	HCG9	HCG10
Perturbed Quadratic	1157	1157	1157	1157	1157	1157	1157	1157	1157	1157
Raydan 2	40	40	40	57	78	81	40	69	NaN	126
Diagonal 2	1584	1581	1542	1488	1500	2110	2193	1843	1475	1453
Extended Tridiagonal 1	805	623	754	2110	2160	10129	1167	966	NaN	270
Diagonal 4	60	60	70	60	70	70	60	70	NaN	113
Diagonal 5	124	39	98	39	120	109	39	141	154	130
Extended Himmelblau	145	139	111	161	181	207	159	381	109	108
Full Hessian FH2	5036	5036	5036	4820	4820	4800	4994	4789	5163	5705
Perturbed quadratic diagonal	1228	1214	1266	934	1093	987	996	1016	NaN	2679
Quadratic QF1	1158	1158	1158	1158	1158	1158	1158	1158	NaN	1158
Quadratic QF2	2125	2098	2174	1995	1991	2425	2378	NaN	2204	2034
TRIDIA (CUTE)	NaN	NaN	NaN	6210	6210	5594	NaN	NaN	6748	7345
Almost Perturbed Quadratic	1158	1158	1158	1158	1158	1158	1158	1158	1158	1158
LIARWHD (CUTE)	1367	817	1592	1024	1831	1774	531	2152	NaN	573
POWER (CUTE)	7782	7782	7782	7779	7779	7802	7781	7780	NaN	7781
NONSCOMP (CUTE)	10092	10746	8896	10466	9972	13390	11029	3520	3988	11411
QUARTC (CUTE)	94	160	145	150	126	95	160	114	165	154
Diagonal 6	40	40	40	57	78	81	40	69	NaN	126
DIXON3DQ (CUTE)	12182	5160	11257	5160	11977	14302	5160	17080	NaN	12264
BIGGSB1 (CUTE)	10664	5160	10479	5160	11082	13600	5160	16192	NaN	11151
Generalized Quartic	129	107	110	107	142	153	107	123	131	145
Diagonal 7	50	NaN	40	NaN	40	50	NaN	50	51	40
Diagonal 8	40	40	40	50	NaN	50	40	NaN	NaN	40
Full Hessian FH3	43	42	42	42	42	43	42	43	NaN	NaN
FLETCHCR (CUTE)	17821	17632	18568	17272	17446	26794	24865	NaN	17315	20813

Label	T1	T2	T3	T4	T5	T6
Value of the scalar $ t $	$ t_k=\alpha_k $	0.05	0.1	0.2	0.5	0.9

Method	T1	T2	T3	T4	T5	T6
DHSDL	32980.14	31281.32	33640.45	32942.36	34448.32	33872.36
DLSDL	30694.00	28701.14	31048.32	30594.77	31926.59	31573.05
MHSDL	29289.73	27653.64	29660.00	29713.50	30491.18	30197.27
MLSDL	25398.82	22941.77	24758.27	24250.68	25722.64	25032.64

Method	T1	T2	T3	T4	T5	T6
DHSDL	1228585.50	1191960.55	1252957.09	1238044.36	1271176.59	1255710.45
DLSDL	1131421.41	1083535.14	1149482.41	1134315.00	1167030.14	1158554.77
MHSDL	1089700.41	1036710.32	1089777.64	1091985.41	1105299.91	1101380.18
MLSDL	904217.14	845017.55	891669.50	879473.14	913165.68	895652.36

Method	T1	T2	T3	T4	T5	T6
DHSDL	902.06	894.73	917.77	930.56	911.28	870.93
DLSDL	816.08	790.63	804.69	816.28	803.84	809.67
MHSDL	770.78	751.65	728.61	749.70	712.64	720.57
MLSDL	573.14	587.41	581.50	576.32	582.62	580.96

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	IT profile			FE profile			CPU time
Test function	DY	FR	CD	DY	FR	CD	DY	FR	CD
Perturbed Quadratic	1157	1157	1157	3481	3481	3481	0.469	0.609	0.531
Raydan 2	86	40	40	192	100	100	0.063	0.016	0.016
Diagonal 2	1636	3440	2058	4774	7982	8063	0.922	1.563	1.297
Extended Tridiagonal 1	2081	690	1140	4639	2022	2984	1.703	1.141	1.578
Diagonal 4	70	70	70	200	200	200	0.047	0.031	0.016
Diagonal 5	40	124	155	100	258	320	0.109	0.141	0.125
Extended Himmelblau	383	339	207	1669	1467	961	0.219	0.172	0.172
Full Hessian FH2	4682	4868	4794	14054	14610	14390	65.938	66.469	65.922
Perturbed quadratic diagonal	1036	1084	1276	3114	3258	3834	0.406	0.422	0.422
Quadratic QF1	1158	1158	1158	3484	3484	3484	0.297	0.297	0.328
Quadratic QF2	NaN	NaN	2349	NaN	NaN	10073	NaN	NaN	1.531
Extended quadratic exponential EP1	NaN	60	60	NaN	310	310	NaN	0.109	0.125
Almost Perturbed Quadratic	1158	1158	1158	3484	3484	3484	0.422	0.453	0.391
LIARWHD (CUTE)	2812	1202	1255	12366	7834	7379	0.938	1.000	1.109
POWER (CUTE)	7779	7781	7782	23347	23353	23356	1.078	1.500	1.328
NONSCOMP (CUTE)	2558	13483	10901	49960	43268	33413	1.203	1.406	1.422
QUARTC (CUTE)	134	94	95	1132	901	916	0.688	0.672	0.563
Diagonal 6	86	40	40	192	100	100	0.047	0.063	0.063
DIXON3DQ (CUTE)	16047	18776	19376	48172	56369	58176	2.266	2.516	2.734
BIGGSB1 (CUTE)	15274	17835	18374	45853	53546	55170	2.875	2.922	2.484
Generalized Quartic	142	214	173	497	712	589	0.078	0.172	0.109
Diagonal 7	50	50	50	160	160	160	0.063	0.047	0.094
Diagonal 8	50	40	40	160	130	130	0.109	0.125	0.063
Full Hessian FH3	43	43	43	139	139	139	0.063	0.109	0.109
FLETCHCR (CUTE)	NaN	NaN	26793	NaN	NaN	240237	NaN	NaN	10.203

	IT profile			FE profile			CPU time
Test function	DY	FR	CD	DY	FR	CD	DY	FR	CD
Perturbed Quadratic	1157	1157	1157	3481	3481	3481	0.469	0.609	0.531
Raydan 2	86	40	40	192	100	100	0.063	0.016	0.016
Diagonal 2	1636	3440	2058	4774	7982	8063	0.922	1.563	1.297
Extended Tridiagonal 1	2081	690	1140	4639	2022	2984	1.703	1.141	1.578
Diagonal 4	70	70	70	200	200	200	0.047	0.031	0.016
Diagonal 5	40	124	155	100	258	320	0.109	0.141	0.125
Extended Himmelblau	383	339	207	1669	1467	961	0.219	0.172	0.172
Full Hessian FH2	4682	4868	4794	14054	14610	14390	65.938	66.469	65.922
Perturbed quadratic diagonal	1036	1084	1276	3114	3258	3834	0.406	0.422	0.422
Quadratic QF1	1158	1158	1158	3484	3484	3484	0.297	0.297	0.328
Quadratic QF2	NaN	NaN	2349	NaN	NaN	10073	NaN	NaN	1.531
Extended quadratic exponential EP1	NaN	60	60	NaN	310	310	NaN	0.109	0.125
Almost Perturbed Quadratic	1158	1158	1158	3484	3484	3484	0.422	0.453	0.391
LIARWHD (CUTE)	2812	1202	1255	12366	7834	7379	0.938	1.000	1.109
POWER (CUTE)	7779	7781	7782	23347	23353	23356	1.078	1.500	1.328
NONSCOMP (CUTE)	2558	13483	10901	49960	43268	33413	1.203	1.406	1.422
QUARTC (CUTE)	134	94	95	1132	901	916	0.688	0.672	0.563
Diagonal 6	86	40	40	192	100	100	0.047	0.063	0.063
DIXON3DQ (CUTE)	16047	18776	19376	48172	56369	58176	2.266	2.516	2.734
BIGGSB1 (CUTE)	15274	17835	18374	45853	53546	55170	2.875	2.922	2.484
Generalized Quartic	142	214	173	497	712	589	0.078	0.172	0.109
Diagonal 7	50	50	50	160	160	160	0.063	0.047	0.094
Diagonal 8	50	40	40	160	130	130	0.109	0.125	0.063
Full Hessian FH3	43	43	43	139	139	139	0.063	0.109	0.109
FLETCHCR (CUTE)	NaN	NaN	26793	NaN	NaN	240237	NaN	NaN	10.203

Test function	HCG1	HCG2	HCG3	HCG4	HCG5	HCG6	HCG7	HCG8	HCG9	HCG10
Perturbed Quadratic	3481	3481	3481	3481	3481	3481	3481	3481	3481	3481
Raydan 2	100	100	100	134	176	182	100	158	NaN	282
Diagonal 2	6136	6217	6006	5923	5944	8281	8594	4822	5711	5636
Extended Tridiagonal 1	2369	1991	2275	4678	4924	22418	3119	2661	NaN	869
Diagonal 4	170	170	200	170	200	200	170	200	NaN	339
Diagonal 5	258	88	206	88	270	228	88	292	338	270
Extended Himmelblau	855	687	583	763	813	961	757	1613	567	594
Full Hessian FH2	15115	15115	15115	14467	14467	14407	14989	14374	15495	17122
Perturbed quadratic diagonal	3686	3647	3805	2805	3282	2967	2993	3053	NaN	8044
Quadratic QF1	3484	3484	3484	3484	3484	3484	3484	3484	NaN	3484
Quadratic QF2	9455	9202	9501	9016	9054	10229	10086	NaN	9531	9085
TRIDIA (CUTE)	NaN	NaN	NaN	18640	18640	16792	NaN	NaN	20260	22051
Almost Perturbed Quadratic	3484	3484	3484	3484	3484	3484	3484	3484	3484	3484
LIARWHD (CUTE)	7712	5931	8275	6165	8113	9395	5854	10305	NaN	4848
POWER (CUTE)	23356	23356	23356	23347	23347	23416	23353	23350	NaN	23353
NONSCOMP (CUTE)	31355	33211	27801	32705	31458	40807	34013	23411	13367	35106
QUARTC (CUTE)	901	1254	1261	1224	1224	916	1254	1041	1347	1305
Diagonal 6	100	100	100	134	176	182	100	158	NaN	282
DIXON3DQ (CUTE)	36508	15534	33759	15534	35926	42952	15534	51284	NaN	36796
BIGGSB1 (CUTE)	31960	15534	31427	15534	33247	40846	15534	48620	NaN	33469
Generalized Quartic	457	371	370	371	481	529	371	439	446	467
Diagonal 7	160	NaN	130	NaN	130	160	NaN	160	142	13
Diagonal 8	130	130	130	160	NaN	160	130	NaN	NaN	130
Full Hessian FH3	139	136	136	136	136	139	136	139	NaN	NaN
FLETCHCR (CUTE)	166463	165774	168739	175309	175845	240240	184939	NaN	174406	215687

Test function	HCG1	HCG2	HCG3	HCG4	HCG5	HCG6	HCG7	HCG8	HCG9	HCG10
Perturbed Quadratic	3481	3481	3481	3481	3481	3481	3481	3481	3481	3481
Raydan 2	100	100	100	134	176	182	100	158	NaN	282
Diagonal 2	6136	6217	6006	5923	5944	8281	8594	4822	5711	5636
Extended Tridiagonal 1	2369	1991	2275	4678	4924	22418	3119	2661	NaN	869
Diagonal 4	170	170	200	170	200	200	170	200	NaN	339
Diagonal 5	258	88	206	88	270	228	88	292	338	270
Extended Himmelblau	855	687	583	763	813	961	757	1613	567	594
Full Hessian FH2	15115	15115	15115	14467	14467	14407	14989	14374	15495	17122
Perturbed quadratic diagonal	3686	3647	3805	2805	3282	2967	2993	3053	NaN	8044
Quadratic QF1	3484	3484	3484	3484	3484	3484	3484	3484	NaN	3484
Quadratic QF2	9455	9202	9501	9016	9054	10229	10086	NaN	9531	9085
TRIDIA (CUTE)	NaN	NaN	NaN	18640	18640	16792	NaN	NaN	20260	22051
Almost Perturbed Quadratic	3484	3484	3484	3484	3484	3484	3484	3484	3484	3484
LIARWHD (CUTE)	7712	5931	8275	6165	8113	9395	5854	10305	NaN	4848
POWER (CUTE)	23356	23356	23356	23347	23347	23416	23353	23350	NaN	23353
NONSCOMP (CUTE)	31355	33211	27801	32705	31458	40807	34013	23411	13367	35106
QUARTC (CUTE)	901	1254	1261	1224	1224	916	1254	1041	1347	1305
Diagonal 6	100	100	100	134	176	182	100	158	NaN	282
DIXON3DQ (CUTE)	36508	15534	33759	15534	35926	42952	15534	51284	NaN	36796
BIGGSB1 (CUTE)	31960	15534	31427	15534	33247	40846	15534	48620	NaN	33469
Generalized Quartic	457	371	370	371	481	529	371	439	446	467
Diagonal 7	160	NaN	130	NaN	130	160	NaN	160	142	13
Diagonal 8	130	130	130	160	NaN	160	130	NaN	NaN	130
Full Hessian FH3	139	136	136	136	136	139	136	139	NaN	NaN
FLETCHCR (CUTE)	166463	165774	168739	175309	175845	240240	184939	NaN	174406	215687

American Institute of Mathematical Sciences

A survey of gradient methods for solving nonlinear optimization

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