A modified scaled memoryless BFGS preconditioned conjugate gradient algorithm for nonsmooth convex optimization

Yigui Ou; Xin Zhou

doi:10.3934/jimo.2017075

Article Contents

2018, Volume 14, Issue 2: 785-801. Doi: 10.3934/jimo.2017075

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A modified scaled memoryless BFGS preconditioned conjugate gradient algorithm for nonsmooth convex optimization

Yigui Ou^, and
Xin Zhou

Department of Applied Mathematics, Hainan University, Haikou 570228, China

* Corresponding author: Yigui Ou
* Corresponding author: Yigui Ou

Received: December 31, 2016

Revised: February 28, 2017

Early access: August 2017

Published: March 2018

The work is supported by NNSF of China (No. 11261015) and NSF of Hainan Province (No. 2016CXTD004; No. 111001; No. 117011)

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Abstract

This paper presents a nonmonotone scaled memoryless BFGS preconditioned conjugate gradient algorithm for solving nonsmooth convex optimization problems, which combines the idea of scaled memoryless BFGS preconditioned conjugate gradient method with the nonmonotone technique and the Moreau-Yosida regularization. The proposed method makes use of approximate function and gradient values of the Moreau-Yosida regularization instead of the corresponding exact values. Under mild conditions, the global convergence of the proposed method is established. Preliminary numerical results and related comparisons show that the proposed method can be applied to solve large scale nonsmooth convex optimization problems.

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Mathematics Subject Classification: Primary: 90C25, 90C30; Secondary: 65K05.

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Figure 1. Performance profiles based on iterations (left) and function evaluations (right)

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Figure 2. Performance profiles based on iterations (left) and function evaluations (right)

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Table 1. Testing functions for small scale problems

No.	Functions	$n$	$x_0$	${{f}_{ops}}\left( x \right)$
1	Rosenbrock	2	(-1.2; 1)	0
2	Crescent	2	(-1.5; 2)	0
3	CB2	2	(1; -0.1)	1.9522245
4	CB3	2	(2; 2)	2.0
5	DEM	2	(1; 1)	-3
6	QL	2	(-1; 5)	7.2
7	LQ	2	(-0.5; -0.5)	-1.4142136
8	Mifflin 1	2	(0.8; 0.6)	-1.0
9	Mifflin 2	2	(-1; -1)	-1.0
10	Wolfe	2	(3; 2)	-8
11	Rosen-Suzuki	4	(0; 0; 0; 0)	-44
12	Shor	5	(0; 0; 0; 0; 1)	22.600162

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Table 2. Numerical results for five algorithms

No.	Algorithn 3.1	LWTR	YWBB	SFTR	RFBFGS
1	3/4/	6/13/	54/56/	48/49/	4/5/
	2.6178e-9	0.2976e-7	3.4484e-7	7.1545e-4	6.2072e-10
2	3/4/	3/7/	14/16/	31/32/	35/36/
	3.3026e-6	6.5430e-4	2.7450e-5	1.6000e-3	3.0590e-7
3	4/5/	5/12/	13/15/	54/55/	5/6/
	1.9522	1.9522	1.9522	1.9573	1.9522
4	2/3/	6/13/	4/8/	55/56	2/3/
	2.0000	2.0000	2.0000	2.0100	2.0076
5	3/4/	8/16/	4/7/	5/6/	3/4
	-3.0000	-3.0000	-3.0000	-3.0000	-3.0000
6	11/12/	4/9/	22/25/	48/49/	10/11/
	7.2000	7.2000	7.2000	7.2003	7.2000
7	3/4/	5/10/	6/7/	3/4/	2/3/
	-1.4142	-1.4142	-1.4142	-1.4118	-1.4033
8	34/35/	15/31/	3/6/	59/60/	57/58/
	-1.0000	-1.0000	-0.9938	-1.0000	-1.0000
9	2/3/	8/16/	12/13/	4/5/	2/3/
	-1.0000	-1.0000	-0.9999	-0.9997	-0.9813
10	3/4/	12/24/	9/12/	43/46/	4/5/
	-8.0000	-8.0000	-8.0000	-8.0000	-8.0000
11	49/106/	20/40/	8/9/	60/61/	25/31/
	-43.9999	-44.0000	-43.9493	-39.9924	-43.9982
12	14/16/	14/28/	9/10/	71/72/	66/152/
	22.6019	22.6002	22.6004	22.6892	22.6017

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Table 3. Testing functions for large scale problems

No.	Functions	Initial points $x_0$
1	Generalization of MAXQ	$(1, 2, \cdots, \frac{n}{2}, -\frac{n}{2}-1, \cdots, -n)$
2	Generalization of MXHILB	$(1, 1, \cdots, 1)$
3	Chained LQ	$(-0.5, -0.5, \cdots, -0.5)$
4	Number of active faces	$(1, 1, \cdots, 1)$
5	Nonsmooth generalization of Brown 2	$(1, 0, 1, 0, \cdots)$
6	Chained Mifflin 2	$(-1, -1, \cdots, -1)$
7	Chained Crescent Ⅰ	$(-1.5, 2, -1.5, 2, \cdots)$
8	Chained Crescent Ⅱ	$(1, 0, 1, 0 \cdots)$

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Table 4. results for Algorithms 3.1 and CG-YWL

No.	n	Algorithn 3.1	CG-YWL
1	1000	186/1601/2.6568e-10	225/4710/6.9354e-8
	5000	242/2725/1.2183e-10	250/5235/6.8798e-8
	10000	253/2997/3.4045e-10	261/5466/6.6528e-8
2	1000	94/1301/4.0582e-9	91/1482/8.2738e-9
	5000	119/1499/2.6731e-9	111/1938/9.7206e-9
	10000	129/1901/2.1905e-9	120/2127/5.8524e-9
3	1000	37/110/2.3278e-9	37/114/7.2687e-9
	5000	39/116/5.9987e-9	39/120/9.0932e-9
	10000	40/121/1.6943e-9	40/123/9.0941e-9
4	1000	71/891/3.6735e-11	77/1026/6.8037e-9
	5000	82/937/7.9864e-11	90/1281/7.8405e-9
	10000	86/1208/5.7163e-11	96/1401/9.9366e-9
5	1000	35/114/1.5021e-11	38/117/7.2687e-9
	5000	39/120/5.2352e-11	40/123/9.0932e-9
	10000	41/126/2.1136e-11	41/125/1.8188e-8
6	1000	38/116/-5.3979e+3	37/114/-2.4975e+4
	5000	41/123/-3.2153e+4	39/120/-1.2498e+5
	10000	43/129/-2.0127e+4	40/123/-2.4998e+5
7	1000	34/101/7.0463e-11	37/114/5.4897e-9
	5000	36/113/3.8145e-11	39/120/6.8294e-9
	10000	39/120/4.3654e-11	40/123/6.8253e-9
8	1000	37/112/6.0424e-11	39/120/6.8185e-9
	5000	39/121/1.8205e-11	41/126/8.5258e-9
	10000	42/125/2.6473e-11	42/129/8.5262e-9

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