A primal-dual interior-point method capable of rapidly detecting infeasibility for nonlinear programs

Yu-Hong Dai; Xin-Wei Liu; Jie Sun

doi:10.3934/jimo.2018190

Article Contents

2020, Volume 16, Issue 2: 1009-1035. Doi: 10.3934/jimo.2018190

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A primal-dual interior-point method capable of rapidly detecting infeasibility for nonlinear programs

1.
LSEC, ICMSEC, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China
2.
School of Mathematical Sciences, University of Chinese Academy of Sciences
3.
Institute of Mathematics, Hebei University of Technology, Tianjin 300401, China
4.
Institute of Mathematics, Hebei University of Technology, Tianjin China
5.
School of Science, Curtin University, Perth, Australia
6.
School of Business, National University of Singapore, Singapore

^*Corresponding author
^*Corresponding author

Received: June 30, 2018

Revised: July 31, 2018

Early access: December 2018

Published: February 2020

The first draft of this paper was completed on December 2, 2014. The first author is supported by the Chinese NSF grants (nos. 11631013, 11331012 and 81173633) and the National Key Basic Research Program of China (no. 2015CB856000). The second author is supported by the Chinese NSF grants (nos. 11671116 and 11271107) and the Major Research Plan of the NSFC (no. 91630202). The third author is supported by Grant DP-160101819 of Australia Research Council

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Abstract

With the help of a logarithmic barrier augmented Lagrangian function, we can obtain closed-form solutions of slack variables of logarithmic-barrier problems of nonlinear programs. As a result, a two-parameter primal-dual nonlinear system is proposed, which corresponds to the Karush-Kuhn-Tucker point and the infeasible stationary point of nonlinear programs, respectively, as one of two parameters vanishes. Based on this distinctive system, we present a primal-dual interior-point method capable of rapidly detecting infeasibility of nonlinear programs. The method generates interior-point iterates without truncation of the step. It is proved that our method converges to a Karush-Kuhn-Tucker point of the original problem as the barrier parameter tends to zero. Otherwise, the scaling parameter tends to zero, and the method converges to either an infeasible stationary point or a singular stationary point of the original problem. Moreover, our method has the capability to rapidly detect the infeasibility of the problem. Under suitable conditions, the method can be superlinearly or quadratically convergent to the Karush-Kuhn-Tucker point if the original problem is feasible, and it can be superlinearly or quadratically convergent to the infeasible stationary point when the problem is infeasible. Preliminary numerical results show that the method is efficient in solving some simple but hard problems, where the superlinear convergence to an infeasible stationary point is demonstrated when we solve two infeasible problems in the literature.

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Mathematics Subject Classification: Primary: 90C30, 90C51; Secondary: 90C26.

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Table 1. Output for test problem (TP1)

$ l $	$ f_l $	$ v_l $	$ \\|\phi_l\\|_{\infty} $	$ \\|\psi_l\\|_{\infty} $	$ \beta_l $	$ \rho_l $	$ k $

0	5	16.6132	129.6234	129.6234	0.1000	3.3226	-
1	0.1606	2.0205	4.8082	0.7313	0.1000	0.0972	3
2	-0.0149	2.0002	0.0989	0.0445	0.1000	0.0020	4
3	-0.0036	2.0000	0.0029	0.0018	0.1000	3.1595e-06	3
4	-0.0029	2.0000	3.1674e-06	2.8185e-06	0.1000	1.0000e-09	1
5	0.0018	2.0000	1.0011e-09	6.7212e-10	-	-	-

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Table 2. Output for test problem (TP2)

$ l $	$ f_l $	$ v_l $	$ \\|\phi_l\\|_{\infty} $	$ \\|\psi_l\\|_{\infty} $	$ \beta_l $	$ \rho_l $	$ k $

0	-20	126.6501	2.8052e+03	2.8052e+03	0.1000	6.3325	-
1	-172.5829	172.7978	1.0948e+03	6.2866	0.1000	0.8719	6
2	0.2155	0.7149	1.4269	0.7894	0.1000	0.3895	1
3	-0.1364	0.5550	0.3865	0.3865	0.0100	0.3895	3
4	-0.1416	0.5223	0.2864	0.2648	0.0100	0.1512	1
5	-0.1472	0.5140	0.1446	0.1446	0.0100	0.0209	4
6	-0.1997	0.4472	0.0084	0.0016	0.0100	2.6880e-06	3
7	-0.1999	0.4472	2.4923e-06	2.4923e-06	0.0100	1.0000e-09	1
8	-0.1999	0.4472	9.2732e-10	9.2732e-10	-	-	-

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Table 3. Output for test problem (TP4)

$ l $	$ f_l $	$ v_l $	$ \\|\phi_l\\|_{\infty} $	$ \\|\psi_l\\|_{\infty} $	$ \beta_l $	$ \rho_l $	$ k $

0	20	2.8284	9.9557	9.9557	0.1000	1	-
1	0.2305	0.4167	0.8900	0.7008	0.0100	1	4
2	0.1652	0.1687	0.1631	0.0771	0.0100	0.3268	4
3	0.1690	0.1630	0.0503	0.0022	0.0100	4.7328e-06	1
4	0.8561	2.9531e-04	3.1379e-06	3.1379e-06	0.0100	1.0000e-09	14
5	0.9028	1.2372e-04	9.3463e-08	9.3463e-08	-	-	-

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Table 4. The last $ 4 $ inner iterations corresponding to $ l = 4 $ for test problem (TP4)

$ k $	$ f_k $	$ v_k $	$ \\|\phi_k\\|_{\infty} $	$ \\|\psi_k\\|_{\infty} $	$ x_{k1} $	$ x_{k2} $

11	0.8500	5.7136e-04	5.6703e-04	5.6703e-04	1.0780	0.0001
12	0.8548	3.0434e-04	1.2222e-05	1.2222e-05	1.0754	-0.0002
13	0.8556	2.9845e-04	6.2125e-06	6.2125e-06	1.0750	-0.0002
14	0.8561	2.9531e-04	3.1379e-06	3.1379e-06	1.0747	-0.0002

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