A merit function method for infinite-dimensional SOCCPs

Yungyen Chiang; Shaohua Pan; Jein Shan Chen

doi:10.1016/j.jmaa.2011.05.019

A merit function method for infinite-dimensional SOCCPs

Yungyen Chiang^*, Shaohua Pan, Jein Shan Chen

^*Corresponding author for this work

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

6 Citations (Scopus)

Abstract

We introduce the Jordan product associated with the second-order cone K into the real Hilbert space H, and then define a one-parametric class of complementarity functions φt on H×H with the parameter t∈[0,2). We show that the squared norm of φt with tφ(0,2) is a continuously F(réchet)-differentiable merit function. By this, the second-order cone complementarity problem (SOCCP) in H can be converted into an unconstrained smooth minimization problem involving this class of merit functions, and furthermore, under the monotonicity assumption, every stationary point of this minimization problem is shown to be a solution of the SOCCP.

Original language	English
Pages (from-to)	159-178
Number of pages	20
Journal	Journal of Mathematical Analysis and Applications
Volume	383
Issue number	1
DOIs	https://doi.org/10.1016/j.jmaa.2011.05.019
Publication status	Published - 2011 Nov 1

Keywords

Complementarity
Hilbert space
Merit functions
Second-order cone

ASJC Scopus subject areas

Analysis
Applied Mathematics

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10.1016/j.jmaa.2011.05.019

Cite this

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title = "A merit function method for infinite-dimensional SOCCPs",

abstract = "We introduce the Jordan product associated with the second-order cone K into the real Hilbert space H, and then define a one-parametric class of complementarity functions φt on H×H with the parameter t∈[0,2). We show that the squared norm of φt with tφ(0,2) is a continuously F(r{\'e}chet)-differentiable merit function. By this, the second-order cone complementarity problem (SOCCP) in H can be converted into an unconstrained smooth minimization problem involving this class of merit functions, and furthermore, under the monotonicity assumption, every stationary point of this minimization problem is shown to be a solution of the SOCCP.",

keywords = "Complementarity, Hilbert space, Merit functions, Second-order cone",

author = "Yungyen Chiang and Shaohua Pan and Chen, {Jein Shan}",

note = "Funding Information: E-mail addresses: chiangyy@math.nsysu.edu.tw (Y. Chiang), shhpan@scut.edu.cn (S. Pan), jschen@math.ntnu.edu.tw (J.-S. Chen). 1 The author{\textquoteright}s work is partially supported by grants from the National Science Council of the Republic of China. 2 The author{\textquoteright}s work is supported by Guangdong Natural Science Foundation (No. 9251802902000001) and the Fundamental Research Funds for the Central Universities (SCUT). 3 Member of Mathematics Division, National Center for Theoretical Sciences, Taipei Office. The author{\textquoteright}s work is partially supported by National Science Council of Taiwan.",

year = "2011",

month = nov,

day = "1",

doi = "10.1016/j.jmaa.2011.05.019",

language = "English",

volume = "383",

pages = "159--178",

journal = "Journal of Mathematical Analysis and Applications",

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T1 - A merit function method for infinite-dimensional SOCCPs

AU - Chiang, Yungyen

AU - Pan, Shaohua

AU - Chen, Jein Shan

N1 - Funding Information: E-mail addresses: chiangyy@math.nsysu.edu.tw (Y. Chiang), shhpan@scut.edu.cn (S. Pan), jschen@math.ntnu.edu.tw (J.-S. Chen). 1 The author’s work is partially supported by grants from the National Science Council of the Republic of China. 2 The author’s work is supported by Guangdong Natural Science Foundation (No. 9251802902000001) and the Fundamental Research Funds for the Central Universities (SCUT). 3 Member of Mathematics Division, National Center for Theoretical Sciences, Taipei Office. The author’s work is partially supported by National Science Council of Taiwan.

PY - 2011/11/1

Y1 - 2011/11/1

N2 - We introduce the Jordan product associated with the second-order cone K into the real Hilbert space H, and then define a one-parametric class of complementarity functions φt on H×H with the parameter t∈[0,2). We show that the squared norm of φt with tφ(0,2) is a continuously F(réchet)-differentiable merit function. By this, the second-order cone complementarity problem (SOCCP) in H can be converted into an unconstrained smooth minimization problem involving this class of merit functions, and furthermore, under the monotonicity assumption, every stationary point of this minimization problem is shown to be a solution of the SOCCP.

AB - We introduce the Jordan product associated with the second-order cone K into the real Hilbert space H, and then define a one-parametric class of complementarity functions φt on H×H with the parameter t∈[0,2). We show that the squared norm of φt with tφ(0,2) is a continuously F(réchet)-differentiable merit function. By this, the second-order cone complementarity problem (SOCCP) in H can be converted into an unconstrained smooth minimization problem involving this class of merit functions, and furthermore, under the monotonicity assumption, every stationary point of this minimization problem is shown to be a solution of the SOCCP.

KW - Complementarity

KW - Hilbert space

KW - Merit functions

KW - Second-order cone

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DO - 10.1016/j.jmaa.2011.05.019

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SN - 0022-247X

VL - 383

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EP - 178

JO - Journal of Mathematical Analysis and Applications

JF - Journal of Mathematical Analysis and Applications

IS - 1

ER -

A merit function method for infinite-dimensional SOCCPs

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Keywords

ASJC Scopus subject areas

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