A feasible primal-dual interior point method for linear semidefinite programming
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- 作者：Imene Touila ; i_touil@yahoo.fr" class="auth_mail" title="E-mail the corresponding author ; Djamel Benterkib ; Adnan Yassinec ; d ; e
- 关键词：90C22 ; 90C51
- 刊名：Journal of Computational and Applied Mathematics
- 出版年：2017
- 出版时间：1 March 2017
- 年：2017
- 卷：312
- 期：Complete
- 页码：216-230
- 全文大小：465 K
文摘
In this paper, we consider a feasible primal–dual interior point method for linear semidefinite programming problem mmlsi1" class="mathmlsrc">mulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0377042716302254&_mathId=si1.gif&_user=111111111&_pii=S0377042716302254&_rdoc=1&_issn=03770427&md5=a9d860af925bdb94da8bbd95618e36a9" title="Click to view the MathML source">(SDP)mathContainer hidden">mathCode"><math altimg="si1.gif" overflow="scroll"><mrow><mo>(mo><mi>Smi><mi>Dmi><mi>Pmi><mo>)mo>mrow>math> based on Alizadeh–Haeberly–Overton (AHO) direction (Monteiro, 1997). Firstly, and by a new and simple technique, we establish the existence and uniqueness of optimal solution of the perturbed problem mmlsi2" class="mathmlsrc">mulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0377042716302254&_mathId=si2.gif&_user=111111111&_pii=S0377042716302254&_rdoc=1&_issn=03770427&md5=63a215222196463013d69bc08131b583" title="Click to view the MathML source">(SDP)μmathContainer hidden">mathCode"><math altimg="si2.gif" overflow="scroll"><msub><mrow><mrow><mo>(mo><mi>Smi><mi>Dmi><mi>Pmi><mo>)mo>mrow>mrow><mrow><mi>μmi>mrow>msub>math> and its convergence to optimal solution of mmlsi1" class="mathmlsrc">mulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0377042716302254&_mathId=si1.gif&_user=111111111&_pii=S0377042716302254&_rdoc=1&_issn=03770427&md5=a9d860af925bdb94da8bbd95618e36a9" title="Click to view the MathML source">(SDP)mathContainer hidden">mathCode"><math altimg="si1.gif" overflow="scroll"><mrow><mo>(mo><mi>Smi><mi>Dmi><mi>Pmi><mo>)mo>mrow>math>. Next, we present new different alternatives to calculate the displacement step. After, we establish the convergence of the obtained algorithm and we show that its complexity is mmlsi4" class="mathmlsrc">mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0377042716302254&_mathId=si4.gif&_user=111111111&_pii=S0377042716302254&_rdoc=1&_issn=03770427&md5=5eb89b208226c49f10c23fc9151a54eb">mg class="imgLazyJSB inlineImage" height="21" width="171" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0377042716302254-si4.gif"> mathContainer hidden">mathCode"><math altimg="si4.gif" overflow="scroll"><mi mathvariant="script">Omi><mrow><mo>(mo><msqrt><mrow><mi>nmi>mrow>msqrt><mo>lnmo><mrow><mo>[mo><msup><mrow><mi>εmi>mrow><mrow><mo>−mo><mn>1mn>mrow>msup><mrow><mo>(mo><mrow><mo>〈mo><msup><mrow><mi>Xmi>mrow><mrow><mn>0mn>mrow>msup><mo>,mo><msup><mrow><mi>Smi>mrow><mrow><mn>0mn>mrow>msup><mo>〉mo>mrow><mo>)mo>mrow><mo>]mo>mrow><mo>)mo>mrow>math>. Finally, we present some numerical simulations which show the effectiveness of the algorithm developed in this work.