AUTHOR=Gong Xiaoyu , Xi Lei , Yuan Bo 

TITLE=A new search direction of IPM for horizontal linear complementarity problems

JOURNAL=Frontiers in Energy Research

VOLUME=10

YEAR=2023

URL=https://www.frontiersin.org/journals/energy-research/articles/10.3389/fenrg.2022.977448

DOI=10.3389/fenrg.2022.977448

ISSN=2296-598X

ABSTRACT=<p>This study presents a new search direction for the horizontal linear complementarity problem. A vector-valued function is applied to the system of <inline-formula id="inf1"><mml:math id="m1" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mtext>xy</mml:mtext><mml:mo>=</mml:mo><mml:mi>μ</mml:mi><mml:mi mathvariant="normal">e</mml:mi></mml:mrow></mml:math></inline-formula>, which defines the central path. Usually, the way to get the equivalent form of the central path is using the square root function. However, in our study, we substitute a new search function formed by a different identity map, which obtains the equivalent shape of the central path using the square root function. We get the new search directions from Newton’s Method. Given this framework, we prove polynomial complexity for the Newton directions. We show that the algorithm’s complexity is <inline-formula id="inf2"><mml:math id="m2" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi mathvariant="normal">O</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msqrt><mml:mi mathvariant="normal">n</mml:mi></mml:msqrt><mml:mi>log</mml:mi><mml:mfrac><mml:mrow><mml:mi mathvariant="normal">n</mml:mi></mml:mrow><mml:mrow><mml:mi>ϵ</mml:mi></mml:mrow></mml:mfrac></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math></inline-formula>, which is the same as the best-given algorithms for the horizontal linear complementarity problem.</p>