lecture11 - Yinyu Ye, MS&E, Stanford...

Info iconThis preview shows pages 1–5. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Yinyu Ye, MS&E, Stanford MS&E211 Lecture Note #11 1 NonlinearOptimizationAlgorithmsII YinyuYe DepartmentofManagementScienceandEngineering StanfordUniversity Stanford,CA94305,U.S.A. http://www.stanford.edu/yyye Yinyu Ye, MS&E, Stanford MS&E211 Lecture Note #11 2 LinearlyConstrainedConvexProblem ( LCCP ) minimize f ( x ) subjectto A x = b x . Weassumethat A hasfullrankand f isa differentiableconvex function. The KKTconditions : X s = A x = b- A T y + f ( x ) T- s = ( x , s ) . Yinyu Ye, MS&E, Stanford MS&E211 Lecture Note #11 3 BarrierFunction ( BLCCP ) minimize f ( x )- n j =1 log x j subjectto A x = b . TheKKTconditionofthe barrieredproblem is f ( x ) T- X- 1 e- A T y = A x = b x > , or X s = e A x = b- A T y + f ( x ) T- s = ( x , s ) > . Yinyu Ye, MS&E, Stanford MS&E211 Lecture Note #11 4 TheBarrierFunctionMethod Wehaveagood approximate solutionfor ( BLCCP ) minimize f ( x )- n j =1 log x j subjectto A x = b . Then,usingNewtonsstepweupdateittoagood approximate solutionto ( BLCCP+ ) minimize f ( x )- + n j =1 log x j subjectto A x = b , where < + = <, where << 1 iscalledthe reductionratio . Andtheprocessrepeats. Yinyu Ye, MS&E, Stanford MS&E211 Lecture Note #11 5 InteriorCentralPathsofLCCP Let ( x ( ) > , y ( ) , s ( ) > 0) bethe(unique)minimizerofthe barriered problem .Then,the centralpath ofLCCPcanbeexpressedas C = { ( x ( ) > , y ( ) , s ( ) > 0):0 << } . Theorem1 LetLCCPhave interiorfeasiblepoints andthebarrierfunctionis bounded forthegivendataset ( A, b ,f ) .Thenforany << ,thecentral pathpoint ( x ( ) , y ( ) , s ( )) existsandisunique. Yinyu Ye, MS&E, Stanford MS&E211 Lecture Note #11 6 y a The objective hyperplanes Figure1:Thecentralpathina(dual)feasibleregion. Yinyu Ye, MS&E, Stanford MS&E211 Lecture Note #11 7 CentralPathProperties Theorem2 Let ( x ( ) , y ( ) , s ( )) beonthe centralpath . i) The centralpath point ( x ( ) , y ( ) , s ( )) is bounded for < and anygiven < < . ii) For < < , f...
View Full Document

Page1 / 14

lecture11 - Yinyu Ye, MS&E, Stanford...

This preview shows document pages 1 - 5. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online