Algorithms and Software for LMI Problems in Control Introduction
Algorithms and Software for LMI Problems in Control Introduction
Algorithms and Software for LMI Problems in Control Introduction
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problemsareatta<strong>in</strong>ed,<strong>and</strong>thesolutionsarecharacterizedbytheoptimalityconditions TrAiX+bi=0;i=1;:::;m XZ+C+Pmi=1yiAi=0 XZ=0: 0;Z 0<br />
TherstthreeconditionsstatefeasibilityofX,Z<strong>and</strong>y.Thelastconditioniscalled (4)<br />
complementaryslackness.<br />
Briefhistory Interior-po<strong>in</strong>tmethods<br />
method<strong>for</strong>LP[24].In1988Nesterov<strong>and</strong>Nemirovskii[25]showedthatthose<strong>in</strong>terior-po<strong>in</strong>t est<strong>in</strong>themwasrevived<strong>in</strong>1984,whenKarmarkar<strong>in</strong>troducedapolynomial-time<strong>in</strong>terior-po<strong>in</strong>t Theideasunderly<strong>in</strong>g<strong>in</strong>terior-po<strong>in</strong>tmethods<strong>for</strong>convexoptimizationcanbetracedbackto<br />
methods<strong>for</strong>l<strong>in</strong>earprogramm<strong>in</strong>gcan,<strong>in</strong>pr<strong>in</strong>ciple,begeneralizedtoallconvexoptimization thesixties;seee.g.,Fiacco<strong>and</strong>McCormick[21],Lieu<strong>and</strong>Huard[22],<strong>and</strong>Dik<strong>in</strong>[23]).Inter-<br />
there<strong>for</strong>e,<strong>in</strong>terior-po<strong>in</strong>tmethodsareapplicable. ertycalledself-concordance.L<strong>in</strong>earmatrix<strong>in</strong>equalitiesareanimportantclassofconvex constra<strong>in</strong>ts<strong>for</strong>whichreadilycomputableself-concordantbarrierfunctionsareknown,<strong>and</strong>, problems.Thekeyelementistheknowledgeofabarrierfunctionwithacerta<strong>in</strong>prop28]generalized<strong>in</strong>terior-po<strong>in</strong>tmethodsfroml<strong>in</strong>earprogramm<strong>in</strong>gtosemideniteprogram-<br />
methods<strong>for</strong>l<strong>in</strong>earprogramm<strong>in</strong>ghavebeenextendedtosemideniteprogramm<strong>in</strong>g.This m<strong>in</strong>g.Vastprogresshasbeenmade<strong>in</strong>thelasttwoyears,<strong>and</strong>todayalmostall<strong>in</strong>terior-po<strong>in</strong>t IndependentlyofNesterov<strong>and</strong>Nemirovskii,Alizadeh[26]<strong>and</strong>Kamath<strong>and</strong>Karmarkar[27,<br />
theexcellentper<strong>for</strong>manceofprimal-dual<strong>in</strong>terior-po<strong>in</strong>tmethods<strong>for</strong>large-scalel<strong>in</strong>earpro- recentresearchhaslargelyconcentratedonprimal-dualmethods<strong>in</strong>thehopeofemulat<strong>in</strong>g<br />
astheprojectivealgorithm<strong>and</strong>themethodofcenters.Wereferto[5,p.80]or[3,x2]<strong>for</strong> gramm<strong>in</strong>g[29,30].Therema<strong>in</strong>derofthissectionwillconcentrateonthisrecentwork.We<br />
surveysoftheseearliermethods. algorithm,themethodofalternat<strong>in</strong>gprojections,<strong>and</strong>primal<strong>in</strong>terior-po<strong>in</strong>tmethodssuch shouldmentionhoweverthatothermethodshavebeenusedsuccessfully,e.g.,theellipsoid<br />
Primal-dualmethods<strong>for</strong>SDP Themostpromis<strong>in</strong>gmethods<strong>for</strong>semideniteprogramm<strong>in</strong>gsolvethetwoproblems(1) <strong>and</strong>(2)simultaneously.Theseprimal-dualmethodsareusually<strong>in</strong>terpretedasmethods<strong>for</strong> y()ofthenonl<strong>in</strong>earequationsX follow<strong>in</strong>gtheprimal-dualcentralpath,whichisdenedasthesetofsolutionsX(),Z(),<br />
TrAiX+bi=0;i=1;:::;m Z+C+Pmi=1yiAi=0 XZ=I; 0;Z 0 (5)<br />
4