<div dir="ltr"><b id="internal-source-marker_0.20565514964982867" style="color:rgb(0,0,0);font-family:'Times New Roman';font-size:medium;font-weight:normal"><h1 dir="ltr" style="text-align:center;margin-top:0pt;margin-bottom:0pt">
<span style="font-size:24px;font-family:Arial;background-color:transparent;vertical-align:baseline;white-space:pre-wrap">Ărdekes kĂŠrdĂŠsek az optimalizĂĄlĂĄsban</span></h1><span style="font-size:19px;font-family:Arial;background-color:transparent;font-weight:bold;vertical-align:baseline;white-space:pre-wrap"></span><br>
<p dir="ltr" style="text-align:center;margin-top:0pt;margin-bottom:0pt"><span style="font-size:19px;font-family:Arial;background-color:transparent;vertical-align:baseline;white-space:pre-wrap">avagy</span></p><p dir="ltr" style="text-align:center;margin-top:0pt;margin-bottom:0pt">
<span style="font-size:19px;font-family:Arial;background-color:transparent;font-weight:bold;vertical-align:baseline;white-space:pre-wrap"> </span></p><p dir="ltr" style="text-align:center;margin-top:0pt;margin-bottom:0pt">
<span style="font-size:24px;font-family:Arial;background-color:transparent;font-weight:bold;vertical-align:baseline;white-space:pre-wrap">ĂtletbĂśrze TDK-ra, BSc/MSc tĂŠzishez </span></p><p dir="ltr" style="text-align:center;margin-top:0pt;margin-bottom:0pt">
<span style="font-size:24px;font-family:Arial;background-color:transparent;font-weight:bold;vertical-align:baseline;white-space:pre-wrap">vagy doktori tĂŠmĂĄnak</span></p><span style="font-size:15px;font-family:Arial;background-color:transparent;vertical-align:baseline;white-space:pre-wrap"></span><br>
<span style="font-size:15px;font-family:Arial;background-color:transparent;vertical-align:baseline;white-space:pre-wrap"></span><br><span style="font-size:15px;font-family:Arial;background-color:transparent;vertical-align:baseline;white-space:pre-wrap"></span><br>
<p dir="ltr" style="text-align:justify;margin-top:0pt;margin-bottom:0pt"><span style="font-size:16px;font-family:Arial;background-color:transparent;vertical-align:baseline;white-space:pre-wrap">FebruĂĄr 28-ĂĄn, csĂźtĂśrtĂśkĂśn, 14:15-tĹl az OptimalizĂĄlĂĄs szeminĂĄriumon 5-10 perces ĂzelĂtĹket hallhattok tĂśbb tĂŠmĂĄban a H306-os teremben. CĂmek:</span></p>
<span style="font-size:16px;font-family:Arial;background-color:transparent;vertical-align:baseline;white-space:pre-wrap"></span><br><ul style="margin-top:0pt;margin-bottom:0pt"><li dir="ltr" style="list-style-type:disc;font-size:16px;font-family:Arial;background-color:transparent;vertical-align:baseline">
<span style="font-size:16px;background-color:transparent;vertical-align:baseline;white-space:pre-wrap">A tĂśbbdimenziĂłs normĂĄlis eloszlĂĄssal kapcsolatos valĂłszĂnĹąsĂŠgek legĂşjabb szĂĄmĂtĂĄsi mĂłdszerei ĂŠs azok hatĂŠkonysĂĄgĂĄnak ĂśsszehasonlĂtĂĄsa (SZT)</span></li>
<li dir="ltr" style="list-style-type:disc;font-size:16px;font-family:Arial;background-color:transparent;vertical-align:baseline"><p dir="ltr" style="text-align:justify;margin-top:0pt;margin-bottom:0pt"><span style="font-size:16px;background-color:transparent;vertical-align:baseline;white-space:pre-wrap">Szimplexek szimplexekkel valĂł fedĂŠse (GTB)</span></p>
</li><li dir="ltr" style="list-style-type:disc;font-size:16px;font-family:Arial;background-color:transparent;vertical-align:baseline"><p dir="ltr" style="text-align:justify;margin-top:0pt;margin-bottom:0pt"><span style="font-size:16px;background-color:transparent;vertical-align:baseline;white-space:pre-wrap">MINLP feladatok megoldĂĄsa megbĂzhatĂł mĂłdszerekkel (GTB)</span></p>
</li><li dir="ltr" style="list-style-type:disc;font-size:16px;font-family:Arial;background-color:transparent;vertical-align:baseline"><p dir="ltr" style="text-align:justify;margin-top:0pt;margin-bottom:0pt"><span style="font-size:16px;background-color:transparent;vertical-align:baseline;white-space:pre-wrap">Stackelberg vagy mĂĄs nĂŠven VezetĹ-KĂśvetĹ feladat megoldĂĄsa (GTB)</span></p>
</li><li dir="ltr" style="list-style-type:disc;font-size:16px;font-family:Arial;background-color:transparent;vertical-align:baseline"><p dir="ltr" style="text-align:justify;margin-top:0pt;margin-bottom:0pt"><span style="font-size:16px;background-color:transparent;vertical-align:baseline;white-space:pre-wrap">TĂśbb cĂŠlfĂźggvĂŠnyes optimalizĂĄlĂĄsi feladatok, teljes Pareto-optimĂĄlis halmazĂĄnak approximĂĄciĂłja (LG)</span></p>
</li><li dir="ltr" style="list-style-type:disc;font-size:16px;font-family:Arial;background-color:transparent;vertical-align:baseline"><p dir="ltr" style="text-align:justify;margin-top:0pt;margin-bottom:0pt"><span style="font-size:16px;background-color:transparent;vertical-align:baseline;white-space:pre-wrap">ĂltalĂĄnos egyensĂşlyi modellek megoldĂĄsa numerikus eszkĂśzĂśkkel (LG)</span></p>
</li><li dir="ltr" style="list-style-type:disc;font-size:16px;font-family:Arial;background-color:transparent;vertical-align:baseline"><p dir="ltr" style="text-align:justify;margin-top:0pt;margin-bottom:0pt"><span style="font-size:16px;background-color:transparent;vertical-align:baseline;white-space:pre-wrap">ĂjsĂĄgĂĄrus feladat problĂŠma ĂŠs variĂĄnsai (LG)</span></p>
</li><li dir="ltr" style="list-style-type:disc;font-size:16px;font-family:Arial;background-color:transparent;vertical-align:baseline"><span style="font-size:16px;color:rgb(34,34,34);background-color:transparent;vertical-align:baseline;white-space:pre-wrap">GrĂĄfszĂnezĂŠsek feljavĂtĂĄsa (HM)</span></li>
<li dir="ltr" style="list-style-type:disc;font-size:16px;font-family:Arial;background-color:transparent;vertical-align:baseline"><p dir="ltr" style="text-align:justify;margin-top:0pt;margin-bottom:0pt"><span style="font-size:16px;color:rgb(34,34,34);background-color:transparent;vertical-align:baseline;white-space:pre-wrap">LineĂĄris programozĂĄs alkalmazĂĄsa bizonyĂtĂĄsokban ĂŠs ellenpĂŠlda-konstrukciĂłkban (HM)</span></p>
</li><li dir="ltr" style="list-style-type:disc;font-size:16px;font-family:Arial;background-color:transparent;vertical-align:baseline"><p dir="ltr" style="text-align:justify;margin-top:0pt;margin-bottom:0pt"><span style="font-size:16px;color:rgb(34,34,34);background-color:transparent;vertical-align:baseline;white-space:pre-wrap">A magyar mĂłdszer kiterjesztĂŠsei (HM)</span></p>
</li><li dir="ltr" style="list-style-type:disc;font-size:16px;font-family:Arial;background-color:transparent;vertical-align:baseline"><span style="font-size:16px;background-color:transparent;vertical-align:baseline;white-space:pre-wrap">KalandozĂĄsok a lineĂĄris komplementaritĂĄsi feladatok vilĂĄgĂĄban:</span></li>
<ul style="margin-top:0pt;margin-bottom:0pt"><li dir="ltr" style="list-style-type:circle;font-size:16px;font-family:Arial;background-color:transparent;vertical-align:baseline"><span style="font-size:16px;background-color:transparent;vertical-align:baseline;white-space:pre-wrap">ElĂŠgsĂŠges mĂĄtrixok jellemzĂŠse (IT)</span></li>
<li dir="ltr" style="list-style-type:circle;font-size:16px;font-family:Arial;background-color:transparent;vertical-align:baseline"><span style="font-size:16px;background-color:transparent;vertical-align:baseline;white-space:pre-wrap">CentrĂĄlis Ăşt lĂŠtezĂŠsĂŠnek ĂŠs egyĂŠrtelmĹąsĂŠgĂŠnek a feltĂŠtele (ENM)</span></li>
<li dir="ltr" style="list-style-type:circle;font-size:16px;font-family:Arial;background-color:transparent;vertical-align:baseline"><span style="font-size:16px;background-color:transparent;vertical-align:baseline;white-space:pre-wrap">Hogyan oldjunk meg ĂĄltalĂĄnos lineĂĄris komplementaritĂĄsi feladatokat (hatĂŠkonyan)? (IT)</span></li>
</ul></ul><span style="font-size:16px;font-family:Arial;background-color:transparent;vertical-align:baseline;white-space:pre-wrap"></span><br><span style="font-size:16px;font-family:Arial;background-color:transparent;vertical-align:baseline;white-space:pre-wrap">Ezek mind:</span><br>
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<span style="font-size:16px;background-color:transparent;vertical-align:baseline;white-space:pre-wrap">egyszerĹąbb ĂŠs nehezebb megoldatlan kĂŠrdĂŠsek az optimalizĂĄlĂĄs terĂźletĂŠrĹl</span></li><li dir="ltr" style="list-style-type:disc;font-size:16px;font-family:Arial;background-color:transparent;vertical-align:baseline">
<span style="font-size:16px;background-color:transparent;vertical-align:baseline;white-space:pre-wrap">TDK-nak, szakdolgozatnak, diplomamunkĂĄnak, vagy akĂĄr doktori tĂŠmĂĄnak alkalmas kutatĂĄsi tĂŠmakĂśrĂśk</span></li></ul><span style="font-size:16px;font-family:Arial;background-color:transparent;vertical-align:baseline;white-space:pre-wrap"></span><br>
<span style="font-size:16px;font-family:Arial;background-color:transparent;vertical-align:baseline;white-space:pre-wrap"></span><br><span style="font-size:16px;font-family:Arial;background-color:transparent;vertical-align:baseline;white-space:pre-wrap">IdĹpont: 2013. februĂĄr 28., csĂźtĂśrtĂśk, 14:15-15:45</span><br>
<span style="font-size:16px;font-family:Arial;background-color:transparent;vertical-align:baseline;white-space:pre-wrap">HelyszĂn: BME H ĂŠpĂźlet 306-os szeminĂĄriumi szoba</span></b><div><b style="color:rgb(0,0,0);font-weight:normal"><font face="Arial"><span style="font-size:15.833333015441895px;white-space:pre-wrap"><br>
</span></font></b></div><div><b style="color:rgb(0,0,0);font-weight:normal"><font face="Arial"><span style="font-size:15.833333015441895px;white-space:pre-wrap"><br></span></font><span style="font-family:Arial;background-color:transparent;font-weight:bold;vertical-align:baseline;white-space:pre-wrap"><font size="4"><u>A tĂŠmĂĄk rĂśvid leĂrĂĄsa:</u></font></span><br>
<font size="3"><span style="font-size:16px;font-family:Arial;background-color:transparent;vertical-align:baseline;white-space:pre-wrap"></span></font><br><br><span style="font-family:Arial;font-size:16px;background-color:transparent;font-weight:bold;vertical-align:baseline;white-space:pre-wrap">A tĂśbbdimenziĂłs normĂĄlis eloszlĂĄssal kapcsolatos valĂłszĂnĹąsĂŠgek legĂşjabb szĂĄmĂtĂĄsi mĂłdszerei ĂŠs azok hatĂŠkonysĂĄgĂĄnak ĂśsszehasonlĂtĂĄsa (SzĂĄntai TamĂĄs)</span><br>
<font size="3"><span style="font-size:16px;font-family:Arial;background-color:transparent;vertical-align:baseline;white-space:pre-wrap"></span></font><br><p dir="ltr" style="font-family:'Times New Roman';font-size:medium;text-align:justify;margin-top:0pt;margin-bottom:0pt">
<span style="font-size:16px;font-family:Arial;background-color:transparent;vertical-align:baseline;white-space:pre-wrap">A tĂśbbdimenziĂłs normĂĄlis eloszlĂĄssal kapcsolatos valĂłszĂnĹąsĂŠgek szĂĄmĂtĂĄsĂĄnak tĂśrtĂŠnete tĂśbb mint 50 esztendĹre nyĂşlik vissza. Az elsĹ ĂśsszefoglalĂł dolgozatot Santhi S. Gupta 1963-ban publikĂĄlta </span><span style="font-size:16px;font-family:Arial;background-color:transparent;font-style:italic;vertical-align:baseline;white-space:pre-wrap">Probability integrals of multivariate normal and multivariate t</span><span style="font-size:16px;font-family:Arial;background-color:transparent;vertical-align:baseline;white-space:pre-wrap"> cĂmmel. Az 1970-es ĂŠvekben DeĂĄk IstvĂĄn ĂŠs SzĂĄntai TamĂĄs dolgoztak ki kĂŠt kĂźlĂśnbĂśzĹ szĂłrĂĄscsĂśkkentĂŠsi mĂłdszert a tĂśbbvĂĄltozĂłs normĂĄlis eloszlĂĄssal kapcsolatos valĂłszĂnĹąsĂŠgek Monte Carlo szimulĂĄciĂłval tĂśrtĂŠnĹ kiĂŠrtĂŠkelĂŠsĂŠre. Alan Genz 1992-ben a tĂśbbdimenziĂłs normĂĄlis eloszlĂĄssal kapcsolatos valĂłszĂnĹąsĂŠgek szĂĄmĂtĂĄsĂĄra egy integrĂĄl transzformĂĄciĂł sorozat vĂŠgrehajtĂĄsa utĂĄni Monte Carlo szimulĂĄciĂłs kiĂŠrtĂŠkelĂŠst dolgozott ki, mely nem tĂşl sok vĂĄltozĂłs egyĂźttes normĂĄlis eloszlĂĄsra igen hatĂŠkonynak bizonyult. BukszĂĄr JĂłzsef PrĂŠkopa AndrĂĄssal ĂŠs SzĂĄntai TamĂĄssal kĂśzĂśsen az 1990-es ĂŠvek vĂŠgĂŠn olyan alsĂł ĂŠs felsĹ korlĂĄtokat adott meg magasabb dimenziĂłs normĂĄlis eloszlĂĄsokkal kapcsolatos valĂłszĂnĹąsĂŠgekre, amelyek hasznosnak bizonyultak SzĂĄntai TamĂĄs korĂĄbban javasolt szimulĂĄciĂłs kiĂŠrtĂŠkelĹ eljĂĄrĂĄsĂĄnak a megjavĂtĂĄsĂĄban. Ezt kĂśvetĹen DeĂĄk IstvĂĄn ĂŠs SzĂĄntai TamĂĄs Horand Gassmann kĂśzremĹąkĂśdĂŠsĂŠvel egyesĂtettĂŠk a kĂźlĂśn-kĂźlĂśn kidolgozott szimulĂĄciĂłs eljĂĄrĂĄsukat, amivel sikerĂźlt azok hatĂŠkonysĂĄgĂĄt megnĂśvelni. A 2000-es ĂŠvek elejĂŠn Tetsusiha Miwa, A.J. Hayter ĂŠs Satoshi Kuriki nem negatĂv ortĂĄns tĂśbbdimenziĂłs normĂĄlis valĂłszĂnĹąsĂŠg eloszlĂĄs melletti valĂłszĂnĹąsĂŠgĂŠnek a szĂĄmĂtĂĄsĂĄra fejlesztettek ki kĂśzvetlen numerikus integrĂĄlĂĄsi mĂłdszert, amely segĂtsĂŠgĂŠvel 6-7 dimenziĂłig jĂłval gyorsabban ĂŠs pontosabban tudtĂĄk szĂĄmolni a tĂśbbdimenziĂłs normĂĄlis eloszlĂĄs eloszlĂĄsfĂźggvĂŠny ĂŠrtĂŠkĂŠt is. Ezt az eljĂĄrĂĄst sikerĂźlt Peter Craig-nek 2008-ban tovĂĄbb javĂtania.</span></p>
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<span style="font-size:16px;font-family:Arial;background-color:transparent;vertical-align:baseline;white-space:pre-wrap">A tĂŠmalabor (TDK munka, szakdolgozat) fĹ cĂŠlkitĹązĂŠse ezeknek a numerikus szĂĄmĂtĂĄsi mĂłdszereknek a megismerĂŠse, szĂĄmĂtĂłgĂŠpes kĂłdok megalkotĂĄsa, amelyek hasznĂĄlatĂĄval gondos statisztikai elemzĂŠsek hajthatĂłk vĂŠgre annak megĂĄllapĂtĂĄsĂĄra, hogy mely mĂłdszerek mely esetekben bizonyulhatnak a legjobbnak. CĂŠlkitĹązĂŠs lehet az egyes mĂłdszerek tovĂĄbbi hatĂŠkonysĂĄg nĂśvelĂŠse, illetve azok Ăşj kombinĂĄciĂłjĂĄnak kidolgozĂĄsa is.</span></p>
<font size="3"><span style="font-size:16px;font-family:Arial;background-color:transparent;font-weight:bold;vertical-align:baseline;white-space:pre-wrap"></span></font><br><font size="3"><span style="font-size:16px;font-family:Arial;background-color:transparent;font-weight:bold;vertical-align:baseline;white-space:pre-wrap"></span></font><br>
<p dir="ltr" style="font-family:'Times New Roman';font-size:medium;text-align:justify;margin-top:0pt;margin-bottom:0pt"><span style="font-size:16px;font-family:Arial;background-color:transparent;font-weight:bold;vertical-align:baseline;white-space:pre-wrap">Szimplexek szimplexekkel valĂł fedĂŠse (G.-TĂłth BoglĂĄrka)</span></p>
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<span style="font-size:16px;font-family:Arial;background-color:transparent;vertical-align:baseline;white-space:pre-wrap">EgysĂŠg szimplexek egyenlĹ oldalĂş szimplexekre bontĂĄsa tĂśbb mint 3 csĂşcs esetĂŠn nem lehetsĂŠges. Van hogy mĂŠgis ilyesmire van szĂźksĂŠg. NĂŠzzĂźk mĂĄskĂŠpp. Hogyan fedjĂźnk le egy szimplexet hasonlĂł szimplexekkel Ăşgy, hogy a legkevesebb legyen az ĂĄtfedĂŠs? Lehet-e tovĂĄbbi bontĂĄsok sorĂĄn az ĂĄtfedĂŠseket kiiktatni?</span></p>
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<span style="font-family:Arial;font-size:16px;background-color:transparent;font-weight:bold;vertical-align:baseline;white-space:pre-wrap">MINLP feladatok megoldĂĄsa megbĂzhatĂł mĂłdszerekkel (G.-TĂłth BoglĂĄrka)</span><br><font size="3"><span style="font-size:16px;font-family:Arial;background-color:transparent;font-weight:bold;vertical-align:baseline;white-space:pre-wrap"></span></font><br>
<p dir="ltr" style="font-family:'Times New Roman';font-size:medium;text-align:justify;margin-top:0pt;margin-bottom:0pt"><span style="font-size:16px;font-family:Arial;background-color:transparent;vertical-align:baseline;white-space:pre-wrap">Egy igen nehĂŠz feladatosztĂĄly, a vegyes egĂŠszĂŠrtĂŠkĹą nemlineĂĄris programozĂĄsi feladatok megoldĂĄsĂĄra nem sok megbĂzhatĂł mĂłdszer szĂźletett eddig, de a szĂĄmĂtĂĄstechnika fejlĹdĂŠsĂŠvel ez lassan elĂŠrhetĹvĂŠ vĂĄlik, ĂŠs Ăgy a mĂłdszerek kidolgozĂĄsa is idĹszerĹą feladat. A feladat izgalmas kihĂvĂĄs mind elmĂŠleti, mind gyakorlati oldalrĂłl: bizonyĂtani kell az eljĂĄrĂĄsok helyessĂŠgĂŠt, ĂŠs pĂŠldafeladatokon bemutatni a mĹąkĂśdĂŠsĂŠt.</span></p>
<font size="3"><span style="font-size:16px;font-family:Arial;background-color:transparent;vertical-align:baseline;white-space:pre-wrap"></span></font><br><font size="3"><span style="font-size:16px;font-family:Arial;background-color:transparent;vertical-align:baseline;white-space:pre-wrap"></span></font><br>
<p dir="ltr" style="font-family:'Times New Roman';font-size:medium;text-align:justify;margin-top:0pt;margin-bottom:0pt"><span style="font-size:16px;font-family:Arial;background-color:transparent;font-weight:bold;vertical-align:baseline;white-space:pre-wrap">Stackelberg vagy mĂĄs nĂŠven VezetĹ-KĂśvetĹ feladat megoldĂĄsa (G.-TĂłth BoglĂĄrka)</span></p>
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<span style="font-size:16px;font-family:Arial;background-color:transparent;vertical-align:baseline;white-space:pre-wrap">Ez egy olyan vĂĄllalatelhelyezĂŠsi feladat, ahol a vĂĄllalatot telepĂtendĹ cĂŠg nem csak a piacon lĂŠvĹ versenytĂĄrsak aktuĂĄlis vĂĄllalatait veszi figyelembe, hanem azt is, hogy a versenytĂĄrsak az Ăşj vĂĄllalat, a VezetĹ megjelenĂŠse utĂĄn vĂĄlaszolnak egy mĂĄsik vĂĄllalat, a KĂśvetĹ felĂŠpĂtĂŠsĂŠvel. A cĂŠl annak a profitnak a maximalizĂĄlĂĄsa, amit a KĂśvetĹ felĂŠpĂtĂŠse utĂĄn a VezetĹ szerez meg. FeltesszĂźk, hogy a KĂśvetĹ a szĂĄmĂĄra optimĂĄlis profitot adĂł helyet vĂĄlasztja.</span></p>
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<span style="font-family:Arial;font-size:16px;background-color:transparent;font-weight:bold;vertical-align:baseline;white-space:pre-wrap">TĂśbb cĂŠlfĂźggvĂŠnyes optimalizĂĄlĂĄsi feladatok, teljes Pareto-optimĂĄlis halmazĂĄnak approximĂĄciĂłja (Lovics GĂĄbor)</span><br>
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<span style="font-size:16px;font-family:Arial;background-color:transparent;vertical-align:baseline;white-space:pre-wrap">Az operĂĄciĂłkutatĂĄs alkalmazĂĄsai gyakran olyan modelleket eredmĂŠnyeznek, ahol egyszerre tĂśbb kĂźlĂśnbĂśzĹ cĂŠlfĂźggvĂŠnyt kell optimalizĂĄlnunk. Klasszikus befektetĂŠsi problĂŠma  is ilyen pĂŠldĂĄul, ahol a lehetĹ legkisebb kockĂĄzat mellett prĂłbĂĄljuk, maximalizĂĄl a vĂĄrhatĂł hozamunkat. Ilyenkor minden olyan megoldĂĄs âoptimĂĄlisâ lehet, ahol az egyik cĂŠlunk javĂtĂĄsa mĂĄr csak a mĂĄsik rovĂĄsĂĄrĂĄra kĂŠpzelhetĹ el. Ezeket a pontokat nevezzĂźk Pareto-optimĂĄlis megoldĂĄsoknak. A legjobb vĂŠgsĹ dĂśntĂŠst gyakran akkor hozhatjuk meg, ha az Ăśsszes Pareto-optimĂĄlis megoldĂĄst, vagy az ilyeneknek legalĂĄbb is valamifĂŠle kĂśzelĂtĂŠse ismert.</span></p>
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<span style="font-family:Arial;font-size:16px;background-color:transparent;font-weight:bold;vertical-align:baseline;white-space:pre-wrap">ĂltalĂĄnos egyensĂşlyi modellek megoldĂĄsa numerikus eszkĂśzĂśkkel (Lovics GĂĄbor) </span><br>
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<span style="font-size:16px;font-family:Arial;background-color:transparent;vertical-align:baseline;white-space:pre-wrap">A mikroĂśkonĂłmia fogyasztĂĄselmĂŠlete szerint az egyĂŠnek az ĂĄltaluk nem befolyĂĄsolhatĂł piacon ĂŠrzĂŠkelt ĂĄrak, sajĂĄt preferenciĂĄik ĂŠs kĂśltsĂŠgvetĂŠsi korlĂĄtjuk alapjĂĄn hoznak dĂśntĂŠst arrĂłl, hogy mennyit vĂĄsĂĄrolnak az egyes javakbĂłl. Â Â Az egyĂŠni dĂśntĂŠsek, ĂśsszegzĂŠse piaci kereslettĂŠ ĂĄll, amely a kĂnĂĄlattal kĂśzĂśsen vĂŠgĂźl is meghatĂĄrozza a kialakult ĂĄrakat. TehĂĄt, az egyĂŠni dĂśntĂŠsek vĂŠgeredmĂŠnyben mĂŠgiscsak befolyĂĄsoljĂĄk az egyensĂşlyi ĂĄrakat. Ennek az ellentmondĂĄsnak feloldĂĄsĂĄra szĂźlettek meg az ĂĄltalĂĄnos egyensĂşlyi modellek. Az egyik legismertebb eredmĂŠny a tĂŠmĂĄban Arrow-Debreu szerzĹpĂĄroshoz kĂśtĹdik, akik meglehetĹsen ĂĄltalĂĄnos kĂśrĂźlmĂŠnyek kĂśzĂśtt bizonyĂtjĂĄk az ĂĄltalĂĄnos egyensĂşlyi modellek megoldĂĄsĂĄnak lĂŠtezĂŠsĂŠt, de semmit nem mondanak, arrĂłl hogyan kell megkeresni ezt a megoldĂĄst. A megoldĂĄsok megkeresĂŠsĂŠvel kapcsolatban sok eredmĂŠny szĂźletett azĂłta, de akadnak mĂŠg ĂŠrdekes, megvĂĄlaszolatlan kĂŠrdĂŠsek is ebben a tĂŠmĂĄban.</span></p>
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<span style="font-family:Arial;font-size:16px;background-color:transparent;font-weight:bold;vertical-align:baseline;white-space:pre-wrap">ĂjsĂĄgĂĄrus feladat problĂŠma ĂŠs variĂĄnsai (Lovics GĂĄbor)</span><br><p dir="ltr" style="font-family:'Times New Roman';font-size:medium;text-align:center;margin-top:0pt;margin-bottom:0pt">
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<span style="font-size:16px;font-family:Arial;background-color:transparent;vertical-align:baseline;white-space:pre-wrap">A gyakorlatban eladĂłkĂŠnt gyakran Ăşgy kell dĂśntenĂźnk egyes termĂŠkek beszerzĂŠsrĹl, hogy nem ismerjĂźk mĂŠg pontosan keresletet. A problĂŠmĂĄt az adja, hogy ki nem szolgĂĄlt vevĹ, vagy a rajtunk maradt termĂŠk, egyarĂĄnt vesztĂŠseget jelent. A tĂŠmĂĄban klasszikusnak szĂĄmĂtĂł Ăşgynevezett ĂşjsĂĄgĂĄrus modell nagyon sok szempontbĂłl speciĂĄlis, mert egy eladĂł ĂĄrusĂt egy olyan termĂŠket, amelynet nincs ĂŠrtelme raktĂĄrozni, ĂŠs amelynek ismert normĂĄlis eloszlĂĄs szerint alakul a kereslete. A valĂłsĂĄgban a kĂŠrdĂŠs sokkal Ăśsszetettebb, mert minden eladĂł tĂśbb termĂŠket ĂĄrul, ezek eloszlĂĄsa nem feltĂŠtlenĂźl normĂĄlis eloszlĂĄsĂş. MĂĄsfelĹl, az eladĂł (bolt) rĂŠsze (lehet) egy ĂźzletlĂĄncnak ĂŠs ebben a helyzetben egy nagyobb piac ellĂĄtĂĄsa a cĂŠl. Ilyenkor a ki nem szolgĂĄlt vevĹk vagy el nem adott ĂĄruk szĂĄma fĂźgghet az ĂĄrĂş mennyisĂŠgĂŠnek helytelen, belsĹ elosztĂĄsi rendszere miatt is. TermĂŠszetesen ebben az utolsĂł esetben nem feledkezhetĂźnk meg a konkurenciĂĄrĂłl sem.</span></p>
<p dir="ltr" style="font-family:'Times New Roman';font-size:medium;text-align:justify;margin-top:0pt;margin-bottom:0pt"><span style="font-size:16px;font-family:Arial;background-color:transparent;vertical-align:baseline;white-space:pre-wrap">BennĂźnket olyan modellek ĂŠrdekelnek, amelyek jĂł kĂśzelĂtĂŠssel leĂrjĂĄk a problĂŠmĂĄt ĂŠs numerikusan meg is oldhatĂłk.</span></p>
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<p dir="ltr" style="font-family:'Times New Roman';font-size:medium;text-align:justify;margin-top:0pt;margin-bottom:0pt"><span style="font-size:16px;font-family:Arial;color:rgb(34,34,34);background-color:transparent;font-weight:bold;vertical-align:baseline;white-space:pre-wrap">GrĂĄfszĂnezĂŠsek feljavĂtĂĄsa (Hujter MihĂĄly)</span></p>
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<span style="font-size:16px;font-family:Arial;color:rgb(34,34,34);background-color:transparent;vertical-align:baseline;white-space:pre-wrap">A grĂĄfok szĂnezĂŠsi kĂŠrdĂŠsei mind matematikatĂśrtĂŠneti ĂŠs algoritmuselmĂŠleti szempontbĂłl, mind a gyakorlati alkalmazĂĄsok oldalĂĄrĂłl tekintve nagyon fontosak. LĂŠnyegĂŠben hĂĄrom fĂŠle megkĂśzelĂtĂŠsi mĂłdszerkĂśr ismeretes:</span></p>
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<span style="font-size:16px;background-color:transparent;vertical-align:baseline;white-space:pre-wrap">A kĂśzvetlenĂźl lĂĄthatĂł szĂźksĂŠgszerĂťsĂŠgek figyelembe vĂŠtelĂŠvel egy menetben szĂnezzĂźk ki a grĂĄfot remĂŠlhetĂľleg kevĂŠs szĂnnel.</span></p>
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<span style="font-size:16px;background-color:transparent;vertical-align:baseline;white-space:pre-wrap">MegkĂsĂŠreljĂźk a grĂĄf viszonylag nagy rĂŠszĂŠt viszonylag kevĂŠs szĂnnek kiszĂnezni, aztĂĄn ezeket a szĂnezĂŠseket lĂŠpĂŠsrĂľl lĂŠpĂŠsre ĂĄtjavĂtgatni.</span></p>
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<span style="font-size:16px;background-color:transparent;vertical-align:baseline;white-space:pre-wrap">Implicit mĂłdon az Ăśsszes lehetĂľsĂŠget leszĂĄmlĂĄljuk.</span></p></li></ol><font size="3"><span style="font-size:16px;font-family:Arial;color:rgb(34,34,34);background-color:transparent;vertical-align:baseline;white-space:pre-wrap"></span></font><br>
<p dir="ltr" style="font-family:'Times New Roman';font-size:medium;text-align:justify;margin-top:0pt;margin-bottom:0pt"><span style="font-size:16px;font-family:Arial;color:rgb(34,34,34);background-color:transparent;vertical-align:baseline;white-space:pre-wrap">Az utolsĂł ĂŠvek eredmĂŠnyei azt mutatjĂĄk, sok kutatnivalĂł van mĂŠg a fenti hĂĄrom mĂłdszerkĂśr kĂśzĂśs hatĂĄrainĂĄl.</span></p>
<font size="3"><span style="font-size:16px;font-family:Arial;color:rgb(34,34,34);background-color:transparent;vertical-align:baseline;white-space:pre-wrap"></span></font><br><font size="3"><span style="font-size:16px;font-family:Arial;color:rgb(34,34,34);background-color:transparent;vertical-align:baseline;white-space:pre-wrap"></span></font><br>
<p dir="ltr" style="font-family:'Times New Roman';font-size:medium;text-align:justify;margin-top:0pt;margin-bottom:0pt"><span style="font-size:16px;font-family:Arial;color:rgb(34,34,34);background-color:transparent;font-weight:bold;vertical-align:baseline;white-space:pre-wrap">LineĂĄris programozĂĄs alkalmazĂĄsa bizonyĂtĂĄsokban ĂŠs ellenpĂŠlda-konstrukciĂłkban (Hujter MihĂĄly)</span></p>
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<span style="font-size:16px;font-family:Arial;color:rgb(34,34,34);background-color:transparent;vertical-align:baseline;white-space:pre-wrap">SzĂĄmos algoritmus jĂł minĂľsĂŠgĂŠnek bizonyĂtĂĄsĂĄban, sok ellenpĂŠlda konstruĂĄlĂĄsĂĄban hasznosak azok a mĂłdszerek, melyekben konkrĂŠt lineĂĄris programozĂĄsi feladatokat kell megoldani, ĂŠs az eredmĂŠnyt a bizonyĂtĂĄsba illetve a konstrukciĂłba beĂŠpĂteni. Ărdemes lenne ezeket az eredmĂŠnyeket rendszerezve ĂśsszegyĂťjteni, tovĂĄbbfejleszteni.</span></p>
<font size="3"><span style="font-size:16px;font-family:Arial;color:rgb(34,34,34);background-color:transparent;vertical-align:baseline;white-space:pre-wrap"></span></font><br><font size="3"><span style="font-size:16px;font-family:Arial;color:rgb(34,34,34);background-color:transparent;vertical-align:baseline;white-space:pre-wrap"></span></font><br>
<p dir="ltr" style="font-family:'Times New Roman';font-size:medium;text-align:justify;margin-top:0pt;margin-bottom:0pt"><span style="font-size:16px;font-family:Arial;color:rgb(34,34,34);background-color:transparent;font-weight:bold;vertical-align:baseline;white-space:pre-wrap">A magyar mĂłdszer kiterjesztĂŠsei (Hujter MihĂĄly)</span><span style="font-size:16px;font-family:Arial;color:rgb(34,34,34);background-color:transparent;vertical-align:baseline;white-space:pre-wrap"></span></p>
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<span style="font-size:16px;font-family:Arial;color:rgb(34,34,34);background-color:transparent;vertical-align:baseline;white-space:pre-wrap">A szĂĄllĂtĂĄsi feladatra bebizonyosodott az egyetemĂźnkĂśn kifejlesztett magyar mĂłdszer hatĂŠkonysĂĄga ĂŠs hasznossĂĄga. KikutatandĂł, hogy a szĂĄllĂtĂĄsi feladat kĂśrĂźlmĂŠnyrendszerĂŠnek</span></p>
<p dir="ltr" style="font-family:'Times New Roman';font-size:medium;text-align:justify;margin-top:0pt;margin-bottom:0pt"><span style="font-size:16px;font-family:Arial;color:rgb(34,34,34);background-color:transparent;vertical-align:baseline;white-space:pre-wrap">mely ĂĄltalĂĄnosĂtĂĄsaira vihetĂľ tovĂĄbb eredmĂŠnyesen a ,,Hungarian Method''.</span></p>
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<font size="3"><span style="font-size:16px;font-family:Arial;background-color:transparent;font-weight:bold;vertical-align:baseline;white-space:pre-wrap"></span></font><br><span style="font-family:Arial;font-size:16px;background-color:transparent;font-weight:bold;vertical-align:baseline;white-space:pre-wrap">KalandozĂĄsok a lineĂĄris komplementaritĂĄsi feladatok vilĂĄgĂĄban </span><br>
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<span style="font-size:16px;font-family:Arial;background-color:transparent;font-weight:bold;vertical-align:baseline;white-space:pre-wrap">ElĂŠgsĂŠges mĂĄtrixok jellemzĂŠse (IllĂŠs Tibor)</span></p><font size="3"><span style="font-size:16px;font-family:Arial;background-color:transparent;font-weight:bold;vertical-align:baseline;white-space:pre-wrap"></span></font><br>
<p dir="ltr" style="font-family:'Times New Roman';font-size:medium;margin-left:36pt;text-align:justify;margin-top:0pt;margin-bottom:0pt"><span style="font-size:16px;font-family:Arial;background-color:transparent;vertical-align:baseline;white-space:pre-wrap">Cottle ĂŠs szerzĹtĂĄrsai 1989-ben bevezettĂŠk az elĂŠgsĂŠges mĂĄtrixok fogalmĂĄt, Kojima ĂŠs tĂĄrsszerzĹi 1991-ben a </span><span style="font-size:16px;font-family:Arial;background-color:transparent;font-style:italic;vertical-align:baseline;white-space:pre-wrap">P* </span><span style="font-size:16px;font-family:Arial;background-color:transparent;vertical-align:baseline;white-space:pre-wrap">mĂĄtrixokĂŠt. Valiaho 1995-ben bizonyĂtotta, hogy a </span><span style="font-size:16px;font-family:Arial;background-color:transparent;font-style:italic;vertical-align:baseline;white-space:pre-wrap">P* </span><span style="font-size:16px;font-family:Arial;background-color:transparent;vertical-align:baseline;white-space:pre-wrap">mĂĄtrixok pontosan az elĂŠgsĂŠges mĂĄtrixok. BizonyĂtĂĄsa nem egyszerĹą, sĹt. IdĹkĂśzben a </span><span style="font-size:16px;font-family:Arial;background-color:transparent;font-style:italic;vertical-align:baseline;white-space:pre-wrap">P* </span><span style="font-size:16px;font-family:Arial;background-color:transparent;vertical-align:baseline;white-space:pre-wrap">mĂĄtrixok szĂĄmos ekvivalens definĂciĂłjĂĄt megismertĂźk ĂŠs ugyanez tĂśrtĂŠnt az elĂŠgsĂŠges mĂĄtrixokkal, de a kapcsolat ezek az ekvivalens jellemzĂŠsek kĂśzĂśtt tovĂĄbbra is Valiaho eredmĂŠnye. SzeretnĂŠk egy olyan bizonyĂtĂĄst lĂĄtni, amelyik ezeknek a jellemzĂŠseknek az ekvivalenciĂĄjĂĄt szĂŠp ĂŠs egyszerĹą (ciklikus mĂłdon) bizonyĂtja, lehetĹsĂŠg szerint kihagyja a Valiaho tĂŠtelt. Ărdekes lenne adott mĂŠretĹą â mondjuk 20x20-as â elĂŠgsĂŠges mĂĄtrixok generĂĄlĂĄsĂĄra hatĂŠkony mĂłdszert kidolgozni.</span></p>
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<span style="font-size:16px;font-family:Arial;background-color:transparent;font-weight:bold;vertical-align:baseline;white-space:pre-wrap">CentrĂĄlis Ăşt lĂŠtezĂŠsĂŠnek ĂŠs egyĂŠrtelmĹąsĂŠgĂŠnek a feltĂŠtele (Eisenberg-Nagy Marianna)</span></p>
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<span style="font-size:16px;font-family:Arial;background-color:transparent;vertical-align:baseline;white-space:pre-wrap">SzĂĄmos alkalmazĂĄs vezet olyan LCP-kre, amelyeknĂŠl a feladat mĂĄtrixa rendelkezik valamilyen speciĂĄlis struktĂşrĂĄval, tulajdonsĂĄggal (pĂŠldĂĄul nemnegatĂvitĂĄs). Ăm ez ĂĄltalĂĄban nem biztosĂtja a centrĂĄlis Ăşt egyĂŠrtelmĹąsĂŠgĂŠt, ezĂŠrt elveszĂtjĂźk a belsĹpontos algoritmus jĂłl definiĂĄltsĂĄgĂĄt. EnnĂŠlfogva a kĂśvetkezĹ kĂŠrdĂŠseket lenne ĂŠrdekes kĂśrbejĂĄrni: </span></p>
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<p dir="ltr" style="text-align:justify;margin-top:0pt;margin-bottom:0pt"><span style="font-size:16px;background-color:transparent;vertical-align:baseline;white-space:pre-wrap">Mikor ĂŠs miĂŠrt nem egyĂŠrtelmĹą? </span></p></li>
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<span style="font-size:16px;background-color:transparent;vertical-align:baseline;white-space:pre-wrap">Ha nem egyĂŠrtelmĹą a centrĂĄlis Ăşt akkor hĂĄny kĂźlĂśnbĂśzĹ lehet? MitĹl fĂźgg a darabszĂĄm? </span></p></li><li dir="ltr" style="list-style-type:lower-roman;font-size:16px;font-family:Arial;background-color:transparent;vertical-align:baseline;margin-left:48px">
<p dir="ltr" style="text-align:justify;margin-top:0pt;margin-bottom:0pt"><span style="font-size:16px;background-color:transparent;vertical-align:baseline;white-space:pre-wrap">A centrĂĄlis utak â ha tĂśbb van â akkor is differenciĂĄlhatĂłk?</span></p>
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<span style="font-size:16px;font-family:Arial;background-color:transparent;font-weight:bold;vertical-align:baseline;white-space:pre-wrap">Hogyan oldjunk meg ĂĄltalĂĄnos lineĂĄris komplementaritĂĄsi feladatokat (hatĂŠkonyan)? (IllĂŠs Tibor)</span></p>
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<span style="font-size:16px;font-family:Arial;background-color:transparent;vertical-align:baseline;white-space:pre-wrap">Az elĹzĹ kĂŠrdĂŠskĂśr â centrĂĄlis Ăşt lĂŠtezĂŠse ĂŠs egyĂŠrtelmĹąsĂŠge â mĂĄr elĹre vetĂti, hogy ha az LCP mĂĄtrixa ĂĄltalĂĄnos vagy nem rendelkezik jĂł tulajdonsĂĄgokkal, akkor az LCP feladat polinom idejĹą megoldĂĄsa nem vĂĄrhatĂł el. Tekintettel arra, hogy szĂĄmos gyakorlati feladat ebbe a kategĂłriĂĄba esik, mĂŠgis fontos lenne megfelelĹ megoldĂł algoritmusokat kidolgozni. Ennek elmĂŠleti alapjait elkezdtĂźk lerakni hĂĄrom cikkĂźnkben, de mĂŠg nagyon sokat kellene dolgozni azon, hogy valĂłs mĂŠretĹą feladatokat meg tudjunk oldani.</span></p>
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