Approksimatsiya i optimizatsiya lipshitsevykh funktsiy: Postroeniya, analiz, metody. (Russian Edition)

Negladkie ili nedostatochno gladkie funktsii, kotorye, naprimer, ne imeyut vtorykh proizvodnykh, stali v nashi dni obychnym instrumentom issledovaniya. Metody optimizatsii takikh funktsiy otlichayutsya ot metodov optimizatsii gladkikh (differentsiruemykh) funktsiy. Obychnogo v nashem ponimanii opredeleniya gradienta u negladkikh funktsiy ne sushchestvuet. Zadacha optimizatsii funktsiy tesno primykaet k zadacham, svyazannym s izucheniem optimal'nykh protsessov v teorii upravleniya. Dlya postroeniya bolee uskorennykh metodov optimizatsii negladkikh funktsiy trebuetsya opredelit' takie konstruktsii, k kotorym primenimy metody optimizatsii vtorogo poryadka dlya dvazhdy differentsiruemykh funktsiy. S pomoshch'yu postroennykh funktsiy my mozhem pereyti ot lokal'noy optimizatsii negladkikh funktsiy k lokal'noy optimizatsii gladkikh funktsiy, a takzhe otsenit' skorost' skhodimosti k tochke ekstremuma, chto bezuslovno vazhno. Dlya etoy tseli vvodyatsya novye metody approksimatsii funktsiy i mnogoznachnykh otobrazheniy. Aktual'nym na segodnyashniy den' yavlyaetsya vvedenie takikh mnogoznachnykh otobrazheniy, kotorye mozhno bylo by ispol'zovat' dlya postroeniya nepreryvnykh rasshireniy subdifferentsiala Klarka i postroeniya na ikh osnove metodov poiska statsionarnykh tochek.