Dvorazinska optimizacija

master's thesis

• Title (english): Bilevel optimization
• Author: Lucija Miško
• Mentor: Marko Vrdoljak (mentor)
• Committee member: Marko Vrdoljak (predsjednik povjerenstva)
• Committee member: Marcela Hanzer (član povjerenstva)
• Committee member: Goran Radunović (član povjerenstva)
• Committee member: Miljenko Huzak (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2020-09-29, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

U ovom radu opisuje se problem dvorazinske optimizacije, s posebnim naglaskom na problem linearne dvorazinske optimizacije gdje pretpostavljamo da su funkcije na obje razine afine. Problemi dvorazinske optimizacije imaju drugi (parametarski) problem optimizacije kao dio ograničenja. Za problem linearne dvorazinske optimizacije navode se svojstva i optimalna rješenja na temelju geometrijske interpretacije. Izučava se egzistencija optimalnih rješenja, te pomoću Karush-Kuhn-Tuckerovih uvjeta optimalnosti za donju razinu problema dobivaju se Karush-Kuhn-Tuckerovi uvjeti optimalnosti za problem linearne dvorazinske optimizacije. Navode se algoritmi za rješavanje danog problema te se postavlja pitanje traženja globalnog optimalnog rješenja. Bez pretpostavki afinosti funkcija na obje razine, dolazi se do općenitijeg modela dvorazinske optimizacije. Navode se nužni i dovoljni uvjeti optimalnosti kao i algoritam za izračunavanje Clarkeove stacionarne točke kao optimalnog rješenja problema. Na kraju ovog rada, navode se razne primjene dvorazinske optimizacije u stvarnom svijetu, koje su imale snažan utjecaj na porast istraživanja ovog optimizacijskog problema.

Abstract (english)

In this paper we study the bilevel optimization problem, with special emphasis on the linear bilevel optimization problem where we assume that the functions on both levels are affine. Bilevel optimization problems have another (parametric) optimization problem as part of the constraints. For the linear bilevel optimization problems, properties and optimal solutions based on geometric interpretation are given. The existence of optimal solutions is studied, and using Karush-Kuhn-Tucker optimality conditions for the lower level problem, Karush- Kuhn-Tucker optimality conditions for the linear bilevel optimization problem are obtained. Algorithms for solving a given problem are given and the question of searching for a global optimal solution is raised. Without the assumptions of the affinity of the functions at both levels, a more general model of bilevel optimization is arrived at. Necessary and suficient optimality conditions are given as well as an algorithm for calculating the Clarke stationary point as the optimal solution to the problem. At the end of this paper, various applications of bilevel optimization in the real world are listed, which have had a strong impact on the growth of research on this optimization problem.

• Keywords: linearna dvorazinska optimizacija; Karush-Kuhn-Tuckerovi uvjeti optimalnosti; Clarkeova stacionarna točka
• Keywords (english): linear bilevel optimization; Karush-Kuhn-Tucker optimality conditions; Clarke stationary poin
• Language: croatian
• URN:NBN: urn:nbn:hr:217:280524
• Study programme: Title: Finance and Business Mathematics; Study programme type: university; Study level: graduate; Academic / professional title: magistar/magistra matematike (magistar/magistra matematike)
• Type of resource: Text
• File origin: Born digital
• Access conditions: Open access
• Created on: 2021-03-01 13:42:05