Penalty methods in constrained optimization

master's thesis

• Author: Lucija Mihalj
• Mentor: Marko Vrdoljak (mentor)
• Committee member: Marko Vrdoljak (predsjednik povjerenstva)
• Committee member: Nenad Antonić (član povjerenstva)
• Committee member: Boris Muha (član povjerenstva)
• Committee member: Goran Radunović (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2019-11-27, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

This master thesis deals with penalty methods in constrained optimization. The approach is used to convert a constrained problem into an equivalent unconstrained problem. At the beginning, the most important definitions and results related to unconstrained, as well as constrained, optimization are presented. In the second chapter we begin with the basic definition of penalty function. Furthermore, the concept of the penalty function is elaborated and geometrically illustrated. In the next chapter, important result which justifies the use of the penalty function is stated and proven. Also, computational difficulties which motivate the next chapter are discussed. In the fourth chapter, exact penalty functions are presented. Finally, in the last chapter all the methods presented are applied on examples. Firstly, a simple convex optimization problem is solved by all of the methods and results are discussed. After that, cardinality constrained portfolio optimization problem is presented and solved with GUROBI and exterior penalty function method.

Abstract (croatian)

Ovaj rad bavi se s metodom kaznenih funkcija u uvjetnoj optimizaciji koja se koristi kako bi se problem uvjetne optimizacije ekvivalentno zapisao kao problem bezuvjetne optimizacije. Na početku rada navedene su najvažnije definicije i rezultati o bezuvjetnoj, kao i uvjetnoj, optimizaciji. U drugom poglavlju počinjemo s osnovnim definicijama o kaznenim funkcijama. Nadalje, prikazan je koncept kaznenih funkcija i njihova geometrijska interpretacija. U sljedećem poglavlju, naveden je i dokazan rezultat koji opravdava korištenje kaznenih funkcija u uvjetnoj optimizaciji. Također, opisane su računske složenosti koje su motivirale sljedeće poglavlje. U četvrtom poglavlju predstavljene su egzaktne kaznene funkcije. Na kraju rada nalazi se primjena opisanih metoda na jednostavnom konveksnom primjeru. Nakon toga, predstavljen je problem optimizacije portfelja s uvjetom kardinalnosti i riješen je pomoću GUROBI-ja i kaznene funkcije te su dobiveni rezultati uspoređeni.

• Keywords: penalty methods; constrained optimization; penalty function; cardinality constrained portfolio optimization problem; GUROBI
• Keywords (english): metoda kaznenih funkcija; uvjetna optimizacija; kaznena funkcija; problem optimizacije portfelja s uvjetom kardinalnosti; GUROBI
• Language: english
• URN:NBN: urn:nbn:hr:217:336877
• Study programme: Title: Mathematical Statistics; 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: 2020-07-10 10:15:45