Aukcijski algoritmi u mrežnoj optimizaciji

master's thesis

• Author: Dora Stoić
• Mentor: Marko Vrdoljak (mentor)
• Committee member: Marko Vrdoljak (predsjednik povjerenstva)
• Committee member: Josip Tambača (član povjerenstva)
• Committee member: Miljenko Marušić (član povjerenstva)
• Committee member: Boris Guljaš (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2018-07-16, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

Aukcijski algoritmi, temeljeni na principu klasične aukcije, djeluju suprotno klasičnim algoritmima mrežne optimizacije jer u svakoj iteraciji ne poboljšavaju nužno ni primarnu ni dualnu funkciju cilja, no na kraju nalaze optimalnu vrijednost. Zasnovani su na metodama nediferencijabilne optimizacije. U početku opisujemo aukcijski algoritam za klasični problem dodjeljivanja, te se bavimo raznim pitanjima kao što su \(\epsilon\)-skaliranje, problem nedopustivosti, obrnuti algoritam i slično. Razlog tomu je što se problem toka minimalnog troška lako može pretvoriti u problem dodjeljivanja, te su ta pitanja zajednička svim vrstama promatranih problema. Stoga, kasnije u radu, promatramo adaptaciju algoritma za problem maksimalnog toka, transportni problem, problem najkraćih putova i problem toka minimalnog troška. Za problem toka minimalnog troška uvodimo generički algoritam, te kao njegov specijalni slučaj, razvijamo metodu \(\epsilon\)-relaksacije.

Abstract (english)

Auction algorithms, based on real auction, work counter to classical network optimization algorithms because in each iteration they don’t necessarily improve the primal or dual cost, although in the end they find an optimal primal solution. They are based on methods of nondifferentiable optimization. Initially, we describe the auction algorithm for classical assignment problem and we deal with various issues such as \(\epsilon\)-scaling, dealing with infeasibility, reverse auction etc.. The reason is that the minimum cost flow problem can be easily transformed to an assignment problem, so these issues are common to all types of observed problems. Therefore, later in the paper, we observe the algorithm adaptation for the max-flow problem, transportation problem, shortest path problem and minimum cost flow problem. For minimum cost flow problem we introduce generic algorithm and develop \(\epsilon\)-relaxation method as a special case of the generic algorithm.

• Keywords: aukcijski algoritam; optimizacija; problem maksimalnog toka; transportni problem; problem najkraćih putova; problem toka minimalnog troška
• Keywords (english): auction algorithm; optimization; max-flow problem; transportation problem; shortest path problem; minimum cost flow problem
• Language: croatian
• URN:NBN: urn:nbn:hr:217:465785
• Study programme: Title: Applied 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: 2019-06-19 10:07:19