Skajaa, Anders

Det overordnede emne for denne afhandling er konveks konisk optimering, et underområde af matematisk optimering som angriber problemer med en bestemt geometrisk struktur. Disse problemer muliggør modellering af en ekstremt bred vifte af virkelige problemer, men tilgængeligheden af løsningsalgoritmer til disse problemer er stadig meget begrænset.Målet for denne afhandling er at undersøge og kaste lys over to beregningsmæssige aspekter af den homogene indre-punkts algoritme til løsning af konvekse koniske optimeringsproblemer: Den første del studerer muligheden for at udvikle en homogen indre-punkts metode målrettet løsning af problemer, der indeholder begrænsninger, som kræver ikke-symmetriske kegler i deres beskrivelse. Den anden del studerer muligheden for at varmstarte den homogene indre-punkts algorithme til koniske problemer.Hovedresultat af den første del er introduktionen af en helt ny homogen indrepunkts algoritme designet til at løse ikke-symmetriske konvekse koniske optimeringsproblemer. Denne algoritme præsenteres i detaljer og derefter analyseret. Vi beviser dens konvergens og kompleksitet. Fra et teoretisk synspunkt er den fuldt kompetitiv med mere generelle metoder og fra et praktisk synspunkt viser vi, at den indeholder stort potentiale; mange gange er den at foretrække frem for andre løsningsmetoder.Hovedresultatet af den anden del af afhandlingen er to nye varmstart metoder til den homogene indre-punkts algorithm til koniske problemer. Som før motiverer og præsenterer vi først metoderne og derefter analyseres de. Det bevises, at de, under visse omstændigheder, resulterer i en forbedret værste-falds kompleksitet når man sammenligner med sædvanlig koldstart. We fortsætter derefter med præsentationen af en omfattende serie af beregningsresultater der understøtter den praktiske anvendelighed af disse varmstart metoder. Eksperimenterne inkluderer standard benchmark problemsamlinger såvel som en anvendelse, der stammer fra smarte energisystemer.The overall topic of this thesis is convex conic optimization, a sub-field of mathematical optimization that attacks optimization problem with a certain geometric structure. These problems allow for modelling of an extremely wide range of real-world problems, but the availability of solution algorithms for these problems is still limited.The goal of this thesis is to investigate and shed light on two computational aspects of homogeneous interior-point algorithms for convex conic optimization:The first part studies the possibility of devising a homogeneous interior-point method aimed at solving problems involving constraints that require nonsymmetric cones in their formulation. The second part studies the possibility of warmstarting the homogeneous interior-point algorithm for conic problems. The main outcome of the first part is the introduction of a completely new homogeneous interior-point algorithm designed to solve nonsymmetric convex conic optimization problems. The algorithm is presented in detail and then analyzed. We prove its convergence and complexity. From a theoretical viewpoint, it is fully competitive with other algorithms and from a practical viewpoint, we show that it holds lots of potential, in several cases being superior to other solution methods.The main outcome of the second part of the thesis is two new warmstarting schemes for the homogeneous interior-point algorithm for conic problems. Again, we first motivate and present the schemes and then analyze them. It is proved that they, under certain circumstances, result in an improved worst-case complexity as compared to a normal coldstart. We then move on to present an extensive series of computational results substantiating the practical usefulness of these warmstarting schemes. These experiments include standard benchmarking problem test sets as well as an application from smart energy systems

Online Research Database In Technology

The Homogeneous Interior-Point Algorithm: Nonsymmetric Cones, Warmstarting, and Applications

http://orbit.dtu.dk/ws/files/91513146/phd311_Skajaa_A.pdf

The Homogeneous Interior-Point Algorithm: Nonsymmetric Cones, Warmstarting, and Applications

Abstract

Similar works

Full text

Available Versions

Online Research Database In Technology