Sekvencijalno kvadratično programiranje

master's thesis

• Author: Armin Čehajić
• Mentor: Marko Vrdoljak (mentor)
• Committee member: Marko Vrdoljak (predsjednik povjerenstva)
• Committee member: Matija Kazalicki (član povjerenstva)
• Committee member: Siniša Slijepčević (član povjerenstva)
• Committee member: Franka Miriam Bruckler (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2014-09-23, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

Sekvencijalno kvadratično programiranje (SQP) koristimo u rješavanju nelinearnih problema optimizacije, na način da dani problem aproksimiramo pripadnom kvadratičnom funkcijom, kao i ograničenja početnog problema. Matematički model koji se pokazao najboljim je upravo Newtonowa metoda, koja iterativnom metodom pokušava doći do rješenja problema. U prvom poglavlju smo se upoznali sa kvadratičnom funkcijom cilja $f(x)=\frac{1}{2}x^{T}Ax-b^{T}x$, te općenito problemom bezuvjetne minimizacije... More kvadratične funkcije. Bitnom strukturom kvadratične funkcije su se upravo pokazale strogo konveksne kvadratične funkcije, jer su nam garantirale postojanje ekstrema. U drugom poglavlju radimo istu analizu ali uz uvjete tipa jednakosti, nejednakosti te smo promotrili i slučaj u kojemu su uvjeti određeni sa oba tipa uvjeta. U analizi problema najkorisnijom se pokazala Lagrangeova funkcija $L_{0}$ koja problematiku početne funkcije cilja svodi upravo na analizu Lagrangeove funkcije. Upravo analiza rješivosti, problema kvadratičnih funkcija, provedena u prva dva poglavlja nam je poslužila kao uvod za analizu našeg problema sekvencijalnog kvadratičnog programiranja, kojime smo se bavili u trećem poglavlju. Uz definicije, dali smo i bitne teoreme konvergerncija koji opravdavaju primjenu Newtonove metode rješavanja. Less

Abstract (english)

Sequential quadratic programming (SQP) is used in solving nonlinear optimization problems, in a way that the basic problem is approximated by associated quadratic function, as well as the constrains of the initial problem. A mathematical model that has proven to be one of the best is Newton method, iterative method that tries to find solutions of the basic problem. In the first chapter we observe quadratic objective cost function $f(x)=\frac{1}{2}x^{T}Ax-b^{T}x$ and its connection with... More unconstrained quadratic programming. Strictly convex quadratic functions has shown to be essential structure of unconstrained quadratic programming problem, because it guarantees the existence of extremes. In the second chapter we do the same analysis but with equality constrains, inequality constrains, and we have also observed case in which the constrains are specified with both equality and inequality constrains. Analyzing the problem, Lagrange function $L_{0}$ has shown to be most useful tool because it issues of the initial objective function reduce to the analysis of the Lagrange function. Solvability of analysis, of the quadratic function problem, implemented in the first two chapters, we took as a prelude to analyze our problems of sequential quadratic programming, which we’ve shown in third chapter. Beside definitions, we have shown important theorems of convergency that justifies the application of Newton’s method. Less

• Keywords: sekvencijalno kvadratično programiranje; SQP; Newtonowa metoda; Lagrangeova funkcija
• Keywords (english): sequential quadratic programming; SQP; Newton method; Lagrange function
• Language: croatian
• URN:NBN: urn:nbn:hr:217:167752
• 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: Access restricted to students and staff of home institution
• Created on: 2019-01-07 12:31:07