Applications of optimization in some complex systems
Date
2018-07-02
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Abstract
Matemaatiline optimeerimine on optimeerimisülesandele optimaalse lahendi leidmine, see tähendab, et leitakse reaalarvuliste väärtustega funktsiooni (eesmärkfunktsiooni) teatud kitsendusi rahuldav maksimaalne või minimaalne väärtus. Matemaatiline optimeerimine on jagatud eesmärkfunktsiooni ja kitsenduste põhjal erinevatesse klassidesse ja need klassid erinevad arvutusliku keerukuse poolest. Praktikas on vaja nende ülesannetega toime tulemiseks robustseid ja kiireid algoritme. Optimeerimisülesanded, mis kuuluvad polünomiaalse keerukusega klassi saab lahendada otseselt. Erinevate optimeerimisülesannete lahendamiseks on olemas algoritmid, eelkõige polünomiaalse keerukusega klassi jaoks. Mõnedel juhtudel pole isegi efektiivsed algoritmid ülesande lahendamiseks piisavalt head, näiteks liiga aeglased. Nendel juhtudel saab optimeerimisprotsessi uurimise abil kindlaks teha probleemi põhjused ja muuta algoritmi selliselt, et neid vältida. Antud töö uurib kumera optimeerimisülesande (Convex Program) lahendust, mille abil on võimalik saavutada osaline informatsiooni liigendamine (partial information decomposition) ning hiljuti kasutati seda teatava keerulise süsteemi analüüsimiseks. Kumerate optimeerimisülesannete lahendamisel tekkivate probleemidega toime tulemiseks loodi antud töös kiire ja robustne algoritm. Lisaks uuritakse antud töös situatsiooni, mis tuli hiljuti esile neuroteaduse valdkonnas, kus osaline informatsiooni liigendamise ülesanne lahendatakse kitsendustega. Viimasena uuritakse antud töös optimeerimisülesannet automatiseeritud süsteemi juhtimiseks, mis on APX keerukusega. Ülesanne modelleeritakse täisarvulise planeerimise ja kehtestatavuse kontrollimise mudelitena, mida saab edaspidi kasutada heuristikute loomiseks, et lahendada ülesanne mõistliku ajaga.
Mathematical Optimization is finding the optimal solution to an optimization problem, i.e., the maximum or minimum value of a real function (objective function) subjected to a set of constraints. Mathematical optimization is divided into classes depending on the nature of the objective function as well as the constraints and these classes vary in their computational complexity. In practice, robust and fast processes are needed to tackle the problems, and so optimization problems which can be modeled into classes of polynomial complexity are applied directly. Several algorithms that solve different optimization problems exist, in particular, efficient ones for the classes with polynomial complexity. But, in some cases, even the efficient algorithms face difficulties in solving optimization problems such as taking immensely long time to find the optimal solution. In this case, studying the optimization, in-depth, leads to the reasons behind these pitfalls and thus modify the algorithms to avoid them. This thesis studies the solution of a Convex Program which was used to obtain partial information decomposition, a tool used recently to analyze particular complex systems. Many difficulties arise in solving the Convex Program and so the study done in this thesis resulted in a fast and robust algorithm to tackle the problem. Then it studies the situation which appeared recently in neuroscience when the obtained partial information decomposition is optimized subject to some constraints. Finally, it studies a fundamental optimization problem in the control of automated systems that is APX-hard. The problem is modeled IP- and CNF-models which can be used, in future, to design good heuristics which tackles the problem in reasonable time.
Mathematical Optimization is finding the optimal solution to an optimization problem, i.e., the maximum or minimum value of a real function (objective function) subjected to a set of constraints. Mathematical optimization is divided into classes depending on the nature of the objective function as well as the constraints and these classes vary in their computational complexity. In practice, robust and fast processes are needed to tackle the problems, and so optimization problems which can be modeled into classes of polynomial complexity are applied directly. Several algorithms that solve different optimization problems exist, in particular, efficient ones for the classes with polynomial complexity. But, in some cases, even the efficient algorithms face difficulties in solving optimization problems such as taking immensely long time to find the optimal solution. In this case, studying the optimization, in-depth, leads to the reasons behind these pitfalls and thus modify the algorithms to avoid them. This thesis studies the solution of a Convex Program which was used to obtain partial information decomposition, a tool used recently to analyze particular complex systems. Many difficulties arise in solving the Convex Program and so the study done in this thesis resulted in a fast and robust algorithm to tackle the problem. Then it studies the situation which appeared recently in neuroscience when the obtained partial information decomposition is optimized subject to some constraints. Finally, it studies a fundamental optimization problem in the control of automated systems that is APX-hard. The problem is modeled IP- and CNF-models which can be used, in future, to design good heuristics which tackles the problem in reasonable time.
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Keywords
complex systems, optimization, convex sets, software