Allmän information om examensarbetet på studentwebben

Specifik information för ämnesvalet matematik

Optimeringslära och systemteori

Optimeringslära och systemteori är ett tillämpat matematiskt ämne som omfattar teori, modeller och metoder för optimering samt systemteoretiska aspekter inom ämnen som biologi, maskinteknik, reglerteknik, robotik, och signalbehandling.

Vårens projektarbete i optimeringslära och systemteori

I VT24 kommer vi att erbjuda följande projekt (OBS listan håller på att uppdateras )

• Coverage path planning for UAVs in search missions
Coverage path planning is a well-studied problem and typically consists of two subtasks: i) use decomposition methods to split the search area into cells to simplify the coverage; ii) use optimization methods to derive an optimal coverage path that minimizes some objectives, e.g., the completion time. In this project, we in particular consider unmanned aerial vehicles (UAVs) that are subject to battery constraints. To cope with this issue, charging piles should be considered and placed inside/outside the area. Therefore, we need to determine the locations of charging piles and derive a corresponding optimal coverage path, allowing UAVs to fully search the area in the shortest possible time. More advanced scenarios include, e.g., i) The terrain of the area consists of ground and water, and charging piles can only be placed on ground. ii) Charing piles are replaced by unmanned ground/surface vehicles with charging capability to shorten the completion time, if possible.
• Hur kan optimering användas för att bidra till klimatomställningen?
Optimering används inom många områden. Ett särskilt aktuellt område just nu är klimatomställning. Arbetet syftar till att hitta konkreta exempel där optimering används för klimatomställning. Eventuellt kan någon enkel egen modell sättas upp.
• Turing's model for pattern formation and network controllability
One of the fundamental questions in developmental biology is how the vast range of patterns and structure we observe in nature emerges from an almost homogeneous fertilized egg. The spirals that originate from chemical reactions and the stripes in fish skin patterning are all examples of the intrinsic ability of seemingly different systems to yield regular structures, both in space and time. Turing's reaction-diffusion model is widely accepted as one of the most fundamental mathematical models to explain the pattern formation. In this project, we investigate the pattern evolution of Turing's reaction-diffusion model by some numerical methods. With suitable spatially discretization, Turing's model is connected to the network controllability. To generate all desired patterns, controls (injection of morphogens) are needed. We focus on determining the controllability of the network and finding as few controls as possible. The energy needed for different set of feasible controls is also compared.
• Optimering av nästa generations flygsystem
Inom några år kommer drönartaxi och olika typer av elektriska och bränslecellsdrivna flygplan att introduceras. Detta innebär att sättet vi reser på kommer att ändras. Med VTOL kommer man att kunna använda mindre flygplatser, så kallade vertiports, och normalt kommer räckvidden och antalet passagerare vara mer begränsat. Därför behöver man studera hur dessa flygsystem ska designas och vi kommer att titta på hur man kan använda optimering för att utforma dessa på bästa sätt.
• Distributed strategies for pursuit-evasion games of networked multiagent systems
Multiple autonomous unmanned systems, or simply multi-agent systems, such as unmanned aerial vehicles and unmanned ground vehicles, are projected to play important roles in many industrial and societal applications. In the community of multi-agent control, the pursuit-evasion game has been a hot topic where the pursuers and evaders have opposite objectives. The pursuers aim to minimize the distance from the evaders so that to capture them while the evaders try to move away from the pursuers. In this project, we will focus on proposing distributed strategies for networked multi-agent systems, by formulating the problem as a distributed optimization problem with a defined performance index, using the relative positions and distances, or bearing information.
• Semidefinite programming and Support vector machines
TThis project is aimed to compare the Sequential Minimal Optimization (SMO)-based methods and optimization-based methods for training the Support Vector Machine (SVM) models. SVM is a supervised machine learning problem where we try to find a hyperplane that best separates the two classes. The most commonly used optimization algorithm for training SVMs is the Sequential Minimal Optimization (SMO) algorithm. Solving Support Vector Machine (SVM) training using semidefinite programming (SDP) is an alternative approach to finding the optimal hyperplane in SVMs. SDP-based methods cast the SVM optimization problem into the framework of semidefinite programming, which is a type of convex optimization problem. These methods offer a different perspective on solving SVMs and can be useful in certain scenarios, particularly when dealing with non-linear kernels and complex SVM formulations.
The requirements of this project: 1. Select at least two dataset with different properties, e.g., suitable for SVM with linear or nonlinear kernels.
2. Use python package to train SVM.
3. Formulate the SDP problems for SVM and solve the problems by SDP solvers, e.g., Mosek.
Specific prerequisites: A basic optimization course, e.g., SF1811 or SF1861, is an advantage but not mandatory.
• Game Players Using Distributional Reinforcement Learning
Reinforcement Learning (RL) is to learn an optimal policy that maximizes the expected cumulative rewards for unknown Markov decision processes. This is achieved through a process of online trial-and-error interactions or by using pre-existing offline running datasets. The remarkable efficacy of RL algorithms has been witnessed across diverse applications, spanning from their triumph in Alpha Go and Alpha Zero to enabling advancements in autonomous driving. However, it is important to recognize that the cumulative reward, as a random variable, is not solely defined by its expected value, but characterized by the entire distribution or other statistics, such as its variance and higher-order moments. These characteristics are essential in risk-sensitive scenarios, for instance constructing a portfolio that aims for not only reasonable expected profit but also a low risk of bankruptcy. In this project, you will (1) learn and explore the RL methods that can assess the entire distribution and/or other statistics of cumulative rewards. (2) Building on this, you can train an optimal policy using objective functions that consider a broader perspective of the rewards distribution, beyond mere expectation.(3) In addition, you will implement and demonstrate the explored distributional RL methods in Python in the Atari environment which consists of 2600 arcade games.
• Portfolio Optimization: robustification and integer constraints
The task of selecting an optimal portfolio from a selection of investment opportunities is a well-studied problem and the work of Markowitz has formed the basis of modern portfolio theory. In this project we will focus on the optimization and computational aspects of selecting an optimal portfolio. Specifically, we will investigate how integer variables can be used in the optimization model to get desirable properties in the selected portfolio and how we can use duality to better account for uncertainty. For example, the task is to model the constraint that the selected portfolio is not allowed to contain more than a specific number of different assets.
The research focus is to study how different modeling techniques impacts the computational challenges (mainly computation time) of determining an optimal portfolio. In the project you will work in Python, and you will learn how to model portfolio optimization problems and use state-of-the-art optimization software.
A basic optimization course, e.g., SF1811 or SF1861, is strongly advised but not absolutely mandatory.
• Origin destination problem for traffic control
In typical flow problems a distribution of particles or agents is transported to another given distribution. However, in many situations, such as traffic control, it is not only the distribution that needs to be matched, but each agent has a desired destination. This type of problems are called origin destination problems. In this project we will study this type of network flow problems and consider methods different methods for solving it based on linear and nonlinear programming.

Flera av projekten relaterar till befintlig forskning inom avdelningen och det finns i Stockholmsområdet ledande industri och forskningsföretag inom dessa tillämpningsområden. Andra projekt behandlar grundläggande matematiska problem inom ämnet vilka kan tillämpas inom många områden.

Projekten skall normalt genomföras i grupper om två studenter men det är även möjligt att arbeta individuellt. I en del projekt kan det finnas flera (2 eller 3) delprojekt. Samtliga projekt har en inläsningsdel och en problemlösningsdel. Inläsningsdelen är gemensam för alla grupperna i varje projekt medan problemlösningsdelen skall utföras självständigt inom de olika delprojekten.

Inläsningsdel

Projekten inleds med en inläsningsdel under LP3. Inläsningen av ämnet sker i form av en informell lärarledd studiecirkel, där deltagarna hjälps åt att lära sig den nödvändiga teorin. Denna delen av kursen avslutas med att studenterna i varje delprojekt presenterar sitt delproblem och sin arbetsplan för projektet. Detaljerna kring upplägget varierar lite grand mellan de olika projekten.

Problemlösningsdel

Problemlösningen utförs i huvudsak under LP4. Här skall grupperna självständigt arbeta med sina problem. Normalt träffas gruppen och lärarna en gång per vecka för att diskutera projektens status.

Kontaktpersoner

För frågor angående inriktningen mot optimeringslära och systemteori: Jan Kronqvist

Ungefärlig tidsplan för projektarbetet

• Januari: Arbetetet påbörjas med inläsning.
• Början av mars: Projektformuleringar och arbetsplan ska finnas färdiga.
• Mitten/slutet av mars: Studenten lämnar disposition och skelett till handledaren.
• Början av maj: Rapport lämnas till handledaren för granskning.
• Mitten/slutet av maj: Redovisning, plagiatgranskning och betygssättning.