Matematik

## Examensarbete i matematik på grundnivå med inriktning mot optimeringslära och systemteori

### 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 VT21 kommer vi att erbjuda följande projekt

• Learning by imitation using machine learning
• Optimal control, reinforcement learning and other optimization-based methods have been widely applied in the domain of robotic manipulation. Compared with other methods such as those based on potential functions, optimization helps resolving the control redundancy and improving task repeatability. However, in many robotics problems with complicated tasks, defining a cost function that can be optimized effectively and encodes the correct task can be challenging in practice. It is always much easier to provide demonstrations of optimal behavior such as human motions. In this project, we study the learning by imitation problem to design controllers for manipulators to mimic human behaviors. Inverse reinforcement learning could be applied to underly intentions of observed demonstrations, thus allowing the synthesis of optimal control in new environments. Other issues that might be studied also include easing the data collection process by considering generative adversarial networks.
• Traffic Planning via Multi-Commodity Network Flow
• In a future with self-driving cars, an important challenge is to coordinate traffic on the streets. A central control system could be used to plan the routes for all cars in order to steer them to their destinations, while optimizing the overall traffic distribution in the street network. The aim of this project is to model traffic in an urban environment as a network flow problem. Each individual car in the environment has a specified point of origin and destination. This can be described by different commodities in the network. The first focus of the project is to formulate the traffic planning problem as a minimum-cost multi-commodity network flow problem. This will lead to a large linear program, and the second focus is to apply different algorithms to solve it.
• Stochastic knapsack problem and its applications
• The knapsack problem can be formally defined as follows: We are given an instance of the knapsack problem with item set , consisting of items with profit and weight , and the capacity value . Then the objective is to select a subset of such that the total profit of the selected items is maximized and the total weight does not exceed . In some practical applications, the profit and weight can be random. In this case, a stochastic approach can be proposed. There is a wide range of real-life applications of the knapsack problem especially in the following topics: transportation, finance, e.g. the purchase of commodities or stocks with a limited budged, schedule planning. One simple example of a one-dimensional knapsack problem can be described as: which boxes should be chosen to maximize the amount of money while still keeping the overall weight under or equal to a given value? A multiple constrained problem could consider both the weight and volume of the boxes. In this project, we will propose a stochastic model for the knapsack problem with random parameters. When the precise information about the random parameters cannot be obtained, a distributionally robust approach will also be taken into consideration. Based on the proposed stochastic model, some practical problems, such as capital budgeting problem and wireless network problem, will be considered.
• Modeling and analysis of immunity in epidemics
• Compartmental models, such as the SIR models, have been widely used for epidemic modeling. These models can be used also for studying the effects of temporary immunity and vaccination strategies. For predicting long term effects of epidemics, such as the current covid19 outbreak, it is important to evaluate the implications of different modeling assumptions about immunity and vaccination strategies. Studying different scenarios and understanding the dynamics involved can help to improve decision making.
• Ultimate pit problem
• A fundamental problem in open-pit mining is the determination of the ultimate pit (UP), which consists in finding the contour of the mine that maximizes the difference between profits obtained from minerals minus extraction costs. The problem is formulated on a representation of the mine into blocks of a certain size, taking into account engineering requirements such as slope and precedence constraints. The UP problem gives an estimate of the total mineral that can be potentially extracted from an economical point of view. Also, its solution allows the generation of the so-called nested pits: by solving the problem for different prices (or revenue factors ) of the mineral it is possible to obtain a sequence of pits such that higher prices generate larger pits that contain the previous ones. This project aims to apply a new method in this problem, such as stochastic programming, and evaluate the outcome to the classical approach. The first step will be to create the model in a deterministic context, and test its outcome against a stochastic version of the model.
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: Xiaoming Hu

### 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.
 Sidansvarig: Xiaoming Hu