Browsing by Subject "Robust Optimization"

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    Optimization of Radiation Dosing Schedules
    (2017-01) Badri, Hamidreza
    Cancer is the second leading cause of death in the United States, claiming the lives of more than half a million people every year. Cancer is aggressively treated with surgery, chemotherapy, and radiotherapy. The primary focus of this thesis is to assist clinicians with hypothesis generation to design novel radiotherapy and chemoradiotherapy fractionation schemes that can improve the results of current clinical practices. We find solutions for some important questions in radiotherapy and chemotherapy fractionation problem. Chapter 2 extends the model developed in the literature to consider radiotherapy fractionated schedules in glioblastomas to best minimize toxicity arising in early- and late-responding tissues. To this end, we decomposed the problem into two separate solvable optimization tasks: optimal radiation schedule or the amount of radiation dose per fraction and optimization of the amount of time that passes between radiation doses. Chapter 3 proposes a method for determining the optimal fractionation in the presence of uncertainties in model parameters. We formulated our problem as a conservative model using robust optimization and a risk adjusted probabilistic formulation. A variable transformation and branch and bound algorithm is implemented to find the optimal regimen. Chapter 4 considers the radiotherapy fractionation problem with a new objective: minimizing production of metastatic cancer cells while keeping normal tissue damage below an acceptable level. A dynamic programming (DP) framework is utilized to determine the optimal fractionation scheme. In Chapter 5, we introduce a mathematical model to obtain optimal drug and radiation protocols in a chemoradiotherapy scheduling problem with two objectives: minimizing metastatic cancer cell populations at multiple potential sites and maintaining a minimum level of control to the primary tumor site. We derive closed-form expressions for optimal chemotherapy fractionation regimens in some special cases. A DP framework is used to determine the optimal radiotherapy fractionation regimen. Using discretization approach, the exact solution of the resulting DP algorithm is computationally intractable. We design efficient DP data structure and use some structural properties of the optimal solution to reduce the complexity of the resulting DP algorithm. In all chapters, we performed substantial numerical experiments to validate our results.
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    Robust Solutions for Geographic Resource Allocation Problems
    (2016-08) Behroozi, Mehdi
    This thesis describes different ways to use robust optimization concepts and techniques in problems that arise in geographic resource allocation. Many problems in geographic resource allocation deal with uncertainty just like many other domains. Some of them, like the k-centers problem, are naturally defined in a {\em minimax} fashion, and some others can be treated under uncertainty where we seek robust solutions. Most geographic resource allocation problems can be settled in one or more of the categories, such as location problems, segmentation (partitioning) problems, assignment problems, routing problems, and backbone network design problems. In all such problems there are parameters that can be unknown in practice. It is sensible to define an uncertainty set for the unknown parameter based on some crude knowledge about that unknown parameter and then to treat the uncertainty like some deterministic variability of the values of the parameter, followed by ultimately solving the problem as that parameter is another variable in a higher dimension. For such problems in geographic resource allocation, we take a robust approach to tackle the uncertainty. Depending on the problem and also geometry of the uncertainty set, the robust optimization model can be tractable or difficult to solve. We deal with both cases in this thesis where we combine elements from computational geometry, geometric probability theory, vector space optimization, and topology, to either solve the problem to optimality or develop fast algorithms to settle with an approximation solution. We also present a divide and conquer type of approach using geometric partitioning to solve robust optimization problems. In a generic robust optimization problem, if the uncertainty set is an infinite set (which is the case in most practical situations), then we will have an infinite or semi-infinite dimensional optimization problem since the model will have infinite number of constraints. We describe the drawbacks of the current approaches to solving such problems and their inability to obtain reasonable solutions for some special but common and practical cases, like clustered data, and then we show that our approach makes these problems easy to solve.