Thèse Année : 2024

Nonlinear Adjustable Robust Optimization

Khademi

Résumé

Robust Optimization (RO) is a popular approach for dealing with uncertain data in optimization. In static robust optimization, decision variables represent here-and-now decisions made without exact knowledge of uncertain parameters but must be feasible when the actual data is within the uncertainty set. However, one mechanism to overcome the limitations of the static RO approach is Adjustable Robust Optimization (ARO), which leverages adaptability. The main difference between static RO and ARO approaches is the decision-making manner. In ARO problems, some variables are here-and-now decisions, while others are wait-and-see decisions made later based on the observed parameters in the uncertainty set. In this PhD thesis, we address two main topics in mathematical optimization. The first topic concerns a class of nonlinear ARO problems with uncertainty in the objective function and constraints. By utilizing Fenchel’s duality, we derive an equivalent dual reformulation that is a nonlinear static robust optimization problem. We then apply perspective relaxation and an alternating method to handle non-concavity and design a new dual-based cutting plane algorithm that can find a reasonable lower bound for the optimal objective value. Through numerical experiments, we show the effectiveness of the cutting plane algorithm in producing locally robust solutions with an acceptable optimality gap. The second topic focuses on the reformulation of quadratic optimization problems using ARO. Quadratic Optimization (QO) has been extensively studied in the literature due to its practical applicability in numerous problems. Despite its practicality, QO problems are generally NP-hard. Consequently, researchers have developed various numerical methods for finding approximate optimal solutions. In this thesis, we analyze QO problems through the lens of robust optimization techniques. We first demonstrate that any QO problem can be reformulated as a disjoint bi-convex QO problem. Subsequently, we present an equivalent ARO reformulation and utilize some methods from the relevant literature to approximate this reformulation. Specifically, we show that employing a so-called decision rule technique to approximate the ARO reformulation can be interpreted as applying a linearization-relaxation technique to its bi-convex reformulation problem. Additionally, we have designed an algorithm capable of finding a solution that is close to optimal based on our new reformulations. Our numerical results highlight the efficiency of our algorithm, particularly for large-sized instances, in comparison with standard off-the-shelf solvers. This work offers a novel perspective on quadratic problems and paves the way for further research in this domain of mathematical optimization.
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tel-04460774 , version 1 (15-02-2024)

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Abbas Khademi. Nonlinear Adjustable Robust Optimization. Mathematics [math]. PhD thesis, University of Tehran, 2024. English. ⟨NNT : ⟩. ⟨tel-04460774⟩
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