Polynomial optimization is concerned with computing the minimum value of a polynomial on a basic semialgebraic set. A well-known methodology is to apply positivity certificates (representations of polynomials positive on basic semialgebraic sets) to design a hierarchy of convex relaxations to solve polynomial optimization problems (POPs). Developed originally by Lasserre in [25], the hierarchy of semidefinite relaxations based on Putinar's Positivstellensatz is called the Moment-SOS hierarchy. We utilize this approach in many applications arising from optimization, operations research, signal processing, computational geometry, probability, statistics, control, PDEs, quantum information, and computer vision. For more details, the interested reader is referred to, e.g., [56, 53, 60, 48, 47, 9, 7, 51, 36, 50] and references therein. However, despite its theoretical efficiency (also observed in practice), the Moment-SOS hierarchy is facing a scalability issue mainly due to the increasing size of the resulting relaxations. Overcoming the scalability and efficiency issues has become a major scientific challenge in polynomial optimization. Many recent efforts in this direction are mainly developed around the following ideas: 1. SDP-relaxations variants with small maximal matrix size solved efficiently by interior point methods. This includes correlative sparsity [54, 26], term sparsity [58, 57, 59], symmetry exploitation [17, 43], Jordan symmetry reduction [6], sublevel relaxations [8]. 2. Exploit low-rank structures of SDP-relaxations; see, e.g., [61, 62]. 3. First-order methods to solve SDP-relaxations involving matrix variables of potentially large size with constant trace [33, 30].

Rank aggregation with pairwise comparisons has shown promising results in elections, sports competitions, recommendations, and information retrieval. However, little attention has been paid to the security issue of such algorithms, in contrast to numerous research work on the computational and statistical characteristics. Driven by huge profits, the potential adversary has strong motivation and incentives to manipulate the ranking list. Meanwhile, the intrinsic vulnerability of the rank aggregation methods is not well studied in the literature. To fully understand the possible risks, we focus on the purposeful adversary who desires to designate the aggregated results by modifying the pairwise data in this paper. From the perspective of the dynamical system, the attack behavior with a target ranking list is a fixed point belonging to the composition of the adversary and the victim. To perform the targeted attack, we formulate the interaction between the adversary and the victim as a game-theoretic framework consisting of two continuous operators while Nash equilibrium is established. Then two procedures against HodgeRank and RankCentrality are constructed to produce the modification of the original data. Furthermore, we prove that the victims will produce the target ranking list once the adversary masters the complete information. It is noteworthy that the proposed methods allow the adversary only to hold incomplete information or imperfect feedback and perform the purposeful attack. The effectiveness of the suggested target attack strategies is demonstrated by a series of toy simulations and several real-world data experiments. These experimental results show that the proposed methods could achieve the attacker's goal in the sense that the leading candidate of the perturbed ranking list is the designated one by the adversary.

Portfolio management is a multi-period multi-objective optimisation problem subject to a wide range of constraints. However, in practice, portfolio management is treated as a single-period problem partly due to the computationally burdensome hyper-parameter search procedure needed to construct a multi-period Pareto frontier. This study presents the \gls{ParDen-Sur} modelling framework to efficiently perform the required hyper-parameter search. \gls{ParDen-Sur} extends previous surrogate frameworks by including a reservoir sampling-based look-ahead mechanism for offspring generation in \glspl{EA} alongside the traditional acceptance sampling scheme. We evaluate this framework against, and in conjunction with, several seminal \gls{MO} \glspl{EA} on two datasets for both the single- and multi-period use cases. Our results show that \gls{ParDen-Sur} can speed up the exploration for optimal hyper-parameters by almost $2\times$ with a statistically significant improvement of the Pareto frontiers, across multiple \glspl{EA}, for both datasets and use cases.

The ability to accelerate the design of biological sequences can have a substantial impact on the progress of the medical field. The problem can be framed as a global optimization problem where the objective is an expensive black-box function such that we can query large batches restricted with a limitation of a low number of rounds. Bayesian Optimization is a principled method for tackling this problem. However, the astronomically large state space of biological sequences renders brute-force iterating over all possible sequences infeasible. In this paper, we propose MetaRLBO where we train an autoregressive generative model via Meta-Reinforcement Learning to propose promising sequences for selection via Bayesian Optimization. We pose this problem as that of finding an optimal policy over a distribution of MDPs induced by sampling subsets of the data acquired in the previous rounds. Our in-silico experiments show that meta-learning over such ensembles provides robustness against reward misspecification and achieves competitive results compared to existing strong baselines.

Distributed pose graph optimization (DPGO) is one of the fundamental techniques of swarm robotics. Currently, the sub-problems of DPGO are built on the native poses. Our validation proves that this approach may introduce an imbalance in the sizes of the sub-problems in real-world scenarios, which affects the speed of DPGO optimization, and potentially increases communication requirements. In addition, the coherence of the estimated poses is not guaranteed when the robots in the swarm fail, or partial robots are disconnected. In this paper, we propose BDPGO, a balanced distributed pose graph optimization framework using the idea of decoupling the robot poses and DPGO. BDPGO re-distributes the poses in the pose graph to the robot swarm in a balanced way by introducing a two-stage graph partitioning method to build balanced subproblems. Our validation demonstrates that BDPGO significantly improves the optimization speed without changing the specific algorithm of DPGO in realistic datasets. What's more, we also validate that BDPGO is robust to robot failure, changes in the wireless network. BDPGO has capable of keeps the coherence of the estimated poses in these situations. The framework also has the potential to be applied to other collaborative simultaneous localization and mapping (CSLAM) problems involved in distributedly solving the factor graph.

We propose a framework for the configuration and operation of expensive-to-evaluate advanced manufacturing methods, based on Bayesian optimization. The framework unifies a tailored acquisition function, a parallel acquisition procedure, and the integration of process information providing context to the optimization procedure. \cmtb{The novel acquisition function is demonstrated, analyzed and compared on state-of-the-art benchmarking problems. We apply the optimization approach to atmospheric plasma spraying and fused deposition modeling.} Our results demonstrate that the proposed framework can efficiently find input parameters that produce the desired outcome and minimize the process cost.

The comparison of local characteristics of two random processes can shed light on periods of time or space at which the processes differ the most. This paper proposes a method that learns about regions with a certain volume, where the marginal attributes of two processes are less similar. The proposed methods are devised in full generality for the setting where the data of interest are themselves stochastic processes, and thus the proposed method can be used for pointing out the regions of maximum dissimilarity with a certain volume, in the contexts of functional data, time series, and point processes. The parameter functions underlying both stochastic processes of interest are modeled via a basis representation, and Bayesian inference is conducted via an integrated nested Laplace approximation. The numerical studies validate the proposed methods, and we showcase their application with case studies on criminology, finance, and medicine.

We study the feature-based newsvendor problem, in which a decision-maker has access to historical data consisting of demand observations and exogenous features. In this setting, we investigate feature selection, aiming to derive sparse, explainable models with improved out-of-sample performance. Up to now, state-of-the-art methods utilize regularization, which penalizes the number of selected features or the norm of the solution vector. As an alternative, we introduce a novel bilevel programming formulation. The upper-level problem selects a subset of features that minimizes an estimate of the out-of-sample cost of ordering decisions based on a held-out validation set. The lower-level problem learns the optimal coefficients of the decision function on a training set, using only the features selected by the upper-level. We present a mixed integer linear program reformulation for the bilevel program, which can be solved to optimality with standard optimization solvers. Our computational experiments show that the method accurately recovers ground-truth features already for instances with a sample size of a few hundred observations. In contrast, regularization-based techniques often fail at feature recovery or require thousands of observations to obtain similar accuracy. Regarding out-of-sample generalization, we achieve improved or comparable cost performance.

Many scenarios where agents with restrictions compete for resources can be cast as maximum matching problems on bipartite graphs. Our focus is on resource allocation problems where agents may have restrictions that make them incompatible with some resources. We assume that a Principle chooses a maximum matching randomly so that each agent is matched to a resource with some probability. Agents would like to improve their chances of being matched by modifying their restrictions within certain limits. The Principle's goal is to advise an unsatisfied agent to relax its restrictions so that the total cost of relaxation is within a budget (chosen by the agent) and the increase in the probability of being assigned a resource is maximized. We establish hardness results for some variants of this budget-constrained maximization problem and present algorithmic results for other variants. We experimentally evaluate our methods on synthetic datasets as well as on two novel real-world datasets: a vacation activities dataset and a classrooms dataset.

Model explainability has become an important problem in machine learning (ML) due to the increased effect that algorithmic predictions have on humans. Explanations can help users understand not only why ML models make certain predictions, but also how these predictions can be changed. In this thesis, we examine the explainability of ML models from three vantage points: algorithms, users, and pedagogy, and contribute several novel solutions to the explainability problem.