389 research outputs found

    Novel Representations of Semialgebraic Sets Arising in Planning and Control

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    The mathematical notion of a set arises frequently in planning and control of autonomous systems. A common challenge is how to best represent a given set in a manner that is efficient, accurate, and amenable to computational tools of interest. For example, ensuring a vehicle does not collide with an obstacle can be generically posed in multiple ways using techniques from optimization or computational geometry. However these representations generally rely on executing algorithms instead of evaluating closed-form expressions. This presents an issue when we wish to represent an obstacle avoidance condition within a larger motion planning problem which is solved using nonlinear optimization. These tools generally can only accept smooth, closed-form expressions. As such our available representations of obstacle avoidance conditions, while accurate, are not amenable to the relevant tools. A related problem is how to represent a set in a compact form without sacrificing accuracy. For example, we may be presented with point-cloud data representing the boundary of an object that our vehicle must avoid. Using the obstacle avoidance conditions directly on the point-cloud data would require performing these calculations with respect to each point individually. A more efficient approach is to first approximate the data with simple geometric shapes and perform later analysis with the approximation. Common shapes include bounding boxes, ellipsoids, and superquadrics. These shapes are convenient in that they have a compact representation and we have good heuristic objectives for fitting the data. However, their primitive nature means accuracy of representation may suffer. Most notably, their inherent symmetry makes them ill-suited for representing asymmetric shapes. In theory we could consider more complicated shapes given by an implicit function. However we lack reliable methods for ensuring a good fit. This thesis proposes novel approaches to these problems based on tools from convex optimization and convex analysis. Throughout, the sets of interest are described by polynomial inequalities, making them semialgebraic

    Flag Aggregator: Scalable Distributed Training under Failures and Augmented Losses using Convex Optimization

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    Modern ML applications increasingly rely on complex deep learning models and large datasets. There has been an exponential growth in the amount of computation needed to train the largest models. Therefore, to scale computation and data, these models are inevitably trained in a distributed manner in clusters of nodes, and their updates are aggregated before being applied to the model. However, a distributed setup is prone to Byzantine failures of individual nodes, components, and software. With data augmentation added to these settings, there is a critical need for robust and efficient aggregation systems. We define the quality of workers as reconstruction ratios ∈(0,1]\in (0,1], and formulate aggregation as a Maximum Likelihood Estimation procedure using Beta densities. We show that the Regularized form of log-likelihood wrt subspace can be approximately solved using iterative least squares solver, and provide convergence guarantees using recent Convex Optimization landscape results. Our empirical findings demonstrate that our approach significantly enhances the robustness of state-of-the-art Byzantine resilient aggregators. We evaluate our method in a distributed setup with a parameter server, and show simultaneous improvements in communication efficiency and accuracy across various tasks. The code is publicly available at https://github.com/hamidralmasi/FlagAggregato

    On Tractable Convex Relaxations of Standard Quadratic Optimization Problems under Sparsity Constraints

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    Standard quadratic optimization problems (StQPs) provide a versatile modelling tool in various applications. In this paper, we consider StQPs with a hard sparsity constraint, referred to as sparse StQPs. We focus on various tractable convex relaxations of sparse StQPs arising from a mixed-binary quadratic formulation, namely, the linear optimization relaxation given by the reformulation-linearization technique, the Shor relaxation, and the relaxation resulting from their combination. We establish several structural properties of these relaxations in relation to the corresponding relaxations of StQPs without any sparsity constraints, and pay particular attention to the rank-one feasible solutions retained by these relaxations. We then utilize these relations to establish several results about the quality of the lower bounds arising from different relaxations. We also present several conditions that ensure the exactness of each relaxation.Comment: Technical Report, School of Mathematics, The University of Edinburgh, Edinburgh, EH9 3FD, Scotland, United Kingdo

    Robust Transceiver Design for Covert Integrated Sensing and Communications With Imperfect CSI

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    We propose a robust transceiver design for a covert integrated sensing and communications (ISAC) system with imperfect channel state information (CSI). Considering both bounded and probabilistic CSI error models, we formulate worst-case and outage-constrained robust optimization problems of joint trasceiver beamforming and radar waveform design to balance the radar performance of multiple targets while ensuring communications performance and covertness of the system. The optimization problems are challenging due to the non-convexity arising from the semi-infinite constraints (SICs) and the coupled transceiver variables. In an effort to tackle the former difficulty, S-procedure and Bernstein-type inequality are introduced for converting the SICs into finite convex linear matrix inequalities (LMIs) and second-order cone constraints. A robust alternating optimization framework referred to alternating double-checking is developed for decoupling the transceiver design problem into feasibility-checking transmitter- and receiver-side subproblems, transforming the rank-one constraints into a set of LMIs, and verifying the feasibility of beamforming by invoking the matrix-lifting scheme. Numerical results are provided to demonstrate the effectiveness and robustness of the proposed algorithm in improving the performance of covert ISAC systems

    Geometric optimization problems in quantum computation and discrete mathematics: Stabilizer states and lattices

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    This thesis consists of two parts: Part I deals with properties of stabilizer states and their convex hull, the stabilizer polytope. Stabilizer states, Pauli measurements and Clifford unitaries are the three building blocks of the stabilizer formalism whose computational power is limited by the Gottesman- Knill theorem. This model is usually enriched by a magic state to get a universal model for quantum computation, referred to as quantum computation with magic states (QCM). The first part of this thesis will investigate the role of stabilizer states within QCM from three different angles. The first considered quantity is the stabilizer extent, which provides a tool to measure the non-stabilizerness or magic of a quantum state. It assigns a quantity to each state roughly measuring how many stabilizer states are required to approximate the state. It has been shown that the extent is multiplicative under taking tensor products when the considered state is a product state whose components are composed of maximally three qubits. In Chapter 2, we will prove that this property does not hold in general, more precisely, that the stabilizer extent is strictly submultiplicative. We obtain this result as a consequence of rather general properties of stabilizer states. Informally our result implies that one should not expect a dictionary to be multiplicative under taking tensor products whenever the dictionary size grows subexponentially in the dimension. In Chapter 3, we consider QCM from a resource theoretic perspective. The resource theory of magic is based on two types of quantum channels, completely stabilizer preserving maps and stabilizer operations. Both classes have the property that they cannot generate additional magic resources. We will show that these two classes of quantum channels do not coincide, specifically, that stabilizer operations are a strict subset of the set of completely stabilizer preserving channels. This might have the consequence that certain tasks which are usually realized by stabilizer operations could in principle be performed better by completely stabilizer preserving maps. In Chapter 4, the last one of Part I, we consider QCM via the polar dual stabilizer polytope (also called the Lambda-polytope). This polytope is a superset of the quantum state space and every quantum state can be written as a convex combination of its vertices. A way to classically simulate quantum computing with magic states is based on simulating Pauli measurements and Clifford unitaries on the vertices of the  Lambda-polytope. The complexity of classical simulation with respect to the polytope   is determined by classically simulating the updates of vertices under Clifford unitaries and Pauli measurements. However, a complete description of this polytope as a convex hull of its vertices is only known in low dimensions (for up to two qubits or one qudit when odd dimensional systems are considered). We make progress on this question by characterizing a certain class of operators that live on the boundary of the  Lambda-polytope when the underlying dimension is an odd prime. This class encompasses for instance Wigner operators, which have been shown to be vertices of  Lambda. We conjecture that this class contains even more vertices of  Lambda. Eventually, we will shortly sketch why applying Clifford unitaries and Pauli measurements to this class of operators can be efficiently classically simulated. Part II of this thesis deals with lattices. Lattices are discrete subgroups of the Euclidean space. They occur in various different areas of mathematics, physics and computer science. We will investigate two types of optimization problems related to lattices. In Chapter 6 we are concerned with optimization within the space of lattices. That is, we want to compare the Gaussian potential energy of different lattices. To make the energy of lattices comparable we focus on lattices with point density one. In particular, we focus on even unimodular lattices and show that, up to dimension 24, they are all critical for the Gaussian potential energy. Furthermore, we find that all n-dimensional even unimodular lattices with n   24 are local minima or saddle points. In contrast in dimension 32, there are even unimodular lattices which are local maxima and others which are not even critical. In Chapter 7 we consider flat tori R^n/L, where L is an n-dimensional lattice. A flat torus comes with a metric and our goal is to approximate this metric with a Hilbert space metric. To achieve this, we derive an infinite-dimensional semidefinite optimization program that computes the least distortion embedding of the metric space R^n/L into a Hilbert space. This program allows us to make several interesting statements about the nature of least distortion embeddings of flat tori. In particular, we give a simple proof for a lower bound which gives a constant factor improvement over the previously best lower bound on the minimal distortion of an embedding of an n-dimensional flat torus. Furthermore, we show that there is always an optimal embedding into a finite-dimensional Hilbert space. Finally, we construct optimal least distortion embeddings for the standard torus R^n/Z^n and all 2-dimensional flat tori

    Wafer Stage Motion Control:from Experiment Design to Robust Performance

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    Optimization under uncertainty and risk: Quadratic and copositive approaches

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    Robust optimization and stochastic optimization are the two main paradigms for dealing with the uncertainty inherent in almost all real-world optimization problems. The core principle of robust optimization is the introduction of parameterized families of constraints. Sometimes, these complicated semi-infinite constraints can be reduced to finitely many convex constraints, so that the resulting optimization problem can be solved using standard procedures. Hence flexibility of robust optimization is limited by certain convexity requirements on various objects. However, a recent strain of literature has sought to expand applicability of robust optimization by lifting variables to a properly chosen matrix space. Doing so allows to handle situations where convexity requirements are not met immediately, but rather intermediately. In the domain of (possibly nonconvex) quadratic optimization, the principles of copositive optimization act as a bridge leading to recovery of the desired convex structures. Copositive optimization has established itself as a powerful paradigm for tackling a wide range of quadratically constrained quadratic optimization problems, reformulating them into linear convex-conic optimization problems involving only linear constraints and objective, plus constraints forcing membership to some matrix cones, which can be thought of as generalizations of the positive-semidefinite matrix cone. These reformulations enable application of powerful optimization techniques, most notably convex duality, to problems which, in their original form, are highly nonconvex. In this text we want to offer readers an introduction and tutorial on these principles of copositive optimization, and to provide a review and outlook of the literature that applies these to optimization problems involving uncertainty

    Algorithmic Regularization in Tensor Optimization: Towards a Lifted Approach in Matrix Sensing

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    Gradient descent (GD) is crucial for generalization in machine learning models, as it induces implicit regularization, promoting compact representations. In this work, we examine the role of GD in inducing implicit regularization for tensor optimization, particularly within the context of the lifted matrix sensing framework. This framework has been recently proposed to address the non-convex matrix sensing problem by transforming spurious solutions into strict saddles when optimizing over symmetric, rank-1 tensors. We show that, with sufficiently small initialization scale, GD applied to this lifted problem results in approximate rank-1 tensors and critical points with escape directions. Our findings underscore the significance of the tensor parametrization of matrix sensing, in combination with first-order methods, in achieving global optimality in such problems.Comment: NeurIPS23 Poste

    Sparse Plus Low Rank Matrix Decomposition: A Discrete Optimization Approach

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    We study the Sparse Plus Low-Rank decomposition problem (SLR), which is the problem of decomposing a corrupted data matrix into a sparse matrix of perturbations plus a low-rank matrix containing the ground truth. SLR is a fundamental problem in Operations Research and Machine Learning which arises in various applications, including data compression, latent semantic indexing, collaborative filtering, and medical imaging. We introduce a novel formulation for SLR that directly models its underlying discreteness. For this formulation, we develop an alternating minimization heuristic that computes high-quality solutions and a novel semidefinite relaxation that provides meaningful bounds for the solutions returned by our heuristic. We also develop a custom branch-and-bound algorithm that leverages our heuristic and convex relaxations to solve small instances of SLR to certifiable (near) optimality. Given an input nn-by-nn matrix, our heuristic scales to solve instances where n=10000n=10000 in minutes, our relaxation scales to instances where n=200n=200 in hours, and our branch-and-bound algorithm scales to instances where n=25n=25 in minutes. Our numerical results demonstrate that our approach outperforms existing state-of-the-art approaches in terms of rank, sparsity, and mean-square error while maintaining a comparable runtime

    Sorta Solving the OPF by Not Solving the OPF: DAE Control Theory and the Price of Realtime Regulation

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    This paper presents a new approach to solve or approximate the AC optimal power flow (ACOPF). By eliminating the need to solve the ACOPF every few minutes, the paper showcases how a realtime feedback controller can be utilized in lieu of ACOPF and its variants. By \textit{(i)} forming the grid dynamics as a system of differential algebraic equations (DAE) that naturally encode the non-convex OPF power flow constraints, \textit{(ii)} utilizing advanced DAE-Lyapunov theory, and \textit{(iii)} designing a feedback controller that captures realtime uncertainty while being uncertainty-unaware, the presented approach demonstrates promises of obtaining solutions that are close to the OPF ones without needing to solve the OPF. The proposed controller responds in realtime to deviations in renewables generation and loads, guaranteeing transient stability, while always yielding feasible solutions of the ACOPF with no constraint violations. As the studied approach herein indeed yields slightly more expensive realtime generator controls, the corresponding price of realtime control and regulation is examined. Cost-comparisons with the traditional ACOPF are also showcased -- all via case studies on standard power networks
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