Showing papers on "Convex optimization published in 2020"

Tensor Robust Principal Component Analysis with a New Tensor Nuclear Norm

Abstract: In this paper, we consider the Tensor Robust Principal Component Analysis (TRPCA) problem, which aims to exactly recover the low-rank and sparse components from their sum. Our model is based on the recently proposed tensor-tensor product (or t-product) [14] . Induced by the t-product, we first rigorously deduce the tensor spectral norm, tensor nuclear norm, and tensor average rank, and show that the tensor nuclear norm is the convex envelope of the tensor average rank within the unit ball of the tensor spectral norm. These definitions, their relationships and properties are consistent with matrix cases. Equipped with the new tensor nuclear norm, we then solve the TRPCA problem by solving a convex program and provide the theoretical guarantee for the exact recovery. Our TRPCA model and recovery guarantee include matrix RPCA as a special case. Numerical experiments verify our results, and the applications to image recovery and background modeling problems demonstrate the effectiveness of our method.

Momentum and stochastic momentum for stochastic gradient, Newton, proximal point and subspace descent methods

Abstract: In this paper we study several classes of stochastic optimization algorithms enriched with heavy ball momentum. Among the methods studied are: stochastic gradient descent, stochastic Newton, stochastic proximal point and stochastic dual subspace ascent. This is the first time momentum variants of several of these methods are studied. We choose to perform our analysis in a setting in which all of the above methods are equivalent: convex quadratic problems. We prove global non-asymptotic linear convergence rates for all methods and various measures of success, including primal function values, primal iterates, and dual function values. We also show that the primal iterates converge at an accelerated linear rate in a somewhat weaker sense. This is the first time a linear rate is shown for the stochastic heavy ball method (i.e., stochastic gradient descent method with momentum). Under somewhat weaker conditions, we establish a sublinear convergence rate for Cesaro averages of primal iterates. Moreover, we propose a novel concept, which we call stochastic momentum, aimed at decreasing the cost of performing the momentum step. We prove linear convergence of several stochastic methods with stochastic momentum, and show that in some sparse data regimes and for sufficiently small momentum parameters, these methods enjoy better overall complexity than methods with deterministic momentum. Finally, we perform extensive numerical testing on artificial and real datasets, including data coming from average consensus problems.

Adaptive Trust Region Policy Optimization: Global Convergence and Faster Rates for Regularized MDPs

Abstract: Trust region policy optimization (TRPO) is a popular and empirically successful policy search algorithm in Reinforcement Learning (RL) in which a surrogate problem, that restricts consecutive policies to be ‘close’ to one another, is iteratively solved. Nevertheless, TRPO has been considered a heuristic algorithm inspired by Conservative Policy Iteration (CPI). We show that the adaptive scaling mechanism used in TRPO is in fact the natural “RL version” of traditional trust-region methods from convex analysis. We first analyze TRPO in the planning setting, in which we have access to the model and the entire state space. Then, we consider sample-based TRPO and establish O(1/√N) convergence rate to the global optimum. Importantly, the adaptive scaling mechanism allows us to analyze TRPO in regularized MDPs for which we prove fast rates of O(1/N), much like results in convex optimization. This is the first result in RL of better rates when regularizing the instantaneous cost or reward.

Multiobjective Fault-Tolerant Control for Fuzzy Switched Systems With Persistent Dwell Time and Its Application in Electric Circuits

Abstract: This article concentrates on the output feedback controller design problem for discrete-time nonlinear switched systems with actuator faults. The Takagi–Sugeno fuzzy model is adopted to approximate the nonlinearity of the plant with a set of local linear models. The persistent dwell-time (DT) switching law, which is more general than DT or average DT switching, is introduced to govern the switching among subsystems. In order to alleviate the effects of actuator failures on system stability and performance, a synthesized fault-tolerant output feedback controller ensuring various performance requirements is designed. Intensive attention is focused on establishing sufficient conditions, which can guarantee the exponential mean-square stability as well as the prescribed extended dissipativity property of the closed-loop system. By virtue of the Lyapunov stability theory and appropriate matrix transformation methods, the desired controller gains can be obtained by solving a convex optimization problem. The developed method is finally applied to address the control issue of a tunnel diode circuit system model to illustrate its efficiency and applicability.

Online Optimization as a Feedback Controller: Stability and Tracking

Abstract: This paper develops and analyzes feedback-based online optimization methods to regulate the output of a linear time invariant (LTI) dynamical system to the optimal solution of a time-varying convex optimization problem. The design of the algorithm is based on continuous-time primal-dual dynamics, properly modified to incorporate feedback from the LTI dynamical system, applied to a proximal augmented Lagrangian function. The resultant closed-loop algorithm tracks the solution of the time-varying optimization problem without requiring knowledge of (time varying) disturbances in the dynamical system. The analysis leverages integral quadratic constraints to provide linear matrix inequality (LMI) conditions that guarantee global exponential stability and bounded tracking error. Analytical results show that under a sufficient time-scale separation between the dynamics of the LTI dynamical system and the algorithm, the LMI conditions can be always satisfied. This paper further proposes a modified algorithm that can track an approximate solution trajectory of the constrained optimization problem under less restrictive assumptions. As an illustrative example, the proposed algorithms are showcased for power transmission systems, to compress the time scales between secondary and tertiary control, and allow to simultaneously power rebalancing and tracking of the DC optimal power flow points.

Optimal Construction of Koopman Eigenfunctions for Prediction and Control

Abstract: This article presents a novel data-driven framework for constructing eigenfunctions of the Koopman operator geared toward prediction and control. The method leverages the richness of the spectrum of the Koopman operator away from attractors to construct a set of eigenfunctions such that the state (or any other observable quantity of interest) is in the span of these eigenfunctions and hence predictable in a linear fashion. The eigenfunction construction is optimization-based with no dictionary selection required. Once a predictor for the uncontrolled part of the system is obtained in this way, the incorporation of control is done through a multistep prediction error minimization, carried out by a simple linear least-squares regression. The predictor so obtained is in the form of a linear controlled dynamical system and can be readily applied within the Koopman model predictive control (MPC) framework of (M. Korda and I. Mezic, 2018) to control nonlinear dynamical systems using linear MPC tools. The method is entirely data-driven and based predominantly on convex optimization. The novel eigenfunction construction method is also analyzed theoretically, proving rigorously that the family of eigenfunctions obtained is rich enough to span the space of all continuous functions. In addition, the method is extended to construct generalized eigenfunctions that also give rise Koopman invariant subspaces and hence can be used for linear prediction. Detailed numerical examples demonstrate the approach, both for prediction and feedback control. * * Code for the numerical examples is available from https://homepages.laas.fr/mkorda/Eigfuns.zip .

Stochastic Conditional Gradient Methods: From Convex Minimization to Submodular Maximization

Abstract: This paper considers stochastic optimization problems for a large class of objective functions, including convex and continuous submodular. Stochastic proximal gradient methods have been widely used to solve such problems; however, their applicability remains limited when the problem dimension is large and the projection onto a convex set is costly. Instead, stochastic conditional gradient methods are proposed as an alternative solution relying on (i) Approximating gradients via a simple averaging technique requiring a single stochastic gradient evaluation per iteration; (ii) Solving a linear program to compute the descent/ascent direction. The averaging technique reduces the noise of gradient approximations as time progresses, and replacing projection step in proximal methods by a linear program lowers the computational complexity of each iteration. We show that under convexity and smoothness assumptions, our proposed method converges to the optimal objective function value at a sublinear rate of $O(1/t^{1/3})$. Further, for a monotone and continuous DR-submodular function and subject to a general convex body constraint, we prove that our proposed method achieves a $((1-1/e)OPT-\eps)$ guarantee with $O(1/\eps^3)$ stochastic gradient computations. This guarantee matches the known hardness results and closes the gap between deterministic and stochastic continuous submodular maximization. Additionally, we obtain $((1/e)OPT -\eps)$ guarantee after using $O(1/\eps^3)$ stochastic gradients for the case that the objective function is continuous DR-submodular but non-monotone and the constraint set is down-closed. By using stochastic continuous optimization as an interface, we provide the first $(1-1/e)$ tight approximation guarantee for maximizing a monotone but stochastic submodular set function subject to a matroid constraint and $(1/e)$ approximation guarantee for the non-monotone case.

Optimal Power Flow Using Graph Neural Networks

Abstract: Optimal power flow (OPF) is one of the most important optimization problems in the energy industry. In its simplest form, OPF attempts to find the optimal power that the generators within the grid have to produce to satisfy a given demand. Optimality is measured with respect to the cost that each generator incurs in producing this power. The OPF problem is non-convex due to the sinusoidal nature of electrical generation and thus is difficult to solve. Using small angle approximations leads to a convex problem known as DC OPF, but this approximation is no longer valid when power grids are heavily loaded. Many approximate solutions have been since put forward, but these do not scale to large power networks. In this paper, we propose using graph neural networks (which are localized, scalable parametrizations of network data) trained under the imitation learning framework to approximate a given optimal solution. While the optimal solution is costly, it is only required to be computed for network states in the training set. During test time, the GNN adequately learns how to compute the OPF solution. Numerical experiments are run on the IEEE-30 and IEEE-118 test cases.

Stability of Stochastic Gradient Descent on Nonsmooth Convex Losses

Abstract: Uniform stability is a notion of algorithmic stability that bounds the worst case change in the model output by the algorithm when a single data point in the dataset is replaced. An influential work of Hardt et al. (2016) provides strong upper bounds on the uniform stability of the stochastic gradient descent (SGD) algorithm on sufficiently smooth convex losses. These results led to important progress in understanding of the generalization properties of SGD and several applications to differentially private convex optimization for smooth losses. Our work is the first to address uniform stability of SGD on {\em nonsmooth} convex losses. Specifically, we provide sharp upper and lower bounds for several forms of SGD and full-batch GD on arbitrary Lipschitz nonsmooth convex losses. Our lower bounds show that, in the nonsmooth case, (S)GD can be inherently less stable than in the smooth case. On the other hand, our upper bounds show that (S)GD is sufficiently stable for deriving new and useful bounds on generalization error. Most notably, we obtain the first dimension-independent generalization bounds for multi-pass SGD in the nonsmooth case. In addition, our bounds allow us to derive a new algorithm for differentially private nonsmooth stochastic convex optimization with optimal excess population risk. Our algorithm is simpler and more efficient than the best known algorithm for the nonsmooth case Feldman et al. (2020).

Sparse high-dimensional regression: Exact scalable algorithms and phase transitions

Abstract: We present a novel binary convex reformulation of the sparse regression problem that constitutes a new duality perspective. We devise a new cutting plane method and provide evidence that it can solve to provable optimality the sparse regression problem for sample sizes $n$ and number of regressors $p$ in the 100,000s, that is, two orders of magnitude better than the current state of the art, in seconds. The ability to solve the problem for very high dimensions allows us to observe new phase transition phenomena. Contrary to traditional complexity theory which suggests that the difficulty of a problem increases with problem size, the sparse regression problem has the property that as the number of samples $n$ increases the problem becomes easier in that the solution recovers 100% of the true signal, and our approach solves the problem extremely fast (in fact faster than Lasso), while for small number of samples $n$, our approach takes a larger amount of time to solve the problem, but importantly the optimal solution provides a statistically more relevant regressor. We argue that our exact sparse regression approach presents a superior alternative over heuristic methods available at present.

Private Stochastic Convex Optimization: Optimal Rates in Linear Time

Abstract: We study differentially private (DP) algorithms for stochastic convex optimization: the problem of minimizing the population loss given i.i.d. samples from a distribution over convex loss functions. A recent work of Bassily et al. (2019) has established the optimal bound on the excess population loss achievable given $n$ samples. Unfortunately, their algorithm achieving this bound is relatively inefficient: it requires $O(\min\{n^{3/2}, n^{5/2}/d\})$ gradient computations, where $d$ is the dimension of the optimization problem. We describe two new techniques for deriving DP convex optimization algorithms both achieving the optimal bound on excess loss and using $O(\min\{n, n^2/d\})$ gradient computations. In particular, the algorithms match the running time of the optimal non-private algorithms. The first approach relies on the use of variable batch sizes and is analyzed using the privacy amplification by iteration technique of Feldman et al. (2018). The second approach is based on a general reduction to the problem of localizing an approximately optimal solution with differential privacy. Such localization, in turn, can be achieved using existing (non-private) uniformly stable optimization algorithms. As in the earlier work, our algorithms require a mild smoothness assumption. We also give a linear-time algorithm achieving the optimal bound on the excess loss for the strongly convex case, as well as a faster algorithm for the non-smooth case.

FedSplit: An algorithmic framework for fast federated optimization

Abstract: Motivated by federated learning, we consider the hub-and-spoke model of distributed optimization in which a central authority coordinates the computation of a solution among many agents while limiting communication. We first study some past procedures for federated optimization, and show that their fixed points need not correspond to stationary points of the original optimization problem, even in simple convex settings with deterministic updates. In order to remedy these issues, we introduce FedSplit, a class of algorithms based on operator splitting procedures for solving distributed convex minimization with additive structure. We prove that these procedures have the correct fixed points, corresponding to optima of the original optimization problem, and we characterize their convergence rates under different settings. Our theory shows that these methods are provably robust to inexact computation of intermediate local quantities. We complement our theory with some simple experiments that demonstrate the benefits of our methods in practice.

Command Filter and Universal Approximator Based Backstepping Control Design for Strict-Feedback Nonlinear Systems With Uncertainty

Abstract: This paper presents an improved backstepping control implementation scheme for a $n$ -dimensional strict-feedback uncertain nonlinear system based on command filtered backstepping and adaptive neural network backstepping. In this approach, $n$ command filters and one neural network are applied to reconstruct the approximations of unknown nonlinearities, which are related to the system uncertainties including the system's unmodeled dynamics and external disturbances. Then, one can use the negative feedback of these approximations to compensate the system uncertainties. Moreover, convex optimization and soft computing technique are adopted to design the update law of the weights of the neural network, and Lyapunov stability criterion is used to prove the stability of the closed-loop system. Finally, simulation results are given to show the effectiveness of the proposed methods.

CVXR: An R Package for Disciplined Convex Optimization

Abstract: CVXR is an R package that provides an object-oriented modeling language for convex optimization, similar to CVX, CVXPY, YALMIP, and Convex.jl. It allows the user to formulate convex optimization problems in a natural mathematical syntax rather than the restrictive form required by most solvers. The user specifies an objective and set of constraints by combining constants, variables, and parameters using a library of functions with known mathematical properties. CVXR then applies signed disciplined convex programming (DCP) to verify the problem's convexity. Once verified, the problem is converted into standard conic form using graph implementations and passed to a cone solver such as ECOS or SCS. We demonstrate CVXR's modeling framework with several applications.

Energy Efficiency Optimization for NOMA UAV Network With Imperfect CSI

Abstract: Unmanned aerial vehicles (UAVs) are developing rapidly owing to flexible deployment and access services as air base stations. However, the channel errors of low-altitude communication links formed by mobile deployment of UAVs cannot be ignored. And the energy efficiency of the UAVs communication with imperfect channel state information (CSI) hasnt been well studied yet. Therefore, we focus on system performance optimization in non-orthogonal multiple access (NOMA) UAV network considering imperfect CSI between the UAV and users. A suboptimal resource allocation scheme including user scheduling and power allocation is designed for maximizing energy efficiency. Because of the nonconvexity of optimization function with an probability constraint for imperfect CSI, the original problem is converted into a non-probability problem and then decoupled into two convex subproblems. First, a user scheduling method is applied in the two-side matching of users and subchannels by the difference of convex programming. Then based on user scheduling, the energy efficiency in UAV cells is optimized through a suboptimal power allocation algorithm by successive convex approximation method. The simulation results prove that the proposed algorithm is effective compared with existing resource allocation schemes.

An Event-Triggered Approach to Sliding Mode Control of Markovian Jump Lur’e Systems Under Hidden Mode Detections

Abstract: The asynchronous sliding mode control (SMC) problem is investigated for networked Markovian jump Lur’e systems, in which the information of system modes is unavailable to the sliding mode controller but could be estimated by a mode detector via a hidden Markov model (HMM). In order to mitigate the burden of data communication, an event-triggered protocol is proposed to determine whether the system state should be released to the controller at certain time-point according to a specific triggering condition. By constructing a novel common sliding surface, this paper designs an event-triggered asynchronous SMC law, which just depends on the hidden mode information. A combination of the stochastic Lur’e-type Lyapunov functional and the HMM approach is exploited to establish the sufficient conditions of the mean square stability with a prescribed ${H}_{\infty }$ performance and the reachability of a sliding region around the specified sliding surface. Moreover, the solving algorithm for the control gain matrices is given via a convex optimization problem. Finally, an example from the dc motor device system is provided.

The O(N) S-matrix monolith

Abstract: We consider the scattering matrices of massive quantum field theories with no bound states and a global O(N) symmetry in two spacetime dimensions. In particular we explore the space of two-to-two S-matrices of particles of mass m transforming in the vector representation as restricted by the general conditions of unitarity, crossing, analyticity and O(N) symmetry. We found a rich structure in that space by using convex maximization and in particular its convex dual minimization problem. At the boundary of the allowed space special geometric points such as vertices were found to correspond to integrable models. The dual convex minimization problem provides a novel and useful approach to the problem allowing, for example, to prove that generically the S-matrices so obtained saturate unitarity and, in some cases, that they are at vertices of the allowed space.

Robust point-to-point iterative learning control with trial-varying initial conditions

Abstract: Iterative learning control (ILC) is a high-performance technique for repeated control tasks with design postulates on a fixed reference profile and identical initial conditions. However, the tracking performance is only critical at few points in point-to-point tasks, and their initial conditions are usually trial-varying within a certain range in practice, which essentially degrades the performance of conventional ILC algorithms. Therefore, this study reformulates the ILC problem setup for point-to-point tasks and considers the effort of trial-varying initial conditions in algorithm design. To reduce the tracking error, it proposes a worst-case norm-optimal problem and reformulates it into a convex optimisation problem using the Lagrange dual approach. In this sense, a robust ILC algorithm is derived based on iteratively solving this problem. The study also shows that the proposed robust ILC is equivalent to conventional norm-optimal ILC with trial-varying parameters. A numerical simulation case study is conducted to compare the performance of this algorithm with that of other control algorithms while performing a given point-to-point tracking task. The results reveal its efficiency for the specific task and robustness against trial-varying initial conditions.

The Sliding Frank-Wolfe Algorithm and its Application to Super-Resolution Microscopy

Abstract: This paper showcases the theoretical and numerical performance of the Sliding Frank-Wolfe, which is a novel optimization algorithm to solve the BLASSO sparse spikes super-resolution problem. The BLASSO is a continuous (i.e. off-the-grid or grid-less) counterpart to the well-known 1 sparse regularisation method (also known as LASSO or Basis Pursuit). Our algorithm is a variation on the classical Frank-Wolfe (also known as conditional gradient) which follows a recent trend of interleaving convex optimization updates (corresponding to adding new spikes) with non-convex optimization steps (corresponding to moving the spikes). Our main theoretical result is that this algorithm terminates in a finite number of steps under a mild non-degeneracy hypothesis. We then target applications of this method to several instances of single molecule fluorescence imaging modalities, among which certain approaches rely heavily on the inversion of a Laplace transform. Our second theoretical contribution is the proof of the exact support recovery property of the BLASSO to invert the 1-D Laplace transform in the case of positive spikes. On the numerical side, we conclude this paper with an extensive study of the practical performance of the Sliding Frank-Wolfe on different instantiations of single molecule fluorescence imaging, including convolutive and non-convolutive (Laplace-like) operators. This shows the versatility and superiority of this method with respect to alternative sparse recovery technics.

Constrained Quaternion-Variable Convex Optimization: A Quaternion-Valued Recurrent Neural Network Approach

Abstract: This paper proposes a quaternion-valued one-layer recurrent neural network approach to resolve constrained convex function optimization problems with quaternion variables. Leveraging the novel generalized Hamilton-real (GHR) calculus, the quaternion gradient-based optimization techniques are proposed to derive the optimization algorithms in the quaternion field directly rather than the methods of decomposing the optimization problems into the complex domain or the real domain. Via chain rules and Lyapunov theorem, the rigorous analysis shows that the deliberately designed quaternion-valued one-layer recurrent neural network stabilizes the system dynamics while the states reach the feasible region in finite time and converges to the optimal solution of the considered constrained convex optimization problems finally. Numerical simulations verify the theoretical results.

Descent-to-Delete: Gradient-Based Methods for Machine Unlearning

Abstract: We study the data deletion problem for convex models. By leveraging techniques from convex optimization and reservoir sampling, we give the first data deletion algorithms that are able to handle an arbitrarily long sequence of adversarial updates while promising both per-deletion run-time and steady-state error that do not grow with the length of the update sequence. We also introduce several new conceptual distinctions: for example, we can ask that after a deletion, the entire state maintained by the optimization algorithm is statistically indistinguishable from the state that would have resulted had we retrained, or we can ask for the weaker condition that only the observable output is statistically indistinguishable from the observable output that would have resulted from retraining. We are able to give more efficient deletion algorithms under this weaker deletion criterion.

Noisy Matrix Completion: Understanding Statistical Guarantees for Convex Relaxation via Nonconvex Optimization

Abstract: This paper studies noisy low-rank matrix completion: given partial and noisy entries of a large low-rank matrix, the goal is to estimate the underlying matrix faithfully and efficiently. Arguably one of the most popular paradigms to tackle this problem is convex relaxation, which achieves remarkable efficacy in practice. However, the theoretical support of this approach is still far from optimal in the noisy setting, falling short of explaining its empirical success. We make progress towards demystifying the practical efficacy of convex relaxation vis-a-vis random noise. When the rank and the condition number of the unknown matrix are bounded by a constant, we demonstrate that the convex programming approach achieves near-optimal estimation errors - in terms of the Euclidean loss, the entrywise loss, and the spectral norm loss - for a wide range of noise levels. All of this is enabled by bridging convex relaxation with the nonconvex Burer-Monteiro approach, a seemingly distinct algorithmic paradigm that is provably robust against noise. More specifically, we show that an approximate critical point of the nonconvex formulation serves as an extremely tight approximation of the convex solution, thus allowing us to transfer the desired statistical guarantees of the nonconvex approach to its convex counterpart.

Dynamic Event-Triggered Sliding Mode Control: Dealing With Slow Sampling Singularly Perturbed Systems

Abstract: This brief endeavors to investigate the sliding mode control (SMC) issue of the networked singularly perturbed systems (SPSs) under slow sampling. For the energy saving purpose in network communication, a dynamic event-triggered mechanism is introduced to SMC design. By considering the structure characteristics of the controlled system, a novel sliding function is constructed with taking the singular perturbed matrix $E_\varepsilon $ into account properly. With the aid of some appropriate Lyapunov functionals, the sufficient conditions are derived to ensure the asymptotic stability of the sliding mode dynamics and the reachability of the specified sliding surface. Besides, it is shown that the quasi-sliding motion is dependent on the dynamic event-triggered parameters and its bound converges to a constant. The solving algorithm for the dynamic event-triggered SMC law is formulated in a convex optimization framework. Finally, an numerical example is provided to illustrate the effectiveness of the proposed results.

First-order optimization algorithms via inertial systems with Hessian driven damping

Abstract: In a Hilbert space setting, for convex optimization, we analyze the convergence rate of a class of first-order algorithms involving inertial features. They can be interpreted as discrete time versions of inertial dynamics involving both viscous and Hessian-driven dampings. The geometrical damping driven by the Hessian intervenes in the dynamics in the form $$ abla ^2 f (x(t)) \dot{x} (t)$$ . By treating this term as the time derivative of $$ abla f (x (t)) $$ , this gives, in discretized form, first-order algorithms in time and space. In addition to the convergence properties attached to Nesterov-type accelerated gradient methods, the algorithms thus obtained are new and show a rapid convergence towards zero of the gradients. On the basis of a regularization technique using the Moreau envelope, we extend these methods to non-smooth convex functions with extended real values. The introduction of time scale factors makes it possible to further accelerate these algorithms. We also report numerical results on structured problems to support our theoretical findings.

Disturbance-Observer-Based Fault Tolerant Control of High-Speed Trains: A Markovian Jump System Model Approach

Abstract: This paper addresses the fault tolerant control problem for high-speed trains in case of multiple possible failures. A new multiple point-mass model with system faults is built based on a stochastic jump system model approach. A novel active fault tolerant composite hierarchical anti-disturbance control strategy based on the disturbance observer is proposed such that the resulting composite system is stochastically stable with position and velocity tracking performance. According to whether the transition probabilities (TPs) of the failure and fault detection and isolation process can be accessed completely, three different cases (TPs are completely known, partially known, and completely unknown) are analyzed. For each case, based on the Lyapunov functional approach, a composite hierarchical controller is synthesized via a convex optimization problem. Finally, the simulations are given to illustrate the performance of the proposed methodologies.

Distributed Online Convex Optimization With Time-Varying Coupled Inequality Constraints

Abstract: This paper considers distributed online optimization with time-varying coupled inequality constraints. The global objective function is composed of local convex cost and regularization functions and the coupled constraint function is the sum of local convex functions. A distributed online primal-dual dynamic mirror descent algorithm is proposed to solve this problem, where the local cost, regularization, and constraint functions are held privately and revealed only after each time slot. Without assuming Slater's condition, we first derive regret and constraint violation bounds for the algorithm and show how they depend on the stepsize sequences, the accumulated dynamic variation of the comparator sequence, the number of agents, and the network connectivity. As a result, under some natural decreasing stepsize sequences, we prove that the algorithm achieves sublinear dynamic regret and constraint violation if the accumulated dynamic variation of the optimal sequence also grows sublinearly. We also prove that the algorithm achieves sublinear static regret and constraint violation under mild conditions. Assuming Slater's condition, we show that the algorithm achieves smaller bounds on the constraint violation. In addition, smaller bounds on the static regret are achieved when the objective function is strongly convex. Finally, numerical simulations are provided to illustrate the effectiveness of the theoretical results.

Distributed Dissipative State Estimation for Markov Jump Genetic Regulatory Networks Subject to Round-Robin Scheduling

Abstract: The distributed dissipative state estimation issue of Markov jump genetic regulatory networks subject to round-robin scheduling is investigated in this paper. The system parameters randomly change in the light of a Markov chain. Each node in sensor networks communicates with its neighboring nodes in view of the prescribed network topology graph. The round-robin scheduling is employed to arrange the transmission order to lessen the likelihood of the occurrence of data collisions. The main goal of the work is to design a compatible distributed estimator to assure that the distributed error system is strictly $(\Lambda _{1},\Lambda _{2},\Lambda _{3}) $ - $\gamma $ -stochastically dissipative. By applying the Lyapunov stability theory and a modified matrix decoupling way, sufficient conditions are derived by solving some convex optimization problems. An illustrative example is given to verify the validity of the provided method.

Joint Trajectory-Resource Optimization in UAV-Enabled Edge-Cloud System With Virtualized Mobile Clone

Abstract: This article studies an unmanned aerial vehicle (UAV)-enabled edge-cloud system, where UAV acts as a mobile edge computing (MEC) server interplaying with remote central cloud to provide computation services to ground terminals (GTs). The UAV-enabled edge-cloud system implements a virtualized network function, namely, mobile clone (MC), for each GT to help execute their offloaded tasks. Through such network function virtualization (NFV) implemented on top of the UAV-enabled edge-cloud system, GTs can have extended computation capability and prolonged battery lifetime. We aim to jointly optimize the allocation of resource and the UAV trajectory in the 3-D spaces to minimize the overall energy consumption of the UAV. The proposed solution, therefore, can extend the endurance of the UAV and support reliable MC functions for GTs. This article solves the complicated optimization problem through a block coordinate descent algorithm in an iterative way. In each iteration, the allocation of resource is modeled as a multiple constrained optimization problem given predefined UAV trajectory, which can be reformulated into a more tractable convex form and solved by successive convex optimization and Lagrange duality. Second, given the allocated resource, the optimization of the trajectory of rotary-wing/fixed-wing UAV can be formulated into a series of convex quadratically constrained quadratically program (QCQP) problems and solved by the standard convex optimization techniques. After the block coordinate descent algorithm converges to a prescribed accuracy, a high-quality suboptimal solution can be found. According to the simulation, the numerical results verify the effectiveness of our proposed solution in contrast to the baseline solutions.

An improved cutting plane method for convex optimization, convex-concave games, and its applications

Abstract: Given a separation oracle for a convex set K ⊂ ℝ n that is contained in a box of radius R, the goal is to either compute a point in K or prove that K does not contain a ball of radius є. We propose a new cutting plane algorithm that uses an optimal O(n log(κ)) evaluations of the oracle and an additional O(n 2) time per evaluation, where κ = nR/є. This improves upon Vaidya’s O( SO · n log(κ) + n ω+1 log(κ)) time algorithm [Vaidya, FOCS 1989a] in terms of polynomial dependence on n, where ω O( SO · n log(κ) + n 3 log O(1) (κ)) time algorithm [Lee, Sidford and Wong, FOCS 2015] in terms of dependence on κ. For many important applications in economics, κ = Ω(exp(n)) and this leads to a significant difference between log(κ) and (log(κ)). We also provide evidence that the n 2 time per evaluation cannot be improved and thus our running time is optimal. A bottleneck of previous cutting plane methods is to compute leverage scores, a measure of the relative importance of past constraints. Our result is achieved by a novel multi-layered data structure for leverage score maintenance, which is a sophisticated combination of diverse techniques such as random projection, batched low-rank update, inverse maintenance, polynomial interpolation, and fast rectangular matrix multiplication. Interestingly, our method requires a combination of different fast rectangular matrix multiplication algorithms. Our algorithm not only works for the classical convex optimization setting, but also generalizes to convex-concave games. We apply our algorithm to improve the runtimes of many interesting problems, e.g., Linear Arrow-Debreu Markets, Fisher Markets, and Walrasian equilibrium.