Convex Optimization Algorithm Research Articles

Identification of time-delay Markov jumps autoregressive system with recursive expectation maximum and convex optimization algorithm

A novel proposed approach of the recursive expectation–maximization (REM)–convex optimization algorithm is developed to solve the parameter identification problem of the Markov jump autoregressive exogenous (MJARX) systems with unknown time delays. First, the REM algorithm is employed for identifying the unknown time delay of the system, based on which the time-delay sequence is obtained. Subsequently, the model parameter vector is determined by the convex optimization algorithm. Finally, the transition probability matrix and variance are calculated based on the minimized mean square error. A numerical example and simulated continuous fermentation reactor process are implemented to demonstrate the effectiveness of the proposed algorithm. The identification results are better than the expectation–maximization algorithm and REM algorithm results.

Theoretical and numerical comparison of first order algorithms for cocoercive equations and smooth convex optimization

This paper provides a theoretical and numerical comparison of classical first-order splitting methods for solving smooth convex optimization problems and cocoercive equations. From a theoretical point of view, we compare convergence rates of gradient descent, forward-backward, Peaceman-Rachford, and Douglas-Rachford algorithms for minimizing the sum of two smooth convex functions when one of them is strongly convex. A similar comparison is given in the more general cocoercive setting under the presence of strong monotonicity and we observe that the convergence rates in optimization are strictly better than the corresponding rates for cocoercive equations for some algorithms. We obtain improved rates with respect to the literature in several instances by exploiting the structure of our problems. Moreover, we indicate which algorithm has the smallest convergence rate depending on strong convexity and cocoercive parameters. From a numerical point of view, we verify our theoretical results by implementing and comparing previous algorithms in well established signal and image inverse problems involving sparsity. We replace the widely used ℓ1 norm with the Huber loss and we observe that fully proximal-based strategies have numerical and theoretical advantages with respect to methods using gradient steps.

Energy management strategy based on convex optimization for fuel cell/battery/ultracapacitor hybrid vehicle

For a fuel cell hybrid electric vehicle (FCHEV) powered by a fuel cell (FC), a battery (BAT) and an ultracapacitor (UC), an improved convex-optimization-based energy management strategy (EMS) is proposed in this article. To protect the fuel cell and the battery from peak power, an adaptive fuzzy filter is used to complete the frequency decoupling of the power demand. Subsequently, an adaptive equivalent consumption minimization strategy (A-ECMS) is introduced to improve fuel economy. To improve the optimality of the EMS obtained, an improved convex optimization method is utilized by introducing a slack variable and a barrier function. Finally, based on experimental data and an improved convex optimization algorithm, the optimal EMS is calculated and verified by a series of simulations under different driving cycles. Simulation results confirm that, compared with the traditional ECMS-based EMS, the proposed EMS can improve fuel economy and achieve lower power fluctuation and optimal FC efficiency for the FCHEV.

Self-paced multi-label co-training

Multi-label learning aims to solve classification problems where instances are associated with a set of labels. In reality, it is generally easy to acquire unlabeled data but expensive or time-consuming to label them, and this situation becomes more serious in multi-label learning as an instance needs to be annotated with several labels. Hence, semi-supervised multi-label learning approaches emerge as they are able to exploit unlabeled data to help train predictive models. This work proposes a novel approach called Self-paced Multi-label Co-Training (SMCT). It leverages the well-known co-training paradigm to iteratively train two classifiers on two views of a dataset and communicate one classifier’s predictions on unlabeled data to augment the other’s training set. As pseudo labels may be false in iterative training, self-paced learning is integrated into SMCT to rectify false pseudo labels and avoid error accumulation. Concretely, the multi-label co-training model in SMCT is formulated as an optimization problem by introducing latent weight variables of unlabeled instances. It is then solved via an alternative convex optimization algorithm. Experimental evaluations are carried out based on six benchmark multi-label datasets and three metrics. The results demonstrate that SMCT is very competitive in each setting when compared with five state-of-the-art methods.

An Effective Optimization Methods for the Suppression of Edge Effects in Ultrawideband Tightly Coupled Antenna Arrays

An effective optimization method is proposed for the suppression of edge effects in finite ultrawideband tightly coupled antenna arrays (TCAAs). By designing a practical finite TCAAs, the influences of edge effects on active voltage standing wave ratio (VSWR) are analyzed. Meanwhile, both the active VSWR and the array gain pattern are formulated as analytical expressions with respect to amplitude and phase excitations for analyzing the design freedom of edge effects. Aimed at suppressing the edge effects within an ultrawideband operation band, a nonconvex optimization problem is established for the synthesis of tightly coupled dipole arrays (TCDAs), where array gains at considered scanning angles and frequencies satisfying given maximum active VSWR and amplitude dynamic range ratio (DRR) are maximized. The nonconvex problem is transformed into an iterative convex optimization problem by the use of an iterative way, and the amplitude and phase excitations are efficiently solved by the convex optimization algorithm accordingly. By comparing to the traditional uniform amplitude and progressive phase excitation (UAPPE) method, the simulated and measured results of the ultrawideband TCDA show that the proposed method can slightly improve array gains at most frequencies, while the active VSWRs are suppressed to 3 over a 10:1 bandwidth when scanning up to 45°.

Positivity-Preserving H∞ Model Reduction for Discrete-Time Positive Systems via a Successive Convex Optimization Algorithm

This paper considers the positivity-preserving model reduction for discrete-time positive systems. Given a stable high-order positive system, we aim to find a reduced-order model such that the approximation error is minimized within a prescribed H∞ performance and positivity is preserved. Regarding the bounded real lemma, the sufficient and necessary condition for the existence of a reduced-order model is established in terms of bilinear matrix inequality and convex semi-definite constraint, which ensures that the reduced-order system is positive and the resulted error system is stable and has an H∞ performance level. Based on the inner-approximation strategy, we approximate the bilinear constraints with convex ones, under which an iterative procedure is provided to calculate the desired reduced-order model. Finally, an example is provided to demonstrate the effectiveness and potential benefits of the presented results.

“FISTA” in Banach spaces with adaptive discretisations

FISTA is a popular convex optimisation algorithm which is known to converge at an optimal rate whenever a minimiser is contained in a suitable Hilbert space. We propose a modified algorithm where each iteration is performed in a subset which is allowed to change at every iteration. Sufficient conditions are provided for guaranteed convergence, although at a reduced rate depending on the conditioning of the specific problem. These conditions have a natural interpretation when a minimiser exists in an underlying Banach space. Typical examples are L1-penalised reconstructions where we provide detailed theoretical and numerical analysis.

Mars Ascent Trajectory Optimization Based on Convex Optimization

In this paper, a Mars ascent trajectory optimization algorithm based on convex optimization is designed for the Mars ascent trajectory optimization problem. The main accomplishment of this paper is the trajectory optimization of the second-stage ignition based on the convex optimization. For the first-stage ignition, open-loop guidance is used. In this stage, the ascender flies using the maximum thrust to track the attitude profile, thus completing the ascent flight control. In the presence of errors in the first-stage ignition, the experiment generates multiple sets of first-stage shutdown point data and completes simulation verification experiments based on them. The simulation verifies that the second-stage ignition trajectory optimization based on convex optimization algorithm can achieve high accuracy into orbit despite the large starting point deviation.

Probabilistic-Constrained Distributed Set-Membership Estimation Over Sensor Networks: A Dynamic Periodic Event-Triggered Approach

This article is concerned with the event-triggered probabilistic-constrained distributed set-membership estimation problem for a discrete-time nonlinear system over sensor networks. For saving communication resource, a novel discrete-time dynamic periodic event-triggered mechanism (ETM) is first developed for the sensor network. Under the proposed method, the sensor node calculates the ETM in a periodic manner and the threshold is adjusted dynamically. Thereafter, a distributed set-membership estimator is constructed, and a probability-based estimated ellipsoidal constraint is put forward to acquire a more flexible set-membership estimation algorithm. Simultaneously, an auxiliary-function-dependent approach is proposed to derive the criterion for the co-design of the probabilistic-constrained set-membership estimator and the event-triggered parameter such that the system states reside in the estimated ellipsoid with a pre-specified probability. The auxiliary function is constructed in a piecewise style aiming to deal with the sawtooth constraint of sampling signals. Furthermore, a recursive convex optimization algorithm with regard to the estimated ellipsoid is presented. Finally, a simulation example is employed to verify the validity of the developed method.

Simple, Reliable, and Noise-Resilient Continuous-Variable Quantum State Tomography with Convex Optimization

Precise reconstruction of unknown quantum states from measurement data, a process commonly called quantum state tomography, is a crucial component in the development of quantum information processing technologies. Many different tomography methods have been proposed over the years. Maximum likelihood estimation is a prominent example, being the most popular method for a long period of time. Recently, more advanced neural network methods have started to emerge. Here, we go back to basics and present a method for continuous variable state reconstruction that is both conceptually and practically simple, based on convex optimization. Convex optimization has been used for process tomography and qubit state tomography, but seems to have been overlooked for continuous variable quantum state tomography. We demonstrate high-fidelity reconstruction of an underlying state from data corrupted by thermal noise and imperfect detection, for both homodyne and heterodyne measurements. A major advantage over other methods is that convex optimization algorithms are guaranteed to converge to the optimal solution.

Identification of Nonlinear State-Space Systems via Sparse Bayesian and Stein Approximation Approach

This paper is concerned with the parameter estimation of non-linear discrete-time systems from noisy state measurements in the state-space form. A novel sparse Bayesian convex optimisation algorithm is proposed for the parameter estimation and prediction. The method fully considers the approximation method, parameter prior and posterior, and adds Bayesian sparse learning and optimization for explicit modeling. Different from the previous identification methods, the main identification challenge resides in two aspects: first, a new objective function is obtained by our improved Stein approximation method in the convex optimization problem, so as to capture more information of particle approximation and convergence; second, another objective function is developed with L1-regularization, which is sparse method based on recursive least squares estimation. Compared with the previous study, the new objective function contains more information and can easily mine more important information from the raw data. Three simulation examples are given to demonstrate the proposed algorithm’s effectiveness. Furthermore, the performances of these approaches are analyzed, including parameter estimation of root mean squared error (RMSE), parameter sparsity and prediction of state and output result.

Static Output Feedback Control for T–S Fuzzy Systems via a Successive Convex Optimization Algorithm

This article focuses on developing a static output feedback (SOF) control scheme for Takagi–Sugeno fuzzy systems. The proposed SOF controller does not share the same premise membership functions with the model, which permits enhancing the flexibility in controller design and implementation. By contrast with the state feedback case, the SOF control generally results in nonconvex design conditions. To circumvent this problem, we develop a successive convex optimization algorithm, which is based on solving a sequence of more tractable convex optimization problems obtained by approximating the nonconvex constraints with some convex ones. As a heuristic algorithm, the validity of the developed successive convex optimization algorithm is highly affected by initial conditions, and, recognizing this, we put forward an iterative procedure for determining the feasible initial condition. Finally, two illustrative examples are presented to validate the efficiency of the proposed algorithms.

An Efficient and Universal Static and Dynamic Convex Optimization for Array Synthesis

In this letter, a novel static-dynamic convex optimization algorithm is proposed for array synthesis, which can be divided into static and dynamic successive optimization stages. At the first stage, the synthetic direction graph is constructed as a second-order cone programming problem so that an initial excitation is quickly obtained. Next, by serving the results obtained previously as the initial excitation, an iterative operation is implemented to obtain a high-precise solution. Moreover, in order to effectively ensure the continuity of the solution process and approach the feasible solution, shrinkage factor and crossover operator are introduced. Four tough examples of synthesizing different kinds of patterns are conducted to validate the feasibility and effectiveness of the proposed method. Results show that compared with the existing representative methods, the proposed algorithm is of high computational efficiency and universality.

Research on Multi-Terminal’s AC Offloading Scheme and Multi-Server’s AC Selection Scheme in IoT

Mobile Edge Computing (MEC) technology and Simultaneous Wireless Information and Power Transfer (SWIPT) technology are important ones to improve the computing rate and the sustainability of devices in the Internet of things (IoT). However, the system models of most relevant papers only considered multi-terminal, excluding multi-server. Therefore, this paper aims at the scenario of IoT with multi-terminal, multi-server and multi-relay, in which can optimize the computing rate and computing cost by using deep reinforcement learning (DRL) algorithm. Firstly, the formulas of computing rate and computing cost in proposed scenario are derived. Secondly, by introducing the modified Actor-Critic (AC) algorithm and convex optimization algorithm, we get the offloading scheme and time allocation that maximize the computing rate. Finally, the selection scheme of minimizing the computing cost is obtained by AC algorithm. The simulation results verify the theoretical analysis. The algorithm proposed in this paper not only achieves a near-optimal computing rate and computing cost while significantly reducing the program execution delay, but also makes full use of the energy collected by the SWIPT technology to improve energy utilization.

A new randomized primal-dual algorithm for convex optimization with fast last iterate convergence rates

We develop a novel unified randomized block-coordinate primal-dual algorithm to solve a class of nonsmooth constrained convex optimization problems, which covers different existing variants and model settings from the literature. We prove that our algorithm achieves and convergence rates in two cases: merely convexity and strong convexity, respectively, where k is the iteration counter and n is the number of block-coordinates. These rates are known to be optimal (up to a constant factor) when n = 1. Our convergence rates are obtained through three criteria: primal objective residual and primal feasibility violation, dual objective residual, and primal-dual expected gap. Moreover, our rates for the primal problem are on the last-iterate sequence. Our dual convergence guarantee requires additionally a Lipschitz continuity assumption. We specify our algorithm to handle two important special cases, where our rates are still applied. Finally, we verify our algorithm on two well-studied numerical examples and compare it with two existing methods. Our results show that the proposed method has encouraging performance on different experiments.

Data-driven distributionally robust joint chance-constrained energy management for multi-energy microgrid

Multi-energy microgrid (MEMG) has the potential to improve the energy utilization efficiency. However, the uncertainty caused by distributed renewable energy resources brings an urgent need for multi-energy co-optimization to ensure secure operation. This paper focuses on the distributionally robust energy management problem for MEMG. Various flexible resources in different energy sectors are utilized for uncertainty mitigation, then, a data-driven Wasserstein distance-based distributionally robust joint chance-constrained (DRJCC) energy management model is proposed. To make the DRJCC model tractable, an optimized conditional value-at-risk (CVaR) approximation (OCA) formulation is proposed to transfer the joint chance-constrained model into a tractable form. Then, an iterative sequential convex optimization algorithm is tailored to reduce the solution conservatism by tuning OCA. Numerical result illustrates the effectiveness of the proposed model.

Robust Interior Point Method for Quantum Key Distribution Rate Computation

Security proof methods for quantum key distribution, QKD, that are based on the numerical key rate calculation problem, are powerful in principle. However, the practicality of the methods are limited by computational resources and the efficiency and accuracy of the underlying algorithms for convex optimization. We derive a stable reformulation of the convex nonlinear semidefinite programming, SDP, model for the key rate calculation problems. We use this to develop an efficient, accurate algorithm. The stable reformulation is based on novel forms of facial reduction, FR, for both the linear constraints and nonlinear quantum relative entropy objective function. This allows for a Gauss-Newton type interior-point approach that avoids the need for perturbations to obtain strict feasibility, a technique currently used in the literature. The result is high accuracy solutions with theoretically proven lower bounds for the original QKD from the FR stable reformulation. This provides novel contributions for FR for general SDP. We report on empirical results that dramatically improve on speed and accuracy, as well as solving previously intractable problems.

Biorthogonal Greedy Algorithms in convex optimization

The study of greedy approximation in the context of convex optimization is becoming a promising research direction as greedy algorithms are actively being employed to construct sparse minimizers for convex functions with respect to given sets of elements. In this paper we propose a unified way of analyzing a certain kind of greedy-type algorithms for the minimization of convex functions on Banach spaces. Specifically, we define the class of Weak Biorthogonal Greedy Algorithms for convex optimization that contains a wide range of greedy algorithms. We analyze the introduced class of algorithms and establish the properties of convergence, rate of convergence, and numerical stability, which is understood in the sense that the steps of the algorithm are allowed to be performed not precisely but with controlled computational inaccuracies. We show that the following well-known algorithms for convex optimization — the Weak Chebyshev Greedy Algorithm (co) and the Weak Greedy Algorithm with Free Relaxation (co) — belong to this class, and introduce a new algorithm — the Rescaled Weak Relaxed Greedy Algorithm (co). Presented numerical experiments demonstrate the practical performance of the aforementioned greedy algorithms in the setting of convex minimization as compared to optimization with regularization, which is the conventional approach of constructing sparse minimizers.

Denoising Generalized Expectation-Consistent Approximation for MR Image Recovery.

To solve inverse problems, plug-and-play (PnP) methods replace the proximal step in a convex optimization algorithm with a call to an application-specific denoiser, often implemented using a deep neural network (DNN). Although such methods yield accurate solutions, they can be improved. For example, denoisers are usually designed/trained to remove white Gaussian noise, but the denoiser input error in PnP algorithms is usually far from white or Gaussian. Approximate message passing (AMP) methods provide white and Gaussian denoiser input error, but only when the forward operator is sufficiently random. In this work, for Fourier-based forward operators, we propose a PnP algorithm based on generalized expectation-consistent (GEC) approximation-a close cousin of AMP-that offers predictable error statistics at each iteration, as well as a new DNN denoiser that leverages those statistics. We apply our approach to magnetic resonance (MR) image recovery and demonstrate its advantages over existing PnP and AMP methods.

Joint Offloading and Resource Allocation Using Deep Reinforcement Learning in Mobile Edge Computing

Mobile edge computation offloading (MECO) has recently emerged as a promising method to support computation-intensive and latency-sensitive applications, significantly saving the battery energy of smart mobile devices (SMDs). However, on the one hand, the energy consumption depends on both the SMD and the MEC server, which makes it necessary to consider these two entities to achieve energy sustainability jointly. On the other hand, for a real-time mobile edge computing (MEC) system, efficient optimization algorithms based on binary offloading have received significant attention, while efficient algorithms for partial offloading under time-varying channels are seldom investigated. In this paper, we propose an energy-efficient algorithm based on deep reinforcement learning to optimize the overall energy cost in a real-time multi-user MEC system. We decompose the energy minimization problem into two sub-problems, where a deep neural network learns the optimal mapping between wireless channels and offloading ratios, and a closed-form solution for the optimal local frequency and a convex optimization algorithm are used to solve the resource allocation sub-problem. Finally, the extensive experiments demonstrate the effectiveness of our proposed algorithm in reducing the total energy consumption of the MECO system against several offloading schemes and achieving low processing latency fit to the time-varying wireless channels.