Nonconvex Terms Research Articles

Real-time dispatch of integrated electricity and thermal system incorporating storages via a stochastic dynamic programming with imitation learning

Coordinated dispatch of integrated electricity and thermal system (IETS) provides extra operation flexibility which is further improved by integration of electrical and thermal storages. However, the problem non-convexity and multiple uncertainties hinder its optimal real-time dispatch. This paper proposes an approximate dynamic programming with imitation learning (ADP-IL) based real-time dispatch policy for IETS with electrical and thermal storages, which is computationally efficient and adaptive to uncertainties while satisfying complex networks constraints. First, real-time dispatch of IETS is reformulated as a multistage stochastic sequential optimization and non-convex terms are addressed by mixed integer programming. Next, an ADP which exploits value function monotonicity is employed to temporally decompose original problem and an off-line pre-learning is incorporated to address dimension complexity. Then, imitation learning, which allows the algorithm to learn from expert demonstration, is introduced to further accelerate off-line pre-learning. After sufficient learning, ADP-IL provides high quality real-time solution with remarkable computation efficiency. Comprehensive studies verify optimality, adaptability, efficiency, and scalability of ADP-IL.

Efficient FMT reconstruction based on L1-αL2 regularization via half-quadratic splitting and a two-probe separation light source strategy.

Fluorescence molecular tomography (FMT) can achieve noninvasive, high-contrast, high-sensitivity three-dimensional imaging in vivo by relying on a variety of fluorescent molecular probes, and has excellent clinical transformation prospects in the detection of tumors in vivo. However, the limited surface fluorescence makes the FMT reconstruction have some ill-posedness, and it is difficult to obtain the ideal reconstruction effect. In this paper, two different emission fluorescent probes and L 1-L 2 regularization are combined to improve the temporal and spatial resolution of FMT visual reconstruction by introducing the weighting factor α and a half-quadratic splitting alternating optimization (HQSAO) iterative algorithm. By introducing an auxiliary variable, the HQSAO method breaks the sparse FMT reconstruction task into two subproblems that can be solved in turn: simple reconstruction and image denoising. The weight factor α (α>1) can increase the weight of nonconvex terms to further promote the sparsity of the algorithm. Importantly, this paper combines two different dominant fluorescent probes to achieve high-quality reconstruction of dual light sources. The performance of the proposed reconstruction strategy was evaluated by digital mouse and nude mouse single/dual light source models. The simulation results show that the HQSAO iterative algorithm can achieve more excellent positioning accuracy and morphology distribution in a shorter time. In vivo experiments also further prove that the HQSAO algorithm has advantages in light source information preservation and artifact suppression. In particular, the introduction of two main emission fluorescent probes makes it easy to separate and reconstruct the dual light sources. When it comes to localization and three-dimensional morphology, the results of the reconstruction are much better than those using a fluorescent probe, which further facilitates the clinical transformation of FMT.

A new nonconvex low-rank tensor approximation method with applications to hyperspectral images denoising

Hyperspectral images (HSIs) are frequently corrupted by mixing noise during their acquisition and transmission. Such complicated noise may reduce the quality of the obtained HSIs and limit the accuracy of the subsequent processing. By using the low-rank prior of the tensor formed by spatial and spectral information and further exploring the intrinsic structure of the underlying HSI from noisy observations, in this paper, we propose a new nonconvex low-rank tensor approximation method including optimization model and efficient iterative algorithm to eliminate multiple types of noise. The proposed mathematical model consists of a nonconvex low-rank regularization term using the γ nuclear norm, which is nonconvex surrogate to Tucker rank, and two data fidelity terms representing sparse and Gaussian noise components, which are regularized by the -norm and the Frobenius norm, respectively. To solve this model, we propose an efficient augmented Lagrange multiplier algorithm. We also study the convergence and parameter setting of the algorithm. Extensive experimental results show that the proposed method has better denoising performance than the state-of-the-art competing methods for low-rank tensor approximation and noise modeling.

Cluster-Free NOMA Communications Toward Next Generation Multiple Access

A generalized downlink multi-antenna non-orthogonal multiple access (NOMA) transmission framework is proposed with the novel concept of cluster-free successive interference cancellation (SIC). In contrast to conventional NOMA approaches, where SIC is successively carried out within the same cluster, the key idea is that the SIC can be flexibly implemented between any arbitrary users to achieve efficient interference elimination. Based on the proposed framework, a sum rate maximization problem is formulated for jointly optimizing the transmit beamforming and the SIC operations between users, subject to the SIC decoding conditions and users’ minimal data rate requirements. To tackle this highly-coupled mixed-integer nonlinear programming problem, an alternating direction method of multipliers-successive convex approximation (ADMM-SCA) algorithm is developed. The original problem is first reformulated into a tractable biconvex augmented Lagrangian (AL) problem by handling the non-convex terms via SCA. Then, this AL problem is decomposed into two subproblems that are iteratively solved by the ADMM to obtain the stationary solution. Furthermore, to reduce the computational complexity and alleviate the parameter initialization sensitivity of ADMM-SCA, a Matching-SCA algorithm is proposed. The intractable binary SIC operations are solved through an extended many-to-many matching, which is jointly combined with an SCA process to optimize the transmit beamforming. The proposed Matching-SCA can converge to an enhanced exchange-stable matching that guarantees the local optimality. Numerical results demonstrate that: i) the proposed Matching-SCA algorithm achieves comparable performance and a faster convergence compared to ADMM-SCA; ii) the proposed generalized framework realizes scenario-adaptive communications and outperforms traditional multi-antenna NOMA approaches in various communication regimes.

Two-Stage Robust Optimal Scheduling of Flexible Distribution Networks Based on Pairwise Convex Hull

With distributed generation (DG) being continuously connected into distribution networks, the stochastic and fluctuating nature of its power generation brings ever more problems than before, such as increasing operating costs and frequent voltage violations. However, existing robust scheduling methods of flexible resources tend to make rather conservative decisions, resulting in high operation costs. In view of this, a two-stage robust optimal scheduling method for flexible distribution networks is proposed in this paper, based on the pairwise convex hull (PWCH) uncertainty set. A two-stage robust scheduling model is first formulated considering coordination among on-load tap changers, energy storage systems and flexible distribution switches. In the first stage, the temporal correlated OLTCs and energy storage systems are globally scheduled using day-ahead forecasted DG outputs. In the second stage, FDSs are scheduled in real time in each time period based on the first-stage decisions and accurate short-term forecasted DG outputs. The spatial correlation and uncertainties of the outputs of multiple DGs are modeled based on the PWCH, such that the decision conservativeness can be reduced by cutting regions in the box with low probability of occurrence. The improved column-and-constraint generation algorithm is then used to solve the robust optimization model. Through alternating iterations of auxiliary variables and dual variables, the nonconvex bilinear terms induced by the PWCH are eliminated, and the subproblem is significantly accelerated. Test results on the 33-bus distribution system and a realistic 104-bus distribution system validate that the proposed PWCH-based method can obtain much less conservative scheduling schemes than using the box uncertainty set.

Group Sparsity Residual Constraint Image Denoising Model with𝒍1/𝒍2Regularization

Group sparse residual constraint with non-local priors (GSRC) has achieved great success in image restoration producing state-of-the-art performance. In the GSRC model, the [see formula in PDF] norm minimization is employed to reduce the group sparse residual. In recent years, non-convex regularization terms have been widely used in image denoising problems, which have achieved better results in denoising than convex regularization terms. In this paper, we use the ratio of the [see formula in PDF] and [see formula in PDF] norm instead of the [see formula in PDF] norm to propose a new image denoising model, i.e., a group sparse residual constraint model with [see formula in PDF] minimization (GSRC-[see formula in PDF]). Due to the computational difficulties arisen from the non-convexity and non-linearity, we focus on a constrained optimization problem that can be solved by alternative direction method of multipliers (ADMM). Experimental results of image denoising show that the pro-posed model outperforms several state-of-the-art image denoising methods both visually and quantitatively.

Autonomous trajectory planning for multi-stage launch vehicles using mass-projection sequential penalized convex relaxation method

Autonomous trajectory planning for multi-stage launch vehicles poses great technical challenges, mainly because of the highly nonlinear flight dynamics and complex multi-phase structure. Here, this problem is solved by a novel, robust, and efficient approach within the framework of convex optimization. Focusing on the multi-phase difficulty, a mass-projection technique that substitutes the original time independent variable with the mass history is introduced to eliminate the problem’s discontinuous nature and phase-linkage event constraints, then a new sequential relaxation-and-penalization method is proposed to address the resulting problem. All non-convex terms involved in the problem are relaxed to formulate a certainly feasible semi-definite problem, the deviation between which and the true problem is penalized by blending the information from the trust region and semi-definite constraint. The combination of the mass-projection technique and the sequential relaxation-and-penalization method is found to be capable of recovering the original problem and providing the optimal solution rapidly and reliably. Three representative simulations are performed to verify the robustness and computational efficiency of the proposed approach, and comparative results show that it achieves a 100% success rate for all cases, while enabling a 69% to 98% saving of computational time compared to typical methods.

An Enhanced McCormick Envelopes to Represent Kron's Loss Formula

Recently, some researchers have employed the McCormick envelopes method to convexify some NP-hard optimization problems with bilinear terms. However, few publications concentrate on its variants to derive a more tight convex relaxation for practical applications. This paper proposes a new viewpoint on Kron’s loss formula, also known as the B-matrix formula, as an equation having bilinear terms. Relying on the perspective, we transform the loss equation to some linear constraints using an enhanced McCormick relaxation. In the technique, the domain of bilinear variables is divided into some smaller parts to improve the relaxation tightness. Some case studies with different nonconvex terms are considered to verify the effectiveness of the enhanced envelopes for capturing Kron’s loss formula. The findings from the numerical simulations suggest that the proposed approach can represent Kron’s loss equation precisely. Moreover, the method performs more effectively than the other methods available in the literature as it usually converges to more optimal solutions.

Stability Verification for Heterogeneous Complex Networks via Iterative SOS Programming

This letter is to investigate the stability verification for heterogeneous polynomial complex networks through iterative sum-of-squares programming approach. With polynomial Lyapunov functions, a global asymptotic stability criterion is established for the heterogeneous complex networks under the directed topology. Based on the proposed criterion, the stability verification problem is then reduced to a sum-of-squares optimization problem for solving polynomial matrix inequalities. To process non-convex terms of the polynomial matrix inequalities, we present an iterative sum-of-squares programming approach to turn them into the convex polynomial matrix inequalities, yielding polynomial Lyapunov functions effectively to realize the stability verification. Finally, a numerical example is given to show the fully automatic verification of global asymptotic stability for the heterogeneous polynomial complex networks, which demonstrates the theoretical and algorithmic advantages of the proposed method.

Portfolio construction as linearly constrained separable optimization

Mean–variance portfolio optimization problems often involve separable nonconvex terms, including penalties on capital gains, integer share constraints, and minimum nonzero position and trade sizes. We propose a heuristic algorithm for such problems based on the alternating direction method of multipliers (ADMM). This method allows for solve times in tens to hundreds of milliseconds with around 1000 securities and 100 risk factors. We also obtain a bound on the achievable performance. Our heuristic and bound are both derived from similar results for other optimization problems with a separable objective and affine equality constraints. We discuss a concrete implementation in the case where the separable terms in the objective are piecewise quadratic, and we empirically demonstrate its effectiveness for tax-aware portfolio construction.

A Coordination Optimization Method for Load Shedding Considering Distribution Network Reconfiguration

Load shedding control is an emergency control measure to maintain the frequency stability of the power system. Most of the existing load shedding methods use the extensive form of directly cutting off the outlet of the substation, featuring low control accuracy and high control cost. A network reconfiguration technique can adjust the topology of the distribution network and offers more optimization space for load shedding control. Therefore, this paper proposes a reconfiguration–load shedding coordination optimization scheme to reduce the power loss caused by load shedding control. In the proposed method, a load shedding mathematical optimization model based on distribution network reconfiguration is first established. The tie switches and segment switches in the distribution network are used to perform the reconfiguration of the distribution network, and the load switches are adopted to realize the load shedding. To improve the solving efficiency of the model, a solving strategy that combined a minimum spanning tree algorithm with an improved genetic algorithm is trailed to address the nonlinear and nonconvex terms. The application of the proposed method and model are finally verified via the IEEE 33 bus system, and the advantages in reducing the loss cost and the number of outage users are accordingly proven.

Robust PCA via non-convex half-quadratic regularization

In this paper, we propose a new non-convex regularization term named half-quadratic function to achieve robustness and sparseness for robust principal component analysis, and derive its proximity operator, indicating that the resultant optimization problem can be solved in computationally attractive manner. In addition, the low-rank matrix component is expressed as the factorization form and proximal block coordinate descent is leveraged to seek its solution, whose convergence is rigorously analyzed. We prove that any limit point of the iterations is a critical point of the objective function. Furthermore, the parameter that controls the robustness and sparseness in our algorithm, is automatically adjusted according to the statistical residual error. Experimental results based on synthetic and real-world data demonstrate that the devised algorithm can effectively extract the low-rank and sparse components. MATLAB code is available at https://github.com/bestzywang.

Robust fuzzy observer‐based fault‐tolerant control: A homogeneous polynomial Lyapunov function approach

In this paper, a homogeneous polynomial Lyapunov function (HPLF) is employed in robust observer-based Integrated Fault-Tolerant Control (IFTC) of nonlinear systems modelled by the polynomial fuzzy model (PFM). This makes the design problem benefit from the conservative reduction property of polynomial Lyapunov functions (PLFs) and simultaneously avoid the emergence of non-convex terms due to differentiation of the polynomial Lyapunov matrix. As a result, the less conservative sum of square (SOS) conditions are obtained in the form of polynomial matrix inequalities (PMI), which are solved via SOSTOOLS. For the first time, a non-quadratic Lyapunov function is used to design a polynomial fuzzy unknown-input observer that estimates the system's states and actuator faults in the presence of model uncertainties and external disturbances. The effectiveness of the proposed approach in providing more relaxed and less conservative results along with the better fault tolerance is illustrated through three simulating examples and compared with the quadratic Lyapunov function (QLF) approach.

An efficient semismooth Newton method for adaptive sparse signal recovery problems

We know that compressive sensing can establish stable sparse recovery results from highly undersampled data under a restricted isometry property condition. In reality, however, numerous problems are coherent, and vast majority conventional methods might work not so well. Recently, it was shown that using the difference between - and -norm as a regularization always has superior performance. In this paper, we consider an adaptive - model where the -norm with measures the data fidelity and the -term measures the sparsity. This proposed model has the ability to deal with different types of noises and extract the sparse property even under high coherent condition. We use a proximal majorization-minimization technique to handle the non-convex regularization term and then employ a semismooth Newton method to solve the corresponding convex relaxation subproblem. We prove that the sequence generated by the semismooth Newton method admits fast local convergence rate to the subproblem under some technical assumptions. Finally, we do some numerical experiments to demonstrate the superiority of the proposed model and the progressiveness of the proposed algorithm.

Robust fault-tolerant control design for polynomial fuzzy systems

This paper focuses on the robust H∞ fault-tolerant control issue for a family of polynomial fuzzy systems subject to actuator faults and external disturbances. Especially, asynchronous fuzzy membership function technique is considered to minimize the expenses of controller implementation. To derive the robust stability conditions, actuator fault factors are taken into the account in the closed-loop system formulation. Moreover, to utilize the exact shape information of fuzzy membership functions in the stability criterion, they are approximated as sum-of-square polynomials with the aid of polynomial curve fitting method. In addition, to reduce the conservatism, a generalized polynomial Lyapunov function is considered. However, there are some non-convex terms in the stability conditions due to the construction of the generalized Lyapunov function that consists of all state-dependent polynomial matrix. Thus, some existing convex methods for general polynomial fuzzy systems are rendered inapplicable. To deal with this issue, the actual non-convex stability conditions converts into a convex optimization problem by introducing some additional convex constraints. Moreover, the membership-function-dependent robust stability conditions of the closed-loop system are provided as sum-of-square convex polynomial constraints. Lastly, the usefulness and superiority of the designed controller are illustrated by the simulation examples.

Static Output Feedback Fault Tolerant Control and Relaxed Stability Analysis of Positive Polynomial Fuzzy Systems

The increased demand for safety in control systems has motivated the research for fault-tolerant controllers (FTCs). This article presents the static output feedback FTC design and relaxed stability analysis for positive nonlinear systems on account of polynomial fuzzy models. A polynomial copositive Lyapunov function candidate is proposed for the stability analysis of positive polynomial fuzzy-model-based (PPFMB) control systems for the first time. This method develops the decision variable on the polynomial copositive Lyapunov function candidate to be a polynomial vector and also makes it independent of the form of the positive polynomial fuzzy model. Thus, it can be applied to a more extensive range of PPFMB control systems and potentially generates more relaxed stability analysis results. To achieve the stability analysis and control synthesis, the following challenges are overcome: 1) approximate the nonconvex terms in stability and positivity conditions into convex ones through the idea of matrix decomposition; 2) eliminate the nonconvex conditions resulting from the derivative of the polynomial Lyapunov function by employing an sum-of-squares (SOS) optimization method; 3) relax the results of stability analysis and control design by introducing the advanced membership-function-dependent technique to develop both the stability conditions and the positivity conditions. The relaxed conditions are obtained on the basis of SOS form so that the MATLAB third-party toolbox, SOSTOOLS, can be used to find feasible solutions. Finally, two simulation examples are presented to illustrate the effectiveness of the proposed method.

Relaxed observer-based stabilization and dissipativity conditions of T-S fuzzy systems with nonhomogeneous Markov jumps via non-PDC scheme

This paper aims to design a robust observer-based dissipative controller for discrete-time Takagi–Sugeno (T-S) fuzzy systems with nonhomogeneous Markov jumps through a non-parallel distributed compensation (non-PDC) scheme. Based on a mode-dependent nonquadratic Lyapunov function, the final form of the stabilization conditions is expressed as linear matrix inequalities in a less conservative manner. To be specific, this paper proposes a decoupling technique that can address the inherent nonconvex terms by extracting them from the stabilization conditions, where all slack variables are set to be fuzzy-basis-dependent for less conservative performance. Furthermore, the proposed stabilization method adopts a one-step design strategy that simultaneously designs the fuzzy observer and control gains without any iteration procedures by employing a positive tuning parameter. In particular, the time-varying transition probabilities included in the stabilization conditions are effectively removed using a modified relaxation technique that can avoid excessive use of free weighting matrices. Finally, based on four examples, the validity of the proposed method is verified through comparison with other studies.

DCA for Sparse Quadratic Kernel-Free Least Squares Semi-Supervised Support Vector Machine

With the development of science and technology, more and more data have been produced. For many of these datasets, only some of the data have labels. In order to make full use of the information in these data, it is necessary to classify them. In this paper, we propose a strong sparse quadratic kernel-free least squares semi-supervised support vector machine (SSQLSS3VM), in which we add a ℓ0norm regularization term to make it sparse. An NP-hard problem arises since the proposed model contains the ℓ0 norm and another nonconvex term. One important method for solving the nonconvex problem is the DC (difference of convex function) programming. Therefore, we first approximate the ℓ0 norm by a polyhedral DC function. Moreover, due to the existence of the nonsmooth terms, we use the sGS-ADMM to solve the subproblem. Finally, empirical numerical experiments show the efficiency of the proposed algorithm.

Distributionally Robust Fair Transit Resource Allocation During a Pandemic

This paper studies the distributionally robust fair transit resource allocation model (DrFRAM) under the Wasserstein ambiguity set to optimize the public transit resource allocation during a pandemic. We show that the proposed DrFRAM is highly nonconvex and nonlinear, and it is NP-hard in general. Fortunately, we show that DrFRAM can be reformulated as a mixed integer linear programming (MILP) by leveraging the equivalent representation of distributionally robust optimization and monotonicity properties, binarizing integer variables, and linearizing nonconvex terms. To improve the proposed MILP formulation, we derive stronger ones and develop valid inequalities by exploiting the model structures. Additionally, we develop scenario decomposition methods using different MILP formulations to solve the scenario subproblems and introduce a simple yet effective no one left-based approximation algorithm with a provable approximation guarantee to solve the model to near optimality. Finally, we numerically demonstrate the effectiveness of the proposed approaches and apply them to real-world data provided by the Blacksburg Transit. History: This paper has been accepted for the Transportation Science Special Issue on Emerging Topics in Transportation Science and Logistics. Funding: This work was supported by the Division of Computing and Communication Foundations [Grant 2153607] and the Division of Civil, Mechanical and Manufacturing Innovation [Grant 2046426]. Supplemental Material: The online appendix is available at https://doi.org/10.1287/trsc.2022.1159 .

Low-rank matrix factorization with nonconvex regularization and bilinear decomposition

Low-rank matrix factorization (LRMF) is a powerful approach that can recover useful information with low-rank features from corrupted data and has been broadly applied in various scenarios such as computer vision and statistics. Existing convex relaxation-based LRMF models' performance and solutions are often limited by unsatisfied recovery accuracy and prolonged execution time due to excessive relaxation bias and heavy computational burden. In this paper, we first explore one class of parameterized non-convex penalty function and generalize it as a non-convex relaxation substitution term for the original rank function in the LRMF, which can effectively enhance the recovery capability for low-rank data. In order to reduce the computational burden of the nuclear-norm-based optimization scheme and make LRMF applicable to large-scale data. We propose to perform bilinear decomposition on the low-rank matrix and derive the bilinear factorization formula under the parameterized non-convex alternative. With the above considering, a low-rank matrix recovery model via bilinear decomposition and non-convex regularization, denoted as NCBF, is formulated. Further, the optimal solution algorithm for the NCBF model is developed in the framework of the block coordinate descent, combining simultaneously three strategies of decoupling, proximal gradient descent, and block prox-linear. Also, the convergence condition for the proposed algorithm has been given. We perform extensive experiments using synthetic data and real-world image datasets. The synthetic data test results demonstrate the dual superiority of the bilinear decomposition scheme in terms of operation time and estimation accuracy. Application of denoising to both generic and CT image datasets, experimental results show that NCBF outperforms several existing methods under both subjective and objective metrics evaluations.