Non-convex Optimization Model Research Articles

SCCLRR: A Robust Computational Method for Accurate Clustering Single Cell RNA-Seq Data.

Single-cell RNA transcriptome data present a tremendous opportunity for studying the cellular heterogeneity. Identifying subpopulations based on scRNA-seq data is a hot topic in recent years, although many researchers have been focused on designing elegant computational methods for identifying new cell types; however, the performance of these methods is still unsatisfactory due to the high dimensionality, sparsity and noise of scRNA-seq data. In this study, we propose a new cell type detection method by learning a robust and accurate similarity matrix, named SCCLRR. The method simultaneously captures both global and local intrinsic properties of data based on a low rank representation (LRR) framework mathematical model. The integrated normalized Euclidean distance and cosine similarity are used to balance the intrinsic linear and nonlinear manifold of data in the local regularization term. To solve the non-convex optimization model, we present an iterative optimization procedure using the alternating direction method of multipliers (ADMM) algorithm. We evaluate the performance of the SCCLRR method on nine real scRNA-seq datasets and compare it with seven state-of-the-art methods. The simulation results show that the SCCLRR outperforms other methods and is robust and effective for clustering scRNA-seq data. (The code of SCCLRR is free available for academic https://github.com/wzhangwhu/SCCLRR).

Designing Unimodular Sequences With Optimized Auto/Cross-Correlation Properties via Consensus-ADMM/PDMM Approaches

Unimodular sequences with good auto/cross-correlation properties are favorable in wireless communication and radar applications. In this paper, we focus on designing these kinds of sequences. The main content is as follows: first, we formulate the designing problem as a quartic polynomial minimization problem with constant modulus constraints; second, by introducing auxiliary phase variables, the polynomial minimization problem is equivalent to a consensus nonconvex optimization problem; third, to achieve its good approximate solution efficiently, we propose two efficient algorithms based on alternating direction method of multipliers (ADMM) and parallel direction method of multipliers (PDMM); fourth, we prove that the consensus-ADMM algorithm can converge to some stationary point of the original nonconvex problem and consensus-PDMM's output is some stationary point of the original nonconvex problem if it is convergent. Moreover, we also analyze the nonconvex optimization model's local optimality and computational complexity of the proposed consensus-ADMM/PDMM approaches. Simulation results demonstrate that the proposed ADMM/PDMM approaches outperform state-of-the-art ones in either computational cost or correlation properties of the designed unimodular sequences.

Forecasting the demand of mobile clinic services at vulnerable communities based on integrated multi-source data

Demand forecasting plays an important role in the deployment of mobile clinic services to vulnerable communities such as school zones and census tracts as it can help the service provider to maximize its coverage under limited resources. In this paper, we consider the issue of how to predict the vaccination delinquency in schools and census tracts. Such an issue is rather challenging as the delinquency is only observed in schools for which very limited information is available; while rich demographic and economic information is available for census tracts, no observations of delinquency have been made at the census tract level. To address the above challenge, we first develop a hierarchical approach to forecast the demand for vaccinations in schools and census tracts. In the first stage of the hierarchical approach, we solve a linear optimization model to compute an association matrix that can align some common features in both census tracts and school zones. Then we use the estimated association to develop a forecasting model to predict the vaccination delinquency in both schools and census tracts. A non-convex quadratic optimization (QO) model is also proposed to find the association matrix and the forecasting model simultaneously. We also introduce an alternative update scheme for the non-convex QO and establish the convergence of the algorithm. Moreover, the two association matrices generated from the proposed approaches can be used to impute the information in the school zone data, which further allows us to apply existing forecasting models to predict the demand in school zones based on the imputed data. A case study from the Houston Independent School District (HISD) and its associated communities is reported to demonstrate the efficacy of the new models and techniques.

Simultaneous Optimization of the Liner Shipping Route and Ship Schedule Designs with Time Windows

This paper proposes to address the problem of the simultaneous optimization of the liner shipping route and ship schedule designs by incorporating port time windows. A mathematical programming model was developed to minimize the carrier’s total operating cost by simultaneously optimizing the port call sequence, ship arrival time per port of call, and sailing speed per shipping leg under port time window constraints. In view of its structure, the nonlinear nonconvex optimization model is further transformed into a mixed-integer linear programming model that can be efficiently solved by extant solvers to provide a global optimal solution. The results of the numerical experiments performed using a real-world case study indicated that the proposed model performs significantly better than the models that handle the design problems separately. The results also showed that different time windows will affect the optimal port call sequence. Moreover, port time windows, bunker price, and port efficiency all affect the total operating cost of the designed shipping route.

Optimal Energy-Hub Planning Based on Dimension Reduction and Variable-Sized Unimodal Searching

Interest in the highly efficient energy hub (EH) model has been growing despite the high computational requirements of planning for a multi-energy, multi-device operation. To address both the device size limitation and the multi-scenario issue, we propose a new solution methodology for solving the EH planning problem. In the method, the decision variables are device sizes. First, a dimension reduction technique is proposed to address the curse of dimensionality based on the correlation of unknown variables such as the capacities of different devices in an EH. Second, to avoid local convergence, a solution method called the variable-sized unimodal searching (VUS) approach is proposed to assure a global optimal planning scheme for the one-dimensional non-convex optimization model obtained from the preceding dimension reduction process. The case study indicates that the proposed approach has a higher computing efficiency than the Benders decomposition (BD) algorithm to deal with a scenario-based stochastic planning problem with a large number of scenarios. Thus, the effectiveness of the EH planning approach is verified.

Optimal shipping schedule for a near-sea liner service route with time windows

This paper proposes a shipping schedule design problem arising in near-sea liner shipping that aims to determine containership deployment plan, the arrival time at each port of call, and sailing speed for a given liner service route with hard time windows. The problem is transformed into a mixed integer nonlinear nonconvex optimization model, and further transformed into a mixed integer linear optimization model, which can be effectively solved by commercial solvers. The case studies based on a real-life service route show the effectiveness and efficiency of the proposed solution method. Results also demonstrate that the port time windows affect the total cost of a ship route, the optimal number of ships to deploy, and the optimal schedule.

A Novel Convex Clustering Method for High-Dimensional Data Using Semiproximal ADMM

Clustering is an important ingredient of unsupervised learning; classical clustering methods include K-means clustering and hierarchical clustering. These methods may suffer from instability because of their tendency prone to sink into the local optimal solutions of the nonconvex optimization model. In this paper, we propose a new convex clustering method for high-dimensional data based on the sparse group lasso penalty, which can simultaneously group observations and eliminate noninformative features. In this method, the number of clusters can be learned from the data instead of being given in advance as a parameter. We theoretically prove that the proposed method has desirable statistical properties, including a finite sample error bound and feature screening consistency. Furthermore, the semiproximal alternating direction method of multipliers is designed to solve the sparse group lasso convex clustering model, and its convergence analysis is established without any conditions. Finally, the effectiveness of the proposed method is thoroughly demonstrated through simulated experiments and real applications.

A Non-convex Optimization Model for Signal Recovery

The electroencephalogram (EEG) signal is one of the most frequently used biomedical signals. In order to accurately exploit the cosparsity and low-rank property which is nature in multichannel EEG signals, motivated by the fact that weighted schatten-p norm and $${l_q}$$ norm can better approximate the matrix rank and $${l_0}$$ norm, in this paper, a non-convex optimization model is proposed to precisely reconstruct the multichannel EEG signal. weighted schatten-p norm and $${l_q}$$ norm are used to enforce low-rank property and cosparsity. In addition, an efficient iterative optimization method based on alternating direction method of multipliers is used to solve the resulting non-convex optimization problem. Experimental results have demonstrated that the proposed algorithm can significantly outperform existing state-of-the-art CS methods for compressive sensing of multichannel EEG signals.

A TV-log nonconvex approach for image deblurring with impulsive noise

In this paper, we study the image deblurring with impulsive noise problem. In order to find a high quality recovery solution, we propose a nonconvex optimization model that combines total variation regularization and nonconvex log penalty for data fitting. The new model can overcome the limitation of the L1-norm penalized data fitting term with total variation regularization model for high noise levels, and is easier to choose the scalar parameter in the data fitting term than the existing methods. For solving the nonconvex optimization problem, a difference of convex (DC) functions algorithm with adaptive proximal parameter is developed. Theoretically, using the Kurdyka-Łojasiewicz property, we establish that the sequence generated by the proposed algorithm converges to a critical point. The experiment results demonstrate the superiority of the new approach against the competing methods.

Low-rank and sparse matrix recovery via inexact Newton-like method with non-monotone search

In this paper, a non-convex optimization model of the low-rank and sparse matrix recovery from partial samples is proposed, and an inexact Newton-like algorithm with non-monotone scheme for solving the non-convex optimization model is suggested, in which the inexact Newton-like algorithm is used for the low-rank matrix part and the non-monotone scheme is used for sparse matrix part. It is proved that the iterative sequence of the new algorithm converges to a stationary point of the model. Finally, numerical experiments show that the proposed algorithm is far superior to the missing low-rank sparse decomposition (MLSD) algorithm in Azghani et al., 2019 and Algorithm 1 in Gu et al., 2016 in CPU time, which indicates that the new algorithm is more efficient.

Non-convex sparse regularization for impact force identification

Many convex regularization methods, such as the classical Tikhonov regularization based on l2-norm penalty and the standard sparse regularization method based on l1-norm penalty, have been widely investigated for impact force identification. However, in many practical applications, these regularization methods commonly underestimate the true solution. In this paper, we propose a non-convex sparse regularization method based on lp-norm (0 < p < 1) penalty, to seek the sufficiently sparse and highly accurate solution of impact force identification. Firstly, a non-convex optimization model based on lp-norm penalty instead of l2-norm penalty or l1-norm penalty is developed for regularizing inverse problems of impact force identification to overcome the mismatch between l0-norm and l1-norm regularizations. Secondly, an iteratively reweighed l1-norm algorithm is introduced to solve such a non-convex model through transforming it into a series of l1-norm regularizations. Finally, numerical simulation and experimental validation are carried out to evaluate the effectiveness and performance of lp-norm regularization when 0 = p ≤ 1. Our numerical results demonstrate that, compared to the state-of-the-art Tikhonov regularization and l1-norm regularization, the solution of lp-norm regularization achieves more stability, sparseness and accuracy when dealing with the heavily noisy response. Additionally, both numerical and experimental results demonstrate that the peak relative error of the identified impact force using lp-norm regularization has a decreasing tendency as p is approaching 0; while the identification of lp-norm regularization with 0 = p ≤ 1/2 bears no significant difference, which always outperforms that the identification with 1/2 = p ≤ 1.

Trajectory planning based on non-convex global optimization for serial manipulators

To perform specific tasks in dynamic environments, robots are required to rapidly update trajectories according to changing factors. A continuous trajectory planning methodology for serial manipulators based on non-convex global optimization is presented in this paper. First, a kinematic trajectory planning model based on non-convex optimization is constructed to balance motion rapidity and safety. Then, a model transformation method for the non-convex optimization model is presented. In this way, the accurate global solution can be obtained with an iterative solver starting from arbitrary initializations, which can greatly improve the computational accuracy and efficiency. Furthermore, an efficient initialization method for the iterative solver based on multivariable-multiple regression is presented, which further speeds up the solution process. The results show that trajectory planning efficiency is significantly enhanced by model transformation and initialization improvement for the iterative solver. Consequently, real-time continuous trajectory planning for serial manipulators with many degrees of freedom can be achieved, which lays a basis for performing dynamic tasks in complex environments.

A Progressive Motion-Planning Algorithm and Traffic Flow Analysis for High-Density 2D Traffic

Unmanned aircraft systems (UASs) are a promising new mode of transportation for cargo delivery. Although control, navigation, and communication technologies are becoming available on individual flight units, system-level motion management for dense UAS traffic remains an open question. This paper presents a motion-planning model based on nonlinear optimization techniques to centrally coordinate paths for all vehicles traversing a shared two-dimensional (2D) space. An exact bound is derived to characterize the discrete-time separation constraints, and a set of new metrics is proposed to measure 2D traffic flow efficiency. The grand goal is to make all vehicles that come under dispatch reach their respective destinations quickly and efficiently while maintaining a safe intervehicle separation at all times. This infinite-horizon operational problem is formulated as a nonlinear nonconvex optimization model that must be solved in a progressive, receding-horizon fashion. To ensure feasibility and overcome path deadlocks attributed to local optima, a series of heuristic measures are developed. By pivoting on artfully coined intermediate feasibility, the algorithm is able to circumvent imminent deadlocks in a predictive manner and progressively construct the solution with guaranteed feasibility. Simulation experiments are performed at various traffic density levels to generate useful insights for airspace regulators and traffic managers.

One-Bit Compressive Sensing With Projected Subgradient Method Under Sparsity Constraints

One-bit compressive sensing theory shows that the sparse signals can be almost exactly reconstructed from a small number of one-bit quantized linear measurements. This paper presents the convergence analysis of the binary iterative hard thresholding (BIHT) algorithm which is a state-of-the-art recovery algorithm in one-bit compressive sensing. The basic idea of the convergence analysis is to view BIHT as a kind of projected subgradient method under sparsity constrains. To the best of our knowledge, this is the first convergence analysis of BIHT. We first consider a general convex function subject to sparsity constraints and connect it with the non-convex model in one-bit compressive sensing literatures. A projected subgradient method is proposed to solve the general model and some convergence results are established. A stronger convergence theorem for $\alpha $ –strongly convex functions without assumption on differentiable condition is also established. Furthermore, the corresponding stochastic projected subgradient method is provided with convergence guarantee. In our settings, BIHT is a special case of the projected subgradient method. Therefore, the convergence analysis can be applied to BIHT naturally. Then, we apply the projected subgradient method to some related non-convex optimization models arising in compressive sensing with $\ell _{1}$ -constraint, sparse support vector machines, and rectifier linear units regression. Finally, some numerical examples are presented to show the validity of our convergence analysis. The numerical experiments also show that the proposed projected subgradient method is very simple to implement, robust to sparse noise, and effective for sparse recovery problems.

Cooperative motion parameter estimation using RSS measurements in robotic sensor networks

When multiple mobile targets start from initial positions at constant velocities in a sensor network, a cooperative motion parameter estimation method is proposed by using the received signal strength (RSS) measurements. Firstly, the nonconvex optimization model is built to estimate the initial positions and velocities of the mobile targets. Then the unconstrained semidefinite programming (USDP) is designed to estimate the positions of the mobile targets by converting the nonconvex optimization model into the convex problem, when the transmit powers of the mobile targets are assumed to be known. To reduce the estimation error, the constrained semidefinite programming (CSDP) is also proposed by exploiting the constraint conditions of motion equation. A new convex semidefinite programming (SDP) with known transmit powers (SDP-KTP) is proposed to directly estimate the initial positions and the velocities of the mobile targets. The USDP, CSDP and SDP algorithms are extended to the situation of unknown transmit power. The simulations and real experiments show that the SDP algorithm provides better performance than the USDP or the CSDP by availing of the movement equation. The proposed USDP, CSDP and SDP algorithms perform better with the increasing of samples, but the computational complexity of the proposed algorithms is higher.

A stochastic mixed-integer convex programming model for long-term distribution system expansion planning considering greenhouse gas emission mitigation

This paper proposes a multistage convex distribution system planning model to find the best reinforcement plan over a specified horizon. This strategy determines planning actions such as reinforcement of existing substations, conductor replacement of overloaded feeders, and siting and sizing of renewable and dispatchable distributed generation units. Besides, the proposed approach aims at mitigating the greenhouse gas emissions of electric power distribution systems via a monetary form. Inherently, this problem is a non-convex optimization model that can be an obstacle to finding the optimal global solution. To remedy this issue, convex envelopes are used to recast the original problem into a mixed integer conic programming (MICP) model. The MICP model guarantees convergence to optimal global solution by using existing commercial solvers. Moreover, to address the prediction errors in wind output power and electricity demands, a two-stage stochastic MICP model is developed. To validate the proposed model, detail analysis is carried out over various case studies of a 34-node distribution system under different conditions, while to show its potential and effectiveness a 135-node system with two substations is used. Numerical results confirm the effectiveness of the proposed planning scheme in obtaining an economic investment plan at the presence of several planning alternatives and to promote an environmentally committed electric power distribution network.

Piecewise linear bounding of univariate nonlinear functions and resulting mixed integer linear programming-based solution methods

Various optimization problems result from the introduction of nonlinear terms into combinatorial optimization problems. In the context of energy optimization for example, energy sources can have very different characteristics in terms of power range and energy demand/cost function, also known as efficiency function or energy conversion function. Introducing these energy sources characteristics in combinatorial optimization problems, such as energy resource allocation problems or energy-consuming activity scheduling problems may result into mixed integer nonlinear problems neither convex nor concave. Approximations via piecewise linear functions have been proposed in the literature. Non-convex optimization models and heuristics exist to compute optimal breakpoint positions under a bounded absolute error-tolerance. We present an alternative solution method based on the upper and lower bounding of nonlinear terms using non necessarily continuous piecewise linear functions with a relative epsilon-tolerance. Conditions under which such approach yields a pair of mixed integer linear programs with a performance guarantee are analyzed. Models and algorithms to compute the non necessarily continuous piecewise linear functions with absolute and relative tolerances are also presented. Computational evaluations performed on energy optimization problems for hybrid electric vehicles show the efficiency of the method with regards to the state of the art.

A variational proximal alternating linearized minimization in a given metric for limited-angle CT image reconstruction

• A nonconvex and nonsmooth optimization model is investigated for limited-angle CT imaging. • The proposed method can avoid computing the inverse of a large system matrix. • We show that each bounded sequence generated by our method globally converges to a critical point. • The results by our method are shown superior than some classical CT reconstruction methods by real data experiments. Due to the restriction of computed tomography (CT) scanning environment, the acquired projection data may be incomplete for exact CT reconstruction. Though some convex optimization methods, such as total variation minimization based method, can be used for incomplete data reconstruction, the edge of reconstruction image may be partly distorted for limited-angle CT reconstruction. To promote the quality of reconstruction image for limited-angle CT imaging, in this paper, a nonconvex and nonsmooth optimization model was investigated. To solve the model, a variational proximal alternating linearized minimization (VPALM) method based on proximal mapping in a given metric was proposed. The proposed method can avoid computing the inverse of a huge system matrix thus can be used to deal with the larger-scale inverse problems. What’s more, we show that each bounded sequence generated by VPALM globally converges to a critical point based on the Kurdyka–Lojasiewicz property . Real data experiments are used to demonstrate the viability and effectiveness of VPALM method, and the results show that the proposed method outperforms two classical CT reconstruction methods.

Optimized Signal Distortion for PAPR Reduction of OFDM Signals With IFFT/FFT Complexity Via ADMM Approaches

In this paper, we propose two low-complexity optimization methods to reduce peak-to-average power ratio (PAPR) values of orthogonal frequency division multiplexing (OFDM) signals via alternating direction method of multipliers (ADMM). First, we formulate a nonconvex signal distortion optimization model based on minimizing data carrier distortion such that the constraints are placed on PAPR and the power of free carriers. Second, to obtain the model's approximate optimal solution efficiently, we design two low-complexity ADMM algorithms, named ADMMDirect and ADMM-Relax respectively. Third, we show that, in ADMM-Direct/-Relax, all the optimization subproblems can be solved semi-analytically and the computational complexity in each iteration is roughly (9(IN log <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> IN), where I and N are oversampling factor and carrier number respectively. Moreover, we show that the resulting solution of ADMM-direct is guaranteed to be some Karush-Kuhn-Tucker (KKT) point of the nonconvex model when the iteration algorithm is convergent. For ADMMRelax, we prove that it has theoretically-guaranteed convergence and can approach arbitrarily close to some KKT point of the model if proper parameters are chosen. Simulation results demonstrate the effectiveness of the proposed approaches.

Half Thresholding Pursuit Algorithm for Fluorescence Molecular Tomography.

Fluorescence Molecular Tomography (FMT) is a promising optical tool for small animal imaging. The l1/2-norm regularization has attracted attention in the field of FMT due to its ability in enhancing sparsity of solution and coping with the high ill-posedness of the inverse problem. However, efficient algorithm for solving the nonconvex regularized model deserve to explore. A Half Thresholding Pursuit Algorithm (HTPA) combined with parameter optimization is proposed in this paper to efficiently solve the nonconvex optimization model. Specifically, the half thresholding iteration method is utilized to solve l1/2-norm model, pursuit strategy is used to accelerate the process of iteration, and the parameter optimization scheme is designed to obtain robust parameter. Analysis and assessment on simulated and experimental data demonstrate that the proposed HTPA performs better in location accuracy and reconstructed fluorescent yield in less time cost, compared with the state-of-the-art reconstruction algorithms. The proposed HTPA combined with the parameter optimization scheme is an efficient and robust reconstruction approach to FMT.