Non-convex Model Research Articles

Nonconvex regularization for blurred images with Cauchy noise

<p style='text-indent:20px;'>In this paper, we propose a nonconvex regularization model for images damaged by Cauchy noise and blur. This model is based on the method of the total variational proposed by Federica, Dong and Zeng [SIAM J. Imaging Sci.(2015)], where a variational approach for restoring blurred images with Cauchy noise is used. Here we consider the nonconvex regularization, namely a weighted difference of <inline-formula><tex-math id="M1">\begin{document}$ l_1 $\end{document}</tex-math></inline-formula>-norm and <inline-formula><tex-math id="M2">\begin{document}$ l_2 $\end{document}</tex-math></inline-formula>-norm coupled with wavelet frame, the alternating direction method of multiplier is carried out for this minimization problem, we describe the details of the algorithm and prove its convergence. Numerical experiments are tested by adding different levels of noise and blur, results show that our method can denoise and deblur the image better.</p>

Multi-Attribute Subspace Clustering via Auto-Weighted Tensor Nuclear Norm Minimization.

Self-expressiveness based subspace clustering methods have received wide attention for unsupervised learning tasks. However, most existing subspace clustering methods consider data features as a whole and then focus only on one single self-representation. These approaches ignore the intrinsic multi-attribute information embedded in the original data feature and result in one-attribute self-representation. This paper proposes a novel multi-attribute subspace clustering (MASC) model that understands data from multiple attributes. MASC simultaneously learns multiple subspace representations corresponding to each specific attribute by exploiting the intrinsic multi-attribute features drawn from original data. In order to better capture the high-order correlation among multi-attribute representations, we represent them as a tensor in low-rank structure and propose the auto-weighted tensor nuclear norm (AWTNN) as a superior low-rank tensor approximation. Especially, the non-convex AWTNN fully considers the difference between singular values through the implicit and adaptive weights splitting during the AWTNN optimization procedure. We further develop an efficient algorithm to optimize the non-convex and multi-block MASC model and establish the convergence guarantees. A more comprehensive subspace representation can be obtained via aggregating these multi-attribute representations, which can be used to construct a clustering-friendly affinity matrix. Extensive experiments on eight real-world databases reveal that the proposed MASC exhibits superior performance over other subspace clustering methods.

A general robust dynamic Bayesian network method for supply chain disruption risk estimation under ripple effect

Robust dynamic Bayesian network (DBN) is a valid tool for disruption propagation estimation in the supply chain under data scarcity. However, one of assumptions in robust DBN is that the Markov transition matrix is fixed and fully known, which is unpractical. To make up this deficiency, a novel and general robust DBN is, for the first time, proposed in this work to assess the worst-case oriented supply chain disruption risk under ripple effect. The study focuses on a supply chain with multiple suppliers and one manufacturer over a time horizon, in which only probability intervals of related probabilities are known. The objective is to obtain the worst-case supply chain disruption risk, measured by the probability of the manufacturer in the fully disrupted state in the final time period. For the problem, a new and general nonconvex programming model is established. Then, a case study is conducted to compare our approach with the classic DBN and robust DBN in the literature.

Controlled Islanding Strategy Considering Uncertainty of Renewable Energy Sources Based on Chance-constrained Model

Controlled islanding plays an essential role in preventing the blackout of power systems. Although there are several studies on this topic in the past, no enough attention is paid to the uncertainty brought by renewable energy sources (RESs) that may cause unpredictable unbalanced power and the observability of power systems after islanding that is essential for back-up black-start measures. Therefore, a novel controlled islanding model based on mixed-integer second-order cone and chance-constrained programming (MISOCCP) is proposed to address these issues. First, the uncertainty of RESs is characterized by their possibility distribution models with chance constraints, and the requirements, e. g., system observ-ability, for rapid back-up black-start measures are also considered. Then, a law of large numbers (LLN) based method is employed for converting the chance constraints into deterministic ones and reformulating the non-convex model into convex one. Finally, case studies on the revised IEEE 39-bus and 118-bus power systems as well as the comparisons among different models are given to demonstrate the effectiveness of the proposed model. The results show that the proposed model can result in less unbalanced power and better observability after islanding compared with other models.

Solving Bilevel AC OPF Problems by Smoothing the Complementary Conditions – Part I: Model Description and the Algorithm

The existing research on market price-affecting agents, i.e. price makers, neglects or simplifies the nature of AC power flows in the power system as it predominantly relies on DC power flows. This paper proposes a novel bilevel formulation based on the smoothing technique, where any price-affecting strategic player can be modelled in the upper level, while the market clearing problem in the lower level uses convex quadratic transmission AC optimal power flow (AC OPF), with the goal of achieving accuracy close to the one of the exact nonlinear formulations. Achieving convexity in the lower level is the foundation for bilevel modeling since traditional single-level reduction techniques do not hold for nonconvex models. The bilevel market participation problem with the AC OPF formulation in the lower level is transformed into a single-level problem and solved using multiple techniques such as the primal-dual counterpart, the strong duality theorem, the McCormick envelopes, the complementary slackness, the penalty factor, the interaction discretization as well as the proposed smoothing techniques. Due to an extensive amount of information and descriptions, the overall work is presented as a two-part paper. This first part provides a literature overview, positions the work and presents the model and the solution algorithm, while the solution techniques and case studies are provided in the accompanying paper.

Distributed Non-Convex Model Predictive Control for Non-Cooperative Collision Avoidance of Networked Differential Drive Mobile Robots

Distributed Non-Convex Model Predictive Control for Non-Cooperative Collision Avoidance of Networked Differential Drive Mobile Robots

Joint image denoising with gradient direction and edge-preserving regularization

Joint image denoising algorithms use the structures of the guidance image as a prior to restore the noisy target image. While the provided guidance images are helpful to improve the denoising performance, the denoised edges are most likely to be blurred especially when the edges of the guidance image are weak or inexistent. To address this weakness, this paper proposes a new gradient-direction-based joint image denoising method in which the absolute cosine value of the angle between two gradient vectors of the guidance image and those of the image to recover is employed as the parallel measurement to ensure that the gradient directions of the denoised image are approximately the same as or opposite to those of the guidance image. Besides, a new edge-preserving regularization term is developed to alleviate the effects of the unreliable prior information from guidance image. To simplify the resultant complex nonconvex and nonlinear fractional model, the logarithm function is employed to convert the multiplication operation into addition operation. Then, we construct the surrogate function for the logarithmic term of l2-norm, and separate the variables to transform the objective function into convex one with high numerical stability while retaining high efficiency. Finally, the optimal solutions can be obtained by directly minimizing the convex functions. Experimental results on public datasets and from nine benchmark methods consistently demonstrate the effectiveness of the proposed method both visually and quantitatively.

Learning Deep Models: Critical Points and Local Openness

With the increasing popularity of nonconvex deep models, developing a unifying theory for studying the optimization problems that arise from training these models becomes very significant. Toward this end, we present in this paper a unifying landscape analysis framework that can be used when the training objective function is the composite of simple functions. Using the local openness property of the underlying training models, we provide simple sufficient conditions under which any local optimum of the resulting optimization problem is globally optimal. We first completely characterize the local openness of the symmetric and nonsymmetric matrix multiplication mapping. Then we use our characterization to (1) provide a simple proof for the classical result of Burer-Monteiro and extend it to noncontinuous loss functions; (2) show that every local optimum of two-layer linear networks is globally optimal. Unlike many existing results in the literature, our result requires no assumption on the target data matrix [Formula: see text], and input data matrix [Formula: see text]; (3) develop a complete characterization of the local/global optima equivalence of multilayer linear neural networks (we provide various counterexamples to show the necessity of each of our assumptions); and (4) show global/local optima equivalence of overparameterized nonlinear deep models having a certain pyramidal structure. In contrast to existing works, our result requires no assumption on the differentiability of the activation functions and can go beyond “full-rank” cases.

Application of the Sine-Cosine Algorithm to the Optimal Design of a Closed Coil Helical Spring

This paper proposes the application of the sinecosine algorithm (SCA) to the optimal design of a closed coil helical spring. The optimization problem addressed corresponds to the minimization of total spring volume subject to physical constraints that represents the closed coil helical spring such as maximum working load, shear stress, and minimum diameter requirements, among other. The resulting mathematical formulation is a complex nonlinear and non-convex optimization model that is typically addressed in literature with trial and error methods or heuristic algorithms. To solve this problem efficiently, the SCA is proposed in this research. This optimization algorithm belongs to the family of the metaheuristic optimization techniques, it works with controlled random processes guided by sine and cosine trigonometric functions, that allows exploring and exploiting the solution space in order to find the best solution to the optimization problem. By presenting as main advantage an easy implementation at any programming language using sequential quadratic programming; eliminating the need to uses specialized and costly software. Numerical results demonstrating that the proposes SCA allows reaching lower spring volume values in comparison with literature approaches, such as genetic algorithms, particle swarm optimization methods, among others. All the numerical simulations have been implemented in the MATLAB software.

PV-EV Integrated Home Energy Management Considering Residential Occupant Behaviors

Rooftop photovoltaics (PV) and electrical vehicles (EV) have become more economically viable to residential customers. Most existing home energy management systems (HEMS) only focus on the residential occupants’ thermal comfort in terms of indoor temperature and humidity while neglecting their other behaviors or concerns. This paper aims to integrate residential PV and EVs into the HEMS in an occupant-centric manner while taking into account the occupants’ thermal comfort, clothing behaviors, and concerns on the state-of-charge (SOC) of EVs. A stochastic adaptive dynamic programming (ADP) model was proposed to optimally determine the setpoints of heating, ventilation, air conditioning (HVAC), occupant’s clothing decisions, and the EV’s charge/discharge schedule while considering uncertainties in the outside temperature, PV generation, and EV’s arrival SOC. The nonlinear and nonconvex thermal comfort model, EV SOC concern model, and clothing behavior model were holistically embedded in the ADP-HEMS model. A model predictive control framework was further proposed to simulate a residential house under the time of use tariff, such that it continually updates with optimal appliance schedules decisions passed to the house model. Cosimulations were carried out to compare the proposed HEMS with a baseline model that represents the current operational practice. The result shows that the proposed HEMS can reduce the energy cost by 68.5% while retaining the most comfortable thermal level and negligible EV SOC concerns considering the occupant’s behaviors.

A Proximal Algorithm with Convergence Guarantee for a Nonconvex Minimization Problem Based on Reproducing Kernel Hilbert Space

The underlying function in reproducing kernel Hilbert space (RKHS) may be degraded by outliers or deviations, resulting in a symmetry ill-posed problem. This paper proposes a nonconvex minimization model with ℓ0-quasi norm based on RKHS to depict this degraded problem. The underlying function in RKHS can be represented by the linear combination of reproducing kernels and their coefficients. Thus, we turn to estimate the related coefficients in the nonconvex minimization problem. An efficient algorithm is designed to solve the given nonconvex problem by the mathematical program with equilibrium constraints (MPEC) and proximal-based strategy. We theoretically prove that the sequences generated by the designed algorithm converge to the nonconvex problem’s local optimal solutions. Numerical experiment also demonstrates the effectiveness of the proposed method.

Optimal Economic–Environmental Operation of BESS in AC Distribution Systems: A Convex Multi-Objective Formulation

This paper deals with the multi-objective operation of battery energy storage systems (BESS) in AC distribution systems using a convex reformulation. The objective functions are CO2 emissions, and the costs of the daily energy losses are considered. The conventional non-linear nonconvex branch multi-period optimal power flow model is reformulated with a second-order cone programming (SOCP) model, which ensures finding the global optimum for each point present in the Pareto front. The weighting factors methodology is used to convert the multi-objective model into a convex single-objective model, which allows for finding the optimal Pareto front using an iterative search. Two operational scenarios regarding BESS are considered: (i) a unity power factor operation and (ii) a variable power factor operation. The numerical results demonstrate that including the reactive power capabilities in BESS reduces 200 kg of CO2 emissions and USD 80 per day of operation. All of the numerical validations were developed in MATLAB 2020b with the CVX tool and the SEDUMI and SDPT3 solvers.

Tensor completion via nonconvex tensor ring rank minimization with guaranteed convergence

In recent studies, the tensor ring (TR) rank has shown high effectiveness in tensor completion due to its ability of capturing the intrinsic structure within high-order tensors. A recently proposed TR rank minimization method is based on the convex relaxation by penalizing the weighted sum of nuclear norm of TR unfolding matrices. However, this method treats each singular value equally and neglects their physical meanings, which usually leads to suboptimal solutions in practice. To alleviate this weakness, this paper proposes an enhanced low-rank sparsity measure, which can more accurately approximate the TR rank and better promote the low-rankness of the solution. In specific, we apply the logdet function onto TR unfolding matrices to shrink less the larger singular values while shrink more the smaller ones. To solve the proposed nonconvex model efficiently, we develop an alternating direction method of multipliers algorithm and theoretically prove that, under some mild assumptions, our algorithm converges to a stationary point. Extensive experiments on color images, multispectral images, and color videos demonstrate that the proposed method outperforms several state-of-the-art competitors in both visual and quantitative comparison.

A non‐convex ternary variational decomposition and its application for image denoising

A non-convex ternary variational decomposition model is proposed in this study, which decomposes the image into three components including structure, texture and noise. In the model, a non-convex total variation (NTV) regulariser is utilised to model the structure component, and the weaker G and E spaces are used to model the texture and noise components, respectively. The proposed model provides a very sparse representation of the structure in total variation (TV) transform domain due to the use of non-convex regularisation and cleanly separates the texture and noise since two different weaker spaces are used to model these two components, respectively. In image denoising application, the proposed model can successfully remove noise while effectively preserving image edges and constructing textures. An alternating direction iteration algorithm combining with iteratively reweighted l1 algorithm, projection algorithm and wavelet soft threshold algorithm is introduced to effectively solve the proposed model. Numerical results validate the model and the algorithm for both synthetic and real images. Furthermore, compared with several state-of-the-art image variational restoration models, the proposed model yields the best performance in terms of the peak signal to noise ratio (PSNR) and the mean structural similarity index (SSIM).

Optimal sizing of residential battery systems with multi-year dynamics and a novel rainflow-based model of storage degradation: An extensive Italian case study

Residential battery storage to perform load shifting and demand side management has become of utmost importance to improve hosting capacity, increase renewable energy penetration and meet environmental targets, especially with energy community policies. As the lifetime of electrochemical batteries depends upon their scheduling and environmental conditions, the multi-year effects of operational strategies can affect the economics of the investment. However, rarely complete long-term simulations of the operation of storage systems are performed to assess the battery profitability including the operational effects of aging, and limited studies account for a large statistics of consumers. In this study, we propose a multi-year sizing methodology for residential applications, where the complete lifetime of batteries is simulated at 15-min time resolution till complete degradation using an improved non-linear non-convex degradation model; the photovoltaic plant aging is also considered. An extensive analysis on the economics and commercial size best suited for 399 real load profiles in Italy is proposed. Results suggest that the break-even price of the storage is about 400 €/kWh, which is lower than the average commercial price, and that, as reviewed, current market components may be unfit for consumers with low energy demand. Net Present Value (NPV) and Discounted PayBack Time (DPBT) can reach 500-1500 € and 8-11 years.

Rates of Expansions for Functional Estimators

AbstractIn this paper, we summarize results on convergence rates of various kernel based non- and semiparametric estimators, focusing on the impact of insufficient distributional smoothness, possibly unknown smoothness and even non-existence of density. In the presence of a possible lack of smoothness and the uncertainty about smoothness, methods of safeguarding against this uncertainty are surveyed with emphasis on nonconvex model averaging. This approach can be implemented via a combined estimator that selects weights based on minimizing the asymptotic mean squared error. In order to evaluate the finite sample performance of these and similar estimators we argue that it is important to account for possible lack of smoothness.

Hyperspectral Image Denoising Based on Nonconvex Low-Rank Tensor Approximation and lp Norm Regularization

A new nonconvex smooth rank approximation model is proposed to deal with HSI mixed noise in this paper. The low-rank matrix with Laplace function regularization is used to approximate the nuclear norm, and its performance is superior to the nuclear norm regularization. A new phase congruency lp norm model is proposed to constrain the spatial structure information of hyperspectral images, to solve the phenomenon of “artificial artifact” in the process of hyperspectral image denoising. This model not only makes use of the low-rank characteristic of the hyperspectral image accurately, but also combines the structural information of all bands and the local information of the neighborhood, and then based on the Alternating Direction Method of Multipliers (ADMM), an optimization method for solving the model is proposed. The results of simulation and real data experiments show that the proposed method is more effective than the competcing state-of-the-art denoising methods.

An Optimization-Based Meta-Learning Model for MRI Reconstruction with Diverse Dataset

This work aims at developing a generalizable Magnetic Resonance Imaging (MRI) reconstruction method in the meta-learning framework. Specifically, we develop a deep reconstruction network induced by a learnable optimization algorithm (LOA) to solve the nonconvex nonsmooth variational model of MRI image reconstruction. In this model, the nonconvex nonsmooth regularization term is parameterized as a structured deep network where the network parameters can be learned from data. We partition these network parameters into two parts: a task-invariant part for the common feature encoder component of the regularization, and a task-specific part to account for the variations in the heterogeneous training and testing data. We train the regularization parameters in a bilevel optimization framework which significantly improves the robustness of the training process and the generalization ability of the network. We conduct a series of numerical experiments using heterogeneous MRI data sets with various undersampling patterns, ratios, and acquisition settings. The experimental results show that our network yields greatly improved reconstruction quality over existing methods and can generalize well to new reconstruction problems whose undersampling patterns/trajectories are not present during training.

Short-term optimal scheduling of cascade hydropower plants shaving peak load for multiple power grids

Cascade hydropower plants which have good regulation performance and are managed by the dispatching center of regional power grids are usually required to simultaneously shave the peak load for multiple provincial power grids, which is an important way of relieving the growing peak shaving pressure on power grids in central and eastern China. This paper establishes an optimization model for the short-term generation scheduling of cascade hydropower plants in regional power grids. In this model, minimizing the peak-valley load difference of multiple power grids is adopted as the objective function. In addition to conventional hydraulic constraints, the operation constraints of individual hydropower units, electrical constraints and head effect on the power generation are all considered to obtain precise scheduling. The original nonlinear and non-convex model is converted into a standard mixed-integer linear programming (MILP) formulation through several linearization strategies. The results for the real-world case study indicate that: 1) the proposed model is computationally efficient with a calculation time of 326 s; 2) the peak-valley differences of the Shanghai Power Grid and Zhejiang Power Grid decreased by 9.71% and 2.29%, respectively; 3) compared with actual operation, the proposed model shows better performance in peak shaving for multiple provincial power grids.

Airline Network Planning: Mixed-integer non-convex optimization with demand–supply interactions

Airlines routinely use analytics tools to support flight scheduling, fleet assignment, revenue management, crew scheduling, and many other operational decisions. However, decision support systems are less prevalent to support strategic planning. This paper fills that gap with an original mixed-integer non-convex optimization model, named Airline Network Planning with Supply and Demand interactions (ANPSD). The ANPSD optimizes network planning (including route selection, flight frequencies and fleet composition), while capturing interdependencies between airline supply and passenger demand. We first estimate a demand model as a function of flight frequencies and network configuration, using a two-stage least-squares procedure fitted to historical data, and then formalize the ANPSD by integrating the empirical demand function into an optimization model. The model is formulated as a non-convex mixed-integer program. To solve it, we develop an exact cutting plane algorithm, named 2αECP, which iteratively generates hyperplanes to develop an outer approximation of the non-linear demand functions. Computational results show that the 2αECP algorithm outperforms state-of-the-art benchmarks and generates tight solution quality guarantees. A case study based on the network of a major European carrier shows that the ANPSD provides much stronger solutions than baselines that ignore – fully or partially – demand–supply interactions.