Non-convex Model Research Articles

A dynamic reconfiguration model and method for load balancing in the snow-shaped distribution network

The snow-shaped distribution network (SDN) is a cable distribution network composed of eight (or six) 10 kV feeders from four (or three) substations in a regular connection. Compared with the traditional 10 kV distribution network, SDN can support a wide range of load transfer among six or eight feeders. Aiming at the problem of load spatial-temporal unbalanced condition caused by the integration of distributed generators (DGs) and different load types in different feeders, this paper proposes a dynamic reconfiguration strategy for load balancing in SDN considering DGs and energy storage system (ESS). Firstly, the basic structure of SDN is analyzed and the power flow model for its dynamic reconfiguration is developed. Secondly, the dynamic reconfiguration optimization model for load balancing in SDN considering DGs and ESS is proposed to utilize the load transfer capability to mitigate the load unbalanced condition and reduce active power loss. Thirdly, the original non-convex model is converted into a mixed-integer second-order cone programming (MISOCP) model by applying the second-order cone relaxation and the big-M method, which is solved by CPLEX solver. Finally, the effectiveness of the proposed model and method are verified by an actual case in Tianjin and IEEE 33-node system. The analysis results show that the proposed method can significantly alleviate the load unbalanced spatial-temporal distribution and improve the economic efficiency by regulating the operation of SDN including ESS optimization and dynamic reconfiguration.

A non-convex low-rank image decomposition model via unsupervised network

Image decomposition is the separation of a given image into two parts with different features, i.e., structure and texture. In order to extract the image information comprehensively, a non-convex unsupervised image decomposition model is proposed in this paper. Firstly, we propose a new non-convex low-rank regular term to describe the low-rank nature of images, which focuses on optimizing the sum of partial singular values of image matrices using a non-convex truncated nuclear norm. Secondly, unsupervised network is used to capture image information by learning the deep semantic information of the image and then extracting the image structure. Finally, the alternative direction method of multipliers is used to divide the model into three sub-problems and optimize them separately. For the non-convex sub-problem, a two-stage solution scheme is given by employing the singular value decomposition technique, and experimental analysis is carried out on both real and synthetic images. The experimental results show that the proposed non-convex image decomposition method outperforms other state-of-the-art methods in terms of subjective visual effects. Meanwhile, in the quantitative evaluation, the proposed method shows a large improvement compared with other methods, with a maximum improvement of 15 % in the structural similarity index measure (SSIM) index.

Top-[formula omitted] discriminative feature selection with uncorrelated and [formula omitted]-norm equation constraints

Supervised feature selection (FS) as an interpretable dimensionality reduction technique has received increasing attention, where linear discriminative analysis (LDA)-based method can select informative features discriminatively and obtain promising performance. When original data has more features than samples, however, LDA-based method generally encounters degradation since the appearance of irreversible scatter matrix. This situation is called the small sample size (SSS) problem. To overcome it and enhance the discriminant power of selected feature subsets, in this paper, we design an elegant LDA-based FS model referred to as Top-k Discriminative FS (TDFS), which is constructed by seamlessly integrating the ℓ2,0-norm equation constraint into uncorrelated LDA model. More concretely, the ℓ2,0-norm equation constraint can explicitly characterize the number of selective features k to ensure the sparsity of projected matrix and select top features. The uncorrelated LDA model aims to improve discriminative ability based on uncorrelated data in projected subspace. Given the formidable nature of solving this non-convex model, a novel optimization algorithm is further developed and the SSS problem can be efficaciously addressed during the optimization process. We first decompose projection matrix into a discrete selection matrix and its corresponding nonzero projection matrix, then concurrently optimize above two matrices by employing a column-by-column update scheme, during which the reversibility of scatter matrix in selective feature subspace can be easily guaranteed to solve SSS problem. The extensive experiments on four synthetic data sets and eight real-world data sets show that the proposed method outperforms eight competitors validated by three classifiers. Moreover, although the theoretical analysis proves that our algorithm has quartic time complexity on the number of selected features k, the running time experiments verify that TDFS is still efficient and applicable in scenarios where only a small number of features need to be selected. From above perspectives, our algorithm shows desirable performance to achieve discriminative FS.

Flexible allocation and optimal configuration of multi-level energy exploitation units for heterogeneous energy systems considering resource distribution

The energy systems are evolving into heterogeneous energy systems (HESs) with complicated integration of sources, networks, loads, and storage towards a decarbonized future of human beings. This paper develops the flexible allocation and optimal configuration of multi-level energy exploitation units (MEEUs) for the HES considering resource distribution including local energy conditions and actual coupled relationship of various energy networks. A bi-level optimal planning model of MEEUs for the HES considering initial investment cost, carbon emission cost, operation cost and line loss is developed to figure out the optimal location, structure, configuration and corresponding operation strategy. To address the non-convex and non-linear model, a novel hybrid solution method considering the mixing characteristics of mixed-integer second-order cone programming (MISOCP) model and quadratic non-convex model is proposed. The solution method adopts MISOCP model for figuring out the optimal location, structure and configuration of MEEUs and quadratic non-convex model for optimal operation strategy of MEEUs, which not only improves the computational efficiency but also ensures the solution accuracy. Finally, the related case studies are conducted to compare the proposed MEEU and distributed energy storage unit (DESU) in terms of energy consumption, renewable energy absorbency, carbon emission and line loss. The results verify the significant improvement of the comprehensive energy efficiency, the investment potentiality and the environmental benefits obtained by the proposed planning model.

Partial reformulation-linearization based optimization models for the Golomb ruler problem

In this paper, we provide a straightforward proof of a conjecture proposed in \cite{Duxbury2021} regarding the optimal solutions of a non-convex mathematical programming model of the Golomb ruler problem. Subsequently, we investigate the computational efficiency of four new binary mixed-integer linear programming models to compute optimal Golomb rulers. These models are derived from a well-known nonlinear integer model proposed in \cite{Kocuk2019}, utilizing the reformulation-linearization technique. \modif{Finally, we provide the correct outputs of the greedy heuristic proposed in \cite{Duxbury2021} and correct some false conclusions stated or implied therein.}

Nonconvex submodule clustering via joint sliced sparse gradient and cluster-aware approach

Most existing subspace clustering methods preprocess image data by converting them into vectors, which lacks exploration of the spatial structure of high-dimensional data. Therefore, we proposes a nonconvex submodule clustering model (NSSGCA) via joint sliced sparse gradient and cluster-aware approach. NSSGCA arranges each 2D image as lateral slices of a 3rd-order tensor, and utilizes the t-product under the model of the union of free submodules to represent 3rd-order tensor samples, thereby exploring the latent spatial structure of samples. To more accurately approximate tensor rank, a nonconvex Schatten p-norm constraint is imposed on the rotated representation tensor. Under the submodule framework, a consistent gradient matrix is derived based on the δ-nearest neighbor adjacency graph to construct sliced sparse gradient (SSG) regularization, which is more conducive to clustering tasks. NSSGCA learns representation tensor with clearer block-like structure based on ℓq norm and cluster-aware attention mechanism. The convergence of the constructed sequence to the stationary Karush–Kuhn–Tucker (KKT) point is proven. Experimental results on real-world image datasets confirm the effectiveness of NSSGCA.

Positive incentive CNN structure coupled nonconvex model for image super-resolution

This paper studies super-resolution (SR) technique to reconstruct high-quality images for deep image analysis. Currently, the convolutional neural networks (CNNs) are well performing methods and the finding that random noise added in the network can have positive incentive effect, we innovatively propose a positive incentive CNNs. However, concerning the uncontrollable characteristic and lack consistency of deep network, we propose a novel framework that joins nonconvex model based on framelet and positive incentive CNN structure, which can impose consistency between the high-resolved image and the given low-resolution image, and depict image information by sparse representation. Furthermore, to overcome the challenge of computing the minimizer of the nonconvex problem, we use proximal linearized minimization (PLM) algorithm to convex the nonconvex term, then apply the alternating direction method of multipliers (ADMM) as the solver which can converge to a stationary point of the nonconvex model. The experimental outcomes on Set5, Set14, BSD100, Urban100, and real-world images demonstrate that the proposed approach outperforms the state-of-the-art methods in terms of peak signal to noise ratio (PSNR) value, structural similarity index (SSIM), and visual quality.

Precise feature selection via non-convex regularized graph embedding and self-representation for unsupervised learning

In graph embedding learning with unsupervised feature selection fields, similarity matrices are usually obtained only from the initial noise-laden samples, and the probable association between various features is often ignored. To tackle the issues described above, this article proposes an effective unsupervised group feature selection approach via non-convex regularized graph embedding and self-representation (NLGMS). NLGMS integrates the similarity matrix and feature selection as a joint system on the basis of graph. It combines self-representation learning into the system to preserve the global structure and learns the nonconvex projection matrix to direct the procedure of feature selection. Specifically, first, NLGMS presents a unified framework for global structure learning, local structure learning, and feature selection process so the data’s intrinsic structure is excellently captured and learned adaptively. Second, nonconvex ℓ2,0-norm is enforced upon the projection matrix to perform the process of feature selection, ensuring that the optimal feature subset can be picked precisely. Nonconvex ℓ2,0-norm constraint considers the probable association between various features so that it can deliver optimal performance by considering the feature subset as an integral whole, which other convex regularized constraints often fail to achieve. Finally, the introduction to self-representation learning can facilitate the reconstruction of samples in low-rank subspaces, strengthening the model’s robustness. To tackle this challenging nonconvex model, an innovative alternative optimization technique is exploited. In addition, to validate the superiority of NLGMS, exhaustive experiments are performed on one synthetic dataset, six benchmark datasets, and one fish image dataset. The source code is available at: https://github.com/hrbai/NLGMS.

Bayesian fusion of multispectral and panchromatic images using a multi-mode and multiorder gradient tensor-based l 1 / 2 sparse model

ABSTRACT In this letter, based on the tensor representation modelling, we propose a multi-mode and multi-order gradient tensor-based non-convex model (M2GTNM) for Bayesian fusion of multispectral (MS) and panchromatic (Pan) images, which aims at producing the high-resolution MS (HRMS) images. Specifically, by modelling the MS image as the order-3 tensor, we mainly develop the multi-mode and multi-order gradient tensor sparse priors of MS image for fusion. For the spectral preservation of low-resolution MS (LRMS) image, the spectral fidelity constraint between HRMS and LRMS images is imposed. For the spatial-mode prior modelling, the multi-order spatial gradient tensor-based non-convex l 1 / 2 sparse prior between HRMS and Pan images is particularly imposed. Moreover, for the spectral-mode prior modelling, the spectral gradient tensor-based non-convex l 1 / 2 sparse prior between HRMS and upsampled LRMS images is further imposed. Then, we apply the alternating direction method of multipliers to optimize the proposed model. Finally, the reduced-scale and full-scale fusion experiments both validate the effectiveness of M2GTNM method.

A unified momentum-based paradigm of decentralized SGD for non-convex models and heterogeneous data

Emerging distributed applications recently boosted the development of decentralized machine learning, especially in IoT and edge computing fields. In real-world scenarios, the common problems of non-convexity and data heterogeneity result in inefficiency, performance degradation, and development stagnation. The bulk of studies concentrate on one of the issues mentioned above without having a more general framework that has been proven optimal. To this end, we propose a unified paradigm called UMP, which comprises two algorithms D-SUM and GT-DSUM based on the momentum technique with decentralized stochastic gradient descent (SGD). The former provides a convergence guarantee for general non-convex objectives, while the latter is extended by introducing gradient tracking, which estimates the global optimization direction to mitigate data heterogeneity (i.e., distribution drift). We can cover most momentum-based variants based on the classical heavy ball or Nesterov's acceleration with different parameters in UMP. In theory, we rigorously provide the convergence analysis of these two approaches for non-convex objectives and conduct extensive experiments, demonstrating a significant improvement in model accuracy up to 57.6% compared to other methods in practice.

Tensor robust PCA with nonconvex and nonlocal regularization

Tensor robust principal component analysis (TRPCA) is a classical way for low-rank tensor recovery, which minimizes the convex surrogate of tensor rank by shrinking each tensor singular value equally. However, for real-world visual data, large singular values represent more significant information than small singular values. In this paper, we propose a nonconvex TRPCA (N-TRPCA) model based on the tensor adjustable logarithmic norm. Unlike TRPCA, our N-TRPCA can adaptively shrink small singular values more and shrink large singular values less. In addition, TRPCA assumes that the whole data tensor is of low rank. This assumption is hardly satisfied in practice for natural visual data, restricting the capability of TRPCA to recover the edges and texture details from noisy images and videos. To this end, we integrate nonlocal self-similarity into N-TRPCA, and further develop a nonconvex and nonlocal TRPCA (NN-TRPCA) model. Specifically, similar nonlocal patches are grouped as a tensor and then each group tensor is recovered by our N-TRPCA. Since the patches in one group are highly correlated, all group tensors have strong low-rank property, leading to an improvement of recovery performance. Experimental results demonstrate that the proposed NN-TRPCA outperforms existing TRPCA methods in visual data recovery. The demo code is available at https://github.com/qguo2010/NN-TRPCA.

Coordinated planning of soft open point and energy store system in active Distribution networks under source-load imbalance

With the increasing penetration rate of renewable energy, the uncertainty of its output has exacerbated power imbalances among different distribution networks. In this paper, we propose collaborative planning of soft open points and energy storage systems to balance a distribution network with source-load imbalance, aiming to improve the economy and reliability of the distribution network. Firstly, considering that distributed generation has uncertainty, the K-means is used to construct a typical day scenario, on the basis of which a collaborative planning model is established with the objective function of minimizing the annual integrated cost. Then, the original nonlinear nonconvex model is transformed into a mixed-integer second-order conical planning model by Big-M and second-order conical relaxation, and a two-stage robust optimization is used to seek the planning solution with the lowest operational cost under the worst-case scenarios, aiming to enhance the robustness of the planning model. Finally, the feasibility of the proposed planning model is verified on two source-load unbalanced IEEE 33-node active distribution network arithmetic cases.

Omnichannel product selection and shelf space planning optimization

As the retail industry is evolving toward an omnichannel paradigm, we study a product selection and shelf space planning problem faced by omnichannel retailers. We aim to maximize aggregated profits across channels by determining the optimal product offerings for the offline and online channels while optimizing the allocation of limited shelf space to these products in brick-and-mortar stores. Furthermore, the showroom effect of physical stores on online sales is also taken into account. We propose a corresponding model that jointly optimizes these decisions. To solve the nonlinear nonconvex model of practical scale, we first reformulate it into a mixed integer quadratic programming by exploring the structural properties. Next, we use the reformulation linearization technique to further improve the computational speed. Our numerical studies validate the efficiency of the proposed solution approach. In addition, we derive some managerial insights, including that retailers should adopt joint optimization and consider how the involved effects and parameters may influence their profits and decisions.

Nonconvex weighted variational metal artifacts removal via convergent primal-dual algorithms

Direct reconstruction through filtered back projection engenders metal artifacts in polychromatic computed tomography images, attributed to highly attenuating implants, which further poses great challenges for subsequent image analysis. Inpainting the metal trace directly in the Radon domain for the extant variational method leads to strong edge diffusion and potential inherent artifacts. With normalization based on pre-segmentation, the inpainted outcome can be notably ameliorated. However, its reconstructive fidelity is heavily contingent on the precision of the pre-segmentation, and highly accurate segmentation of images with metal artifacts is non-trivial in actuality. In this paper, we propose a nonconvex weighted variational approach for metal artifact reduction. Specifically, in lieu of employing a binary function with zeros in the metal trace, an adaptive weight function is designed in the Radon domain, with zeros in the overlapping regions of multiple disjoint metals as well as areas of highly attenuated projections, and the inverse square root of the measured projection in other regions. A nonconvex L1−αL2 regularization term is incorporated to further enhance edge contrast, alongside a box-constraint in the image domain. Efficient first-order primal-dual algorithms, proven to be globally convergent and of low computational cost owing to the closed-form solution of all subproblems, are devised to resolve such a constrained nonconvex model. Both simulated and real experiments are conducted with comparisons to other variational algorithms, validating the superiority of the presented method. Especially in comparison to Reweighted JSR, our proposed algorithm can curtail the total computational cost to at most one-third, and for the case of inaccurate pre-segmentation, the recovery outcomes by the proposed algorithms are notably enhanced.

Dealing with congestion in the optimization of locating single-server battery swapping stations

This paper presents a study on the location problem of single-server battery swap stations, identifying instances of excessively long waiting times at certain stations during their operation in a real-world company scenario. This study innovatively transforms the problem into an extended version of the classic maximal covering location problem, incorporating technology selection and three sets of additional constraints: budget, closest-assignment, and average waiting time constraints. In particular, the closest-assignment constraint is specifically designed to mimic human behavioral patterns. A non-convex integer programming model is proposed and then reformulated thus enabling it to be tackled by some general-purpose solvers. The model is validated considering a test bed of randomly generated instances. The model comprehensiveness added-value is assessed. Finally, real-world data is used which allows discussing the extraction of information in the studied context. That information is used to build an instance that is solved and analyzed to provide valuable managerial insights, emphasizing the importance of technological improvement to the management.

Fabrication uncertainty guided design optimization of a photonic crystal cavity by using Gaussian processes

We present a fabrication uncertainty aware and robust design optimization approach that can be used to obtain robust design estimates for nonlinear, nonconvex, and expensive model functions. It is founded on Gaussian processes and a Monte Carlo sampling procedure, and assumes knowledge about the uncertainties associated with a manufacturing process. The approach itself is iterative. First, a large parameter domain is sampled in a coarse fashion. This coarse sampling is used primarily to determine smaller candidate regions to investigate in a second, more refined sampling pass. This finer step is used to obtain an estimate of the expected performance of the found design parameter under the assumed manufacturing uncertainties. We apply the presented approach to the robust optimization of the Purcell enhancement of a photonic crystal nanobeam cavity. We obtain a predicted robust Purcell enhancement of F¯P≈3.6. For comparison we also perform an optimization without robustness. We find that an unrobust optimum of FP≈256.5 dwindles to only F¯P≈0.7 when fabrication uncertainties are taken into account. We thus demonstrate that the presented approach is able to find designs of significantly higher performance than those obtained with conventional optimization.

Matheuristics vs. metaheuristics for joint lot-sizing and dynamic pricing problem with nonlinear demands

This paper considers a joint dynamic pricing and production planning decisions problem for a profit-maximizing firm that produces and sells multiple products. The objective is to develop a coordinated decision approach for multi-product pricing and lot sizing decisions for a manufacturer considering a limited production capacity. The demand for each product is assumed to be iso-elastic and integrates the complementarity and the substitution effects between the products. First, the problem is formulated as a non-convex mixed integer nonlinear programming model (MINLP) incorporating capacity constraints, setup costs, and nonlinear demand functions. Then, since the model is nonlinear and non-convex, a set of approximate approaches based on the Genetic algorithm, Late Acceptance Hill Climbing and Simulated Annealing methods are designed to solve this problem. Based on this study, the performances of two variants of approximate methods: matheuristics and metaheuristics are discussed and analyzed. The extensive experimental study, performed on real-world inspired instances, shows that matheuristic methods with setup-variables encoding scheme outperform the rest of the methods. The research outcomes show that coordinating a decision-making process by optimizing both prices and production plans simultaneously can result in significant profit for a company. However, one must consider the joint effect of the parameters of the demand function as well as the impact of the production capacity. This comprehensive understanding enables the company to avoid excessive investments in less lucrative products with lower sales potential, thereby ensuring resource allocation aligns with profitability and market demand.

Multi-Objective Battery Coordination in Distribution Networks to Simultaneously Minimize CO2 Emissions and Energy Losses

The techno–environmental analysis of distributed energy resources in electrical distribution networks is a complex optimization task due to the non-convexities of its nonlinear programming formulation. This research employs convex optimization to address this issue while minimizing the expected carbon dioxide emissions and daily energy losses of a distribution grid via the optimal dispatch of battery energy storage units (BESUs) and renewable energy units (REUs). The exact non-convex model is approximated via semi-definite programming in the complex variable domain. The optimal Pareto front is constructed using a weighting-based optimization approach. Numerical results using an IEEE 69-bus grid confirm the effectiveness of our proposal when considering unitary and variable power factor operation for the BESUs and the REUs. All numerical simulations were carried out using MATLAB software (version 2022b), a convex disciplined tool (CVX), and the semi-definite programming solvers SEDEUMI and SDPT3.

Quadratic Optimization Models for Balancing Preferential Access and Fairness: Formulations and Optimality Conditions

Typically, within facility location problems, fairness is defined in terms of accessibility of users. However, for facilities perceived as undesirable by communities hosting them, fairness between the usage of facilities becomes especially important. Limited research exists on this notion of fairness. To close this gap, we develop a series of optimization models for the allocation of populations of users to facilities such that access for users is balanced with a fair utilization of facilities. The optimality conditions of the underlying nonconvex quadratic models state the precise balance between accessibility and fairness. We define new classes of fairness and a metric to quantify the extent to which fairness is achieved in both optimal and suboptimal allocations. We show that a continuous relaxation of our central model is sufficient to achieve a perfect extent of fairness, while a special case reduces to the classical notion of proportional fairness. Our work is motivated by pervasive ecological challenges faced by the waste management community as policymakers seek to reduce the number of recycling centers in the last few years. As a computational case study, applying our models on data for the state of Bavaria in Germany, we find that even after the closure of a moderate number of recycling centers, large degrees of access can be ensured, provided that the closures are conducted optimally. Fairness, however, is impacted more, with facilities in rural regions shouldering larger loads of visiting populations than those in urban regions. History: Accepted by Pascal Van Hentenryck, Area Editor for Computational Modeling: Methods & Analysis. Funding: Computer resources and support provided by the Erlangen Regional Computing Center are gratefully acknowledged. B. Singh was partially financially supported by the Bavarian State Ministry for Science and Art (Bayerisches Staatsministerium für Wissenschaft und Kunst) under the Competence Network for Scientific High Performance Computing in Bavaria. Supplemental Material: The software that supports the findings of this study is available within the paper and its Supplemental Information ( https://pubsonline.informs.org/doi/suppl/10.1287/ijoc.2022.0308 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2022.0308 ). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/ .

An MTL1TV non-convex regularization model for MR Image reconstruction using the alternating direction method of multipliers

<abstract><p>The acquisition time of magnetic resonance imaging (MRI) is relatively long. To achieve high-quality and fast reconstruction of magnetic resonance (MR) images, we proposed a non-convex regularization model for MR image reconstruction with the modified transformed $ {l_1} $ total variation (MTL1TV) regularization term. We addressed this new model using the alternating direction method of multipliers (ADMM). To evaluate the proposed MTL1TV model, we performed numerical experiments on several MR images. The numerical results showed that the proposed model gives reconstructed images of improved quality compared with those obtained from state of the art models. The results indicated that the proposed model can effectively reconstruct MR images.</p></abstract>