Minimization Problem Research Articles

Results of Computational Experiment for Solving the Minimum Traveling Salesman Problem by the Cycle Merging Algorithm

The traveling salesman problem is an important combinatorial optimization problem, which consists in finding the shortest path that passes through each city exactly once and ends at the starting point. Despite its simple formulation, this problem turns out to be computationally complex, and its exact solution requires significant computational resources. In this regard, the search for efficient algorithms to solve the traveling salesman problem remains an urgent task in the context of modern technological challenges. In this paper we present the results of a computational experiment that gives an insight into the quality of the cycle merging algorithm for the minimum traveling salesman problem. The quality of the cycle merging algorithm is investigated on seven sets of instances of the traveling salesman problem. The results obtained are compared with the results of computational experiments for some known heuristics. The analysis allows us to say that the cycle merging algorithm shows relatively good results for all tested sets and can be used for a wide range of instances of the traveling salesman problem as a relatively fast heuristic of good enough quality.

ℓp-Sphere Covering and Approximating Nuclear p-Norm

The spectral p-norm and nuclear p-norm of matrices and tensors appear in various applications, albeit both are NP-hard to compute. The former sets a foundation of [Formula: see text]-sphere-constrained polynomial optimization problems, and the latter has been found in many rank minimization problems in machine learning. We study approximation algorithms of the tensor nuclear p-norm with an aim to establish the approximation bound matching the best one of its dual norm, the tensor spectral p-norm. Driven by the application of sphere covering to approximate both tensor spectral and nuclear norms ([Formula: see text]), we propose several types of hitting sets that approximately represent the [Formula: see text]-sphere with adjustable parameters for different levels of approximations and cardinalities, providing an independent toolbox for decision making on [Formula: see text]-spheres. Using the idea in robust optimization and second-order cone programming, we obtain the first polynomial-time algorithm with an [Formula: see text]-approximation bound for the computation of the matrix nuclear p-norm when [Formula: see text] is a rational, paving a way for applications in modeling with the matrix nuclear p-norm. These two new results enable us to propose various polynomial-time approximation algorithms for the computation of the tensor nuclear p-norm using tensor partitions, convex optimization, and duality theory, attaining the same approximation bound to the best one of the tensor spectral p-norms. Effective performance of the proposed algorithms for the tensor nuclear p-norm is shown by numerical implementations. We believe the ideas of [Formula: see text]-sphere covering with its applications in approximating nuclear p-norm would be useful for tackling optimization problems on other sets, such as the binary hypercube with its applications in graph theory and neural networks and the nonnegative sphere with its applications in copositive programming and nonnegative matrix factorization. Funding: This work was supported by the National Natural Science Foundation of China [Grants 71825003, 72394360, 72394364, 72171141, 72192830, and 72192832], and the program for Innovative Research Team of Shanghai University of Finance and Economics.

Adaptive Dictionary Identification Framework and its Application to Sparsity-Optimized Harmonic Noise Separation

Coherent noise separation stands as a critical step in seismic data processing. The Morphological Component Analysis (MCA)-based separation method, treating coherent noise and signal as distinct components and representing them sparsely with dictionaries, has found widespread application in the effective suppression of coherent noise. In practice, when constructing effective fixed dictionaries for MCA-based separation methods, a common approach involves conducting numerous experiments to meticulously select appropriate transform basis functions from an extensive dictionary library and tune their parameters. To mitigate time consumption and ensure optimal dictionary construction, we introduce a framework for adaptive identification of the optimal dictionaries used in MCA-based coherent noise separation. Specifically, we initially characterize the notion of a fixed dictionary library, which encompasses dictionaries constructed using various transform basis functions and their corresponding parameters. Then, we present a Relative Sparsity Minimization Problem (RSMP), formulated to identify the optimal fixed dictionaries that lead to minimum relative sparsity within the predefined library. Finally, we design a genetic algorithm to solve RSMP. The optimal dictionaries obtained for representing signal and coherent noise, respectively, are applied to MCA-based coherent noise separation. Synthetic and field data examples demonstrate the effectiveness of the proposed method.

An optimisation method for coplanar array capacitor image reconstruction based on Chambolle-Pock framework with automatic parameter selection

ABSTRACT Coplanar array capacitance imaging is an attractive imaging technique in industrial and detection fields, but low-quality images reduce its practicality and reliability. The inverse problem is difficult to solve and the image reconstruction quality is low, in this paper, an auxiliary optimisation strategy is proposed to transform the inverse problem of image reconstruction into a general minimisation problem, combined with the threshold of automatic parameter selection for feature extraction of capacitive data. The Kullback-Leibler(KL) data error term is combined with the Total Variation (TV) semi-norm to form a new imaging model. Utilising the Chambolle-Pock framework, the image reconstruction problem is transformed into a general minimisation problem to obtain high-quality images. Capacitance data from a 3 × 4 coplanar array capacitance sensor is used for feature extraction, serving as the basis for automatic parameter selection in the auxiliary optimisation strategy. The proposed Chambolle-Pock framework with the support of the auxiliary optimisation strategy, validated through simulation and experiments, outperforms popular imaging algorithms in terms of performance. The novel automatic parameter selection optimization algorithm introduced in this paper enhances the accuracy of capacitor image reconstruction in coplanar arrays.

OPTIMIZATION APPROACH FOR SOLVING THE CONTINUATION PROBLEM IN FREQUENCY DOMAIN

The paper presents analytical expressions for solving the problem of continuation of the electromagnetic field in the frequency domain for a horizontally layered medium. The numerical solution algorithm uses layerwise recalculation of the required quantities, for which analytical expressions are presented in a form that allows calculations and layerwise recalculation without the accumulation of rounding errors. The solution to the continuation problem is obtained as a solution to the cost functional minimization problem. The strong convexity of the functional is proven, which implies the existence of a unique solution to the problem posed. Before solving the continuation problem, it is proposed to apply the Butterworth filter to smooth the radargrams and the Hilbert transform technique to clarify the coordinates of the discontinuity points of the medium.

Covariance-Modulated Optimal Transport and Gradient Flows

We study a variant of the dynamical optimal transport problem in which the energy to be minimised is modulated by the covariance matrix of the distribution. Such transport metrics arise naturally in mean-field limits of certain ensemble Kalman methods for solving inverse problems. We show that the transport problem splits into two coupled minimization problems: one for the evolution of mean and covariance of the interpolating curve and one for its shape. The latter consists in minimising the usual Wasserstein length under the constraint of maintaining fixed mean and covariance along the interpolation. We analyse the geometry induced by this modulated transport distance on the space of probabilities as well as the dynamics of the associated gradient flows. Those show better convergence properties in comparison to the classical Wasserstein metric in terms of exponential convergence rates independent of the Gaussian target. On the level of the gradient flows a similar splitting into the evolution of moments and shapes of the distribution can be observed.

DeepMulticut: Deep Learning of Multicut Problem for Neuron Segmentation From Electron Microscopy Volume.

Superpixel aggregation is a powerful tool for automated neuron segmentation from electron microscopy (EM) volume. However, existing graph partitioning methods for superpixel aggregation still involve two separate stages-model estimation and model solving, and therefore model error is inherent. To address this issue, we integrate the two stages and propose an end-to-end aggregation framework based on deep learning of the minimum cost multicut problem called DeepMulticut. The core challenge lies in differentiating the NP-hard multicut problem, whose constraint number is exponential in the problem size. With this in mind, we resort to relaxing the combinatorial solver-the greedy additive edge contraction (GAEC)-to a continuous Soft-GAEC algorithm, whose limit is shown to be the vanilla GAEC. Such relaxation thus allows the DeepMulticut to integrate edge cost estimators, Edge-CNNs, into a differentiable multicut optimization system and allows a decision-oriented loss to feed decision quality back to the Edge-CNNs for adaptive discriminative feature learning. Hence, the model estimators, Edge-CNNs, can be trained to improve partitioning decisions directly while beyond the NP-hardness. Also, we explain the rationale behind the DeepMulticut framework from the perspective of bi-level optimization. Extensive experiments on three public EM datasets demonstrate the effectiveness of the proposed DeepMulticut.

TOA based mobile source localization with unknown start transmission time and sensor position uncertainty in asynchronous networks

ABSTRACT This letter addresses the issue of mobile source localization using time of arrival (TOA) measurements in scenarios where the source and sensors are not synchronized, and the start transmission time is unknown. We propose a non-convex weighted least squares (WLS) minimization problem for estimating the source position, velocity and start transmission time from the distance measurement equation, without the need for Taylor approximation commonly used in traditional processing approaches. To effectively solve the non-convex WLS optimization problem, we use the semi-positive definite relaxation (SDR) technique to transform the original problem into a convex semi-positive definite programming (SDP) problem. Additionally, we further extend the proposed method to scenarios where the positions of sensors are uncertain. Computer simulation results demonstrate that the proposed method surpasses the existing methods, whether the positions of sensors are known or uncertain.

Evolutionary multitasking algorithm based on a dynamic solution encoding strategy for the minimum s-club cover problem

Evolutionary multitasking algorithm based on a dynamic solution encoding strategy for the minimum s-club cover problem

Joint edge caching and computation offloading for heterogeneous tasks in MEC-enabled vehicular networks

Edge caching is an effective paradigm that can significantly reduce the computation task offloading latency for mobile edge computing (MEC) in vehicular networks, while also alleviating the backhaul transmission pressure for retrieving content data from the cloud server. However, most existing works fail to address how to handle heterogeneous tasks generated by vehicle terminals (VTs), especially in complex scenarios where both computation and content tasks are generated simultaneously. In this paper, we consider a mobility-aware vehicular network model where VTs simultaneously generate heterogeneous task requests, i.e., a computation task and a content task, and investigate joint optimization of caching for heterogeneous tasks data, computation offloading, and computing resource allocation. In order to optimize the latency for processing the heterogeneous tasks, an average execution latency minimization problem with sojourn time and caching capacity constraints is formulated. We decompose this problem into two tractable subproblems, i.e., caching optimization subproblem, and computation offloading and resource allocation optimization subproblem. We first develop a dynamic programming (DP) algorithm to obtain the optimal caching strategies for heterogeneous tasks data. We compare the obtained content retrieval latency with the local computing latency, and derive the optimal computation offloading and edge computing resource allocation solutions. On this basis, we propose a joint computation offloading and resource allocation (JCORA) algorithm to determine the computing resources allocated to each VT and corresponding computation offloading strategy. Numerical results indicate that the proposed algorithm, which integrates DP algorithm and JCORA algorithm, can achieve lower execution latency for heterogeneous tasks compared to the benchmark schemes. Additionally, for task loss scenarios where the sojourn time constraint cannot be met, the impact of VT mobility on the task loss probability is also revealed.

Optimizing Transmit Power for User-Cooperative Backscatter-Assisted NOMA-MEC: A Green IoT Perspective

Non-orthogonal multiple access (NOMA) enables the parallel offloading of multiuser tasks, effectively enhancing throughput and reducing latency. Backscatter communication, which passively reflects radio frequency (RF) signals, improves energy efficiency and extends the operational lifespan of terminal devices. Both technologies are pivotal for the next generation of wireless networks. However, there is little research focusing on optimizing the transmit power in backscatter-assisted NOMA-MEC systems from a green IoT perspective. In this paper, we aim to minimize the transmit energy consumption of a Hybrid Access Point (HAP) while ensuring task deadlines are met. We consider the integration of Backscatter Communication (BackCom) and Active Transmission (AT), and leverage NOMA technology and user cooperation to mitigate the double near–far effect. Specifically, we formulate a transmit energy consumption minimization problem, accounting for task deadline constraints, task offloading decisions, transmit power allocation, and energy constraints. To tackle the non-convex optimization problem, we employ variable substitution and convex optimization theory to transform the original non-convex problem into a convex one, which is then efficiently solved. We deduce the semi-closed form expression of the optimal solution and propose an energy-efficient algorithm to minimize the transmit power of the entire wireless powered MEC. The extensive simulation results demonstrate that our proposed scheme significantly reduces the HAP transmit power by around 8% compared to existing schemes, validating the effectiveness of our approach. This study provides valuable insights for the design of green IoT systems by optimizing the transmit power in NOMA-MEC networks.

Generalized MPD and DMP Inverses

For a generalized Drazin invertible operator A and its core–EP inverse A ◯ ​​​​ d , the expression A A ◯ ​​​​ d A plays important role in the representations of well-known generalized inverses, in theory of partial orders and for decompositions of operator A. Solving certain system of operator equations leads to introduction of new generalized inverse of A A ◯ ​​​​ d A . Since this new inverse can be expressed by the generalized Moore–Penrose inverse and generalized Drazin inverse of A, it is called a generalized MPD inverse of A A ◯ ​​​​ d A . As a dual version of generalized MPD inverse, we also consider a generalized DMP inverse. The dual core inverse and core inverse of A, respectively, are special cases of the generalized MPD inverse and generalized DMP inverse. Several properties, representations and characterizations of generalized MPD and DMP inverses are established. Applying the generalized MPD and DMP inverses, we solve corresponding systems of linear equations and minimization problems. One well-known minimization problem is a particular case of our results.

Deep learning-enhanced atomic norm minimization for DOA estimation of coherent and incoherent sources using coprime array

Abstract In the field of array signal processing, accurate direction of arrival (DOA) estimation is crucial for handling both coherent and incoherent mixed signal sources. This paper proposes a deep learning-enhanced atomic norm minimization (DLEANM) algorithm, designed to improve DOA estimation for coprime arrays. The algorithm first models the mixed signal scenarios realistically and employs interpolation to transform coprime arrays into virtual uniform linear arrays, which facilitates the formulation of the atomic norm minimization problem. Solving this problem reconstructs an augmented Hermitian Toeplitz covariance matrix. To further enhance the accuracy of DOA estimation, we developed a multi-scale convolutional neural network with an attention mechanism, which uses the covariance matrix as input to better capture signal variations across different frequencies and spatial distributions. By processing multi-scale inputs in parallel, the model improves adaptability and robustness in estimating mixed signal sources. Simulation results show that, under equivalent SNR and snapshot conditions, the DLEANM algorithm achieves lower estimation errors, more accurate multi-target DOA estimation, and relatively faster computation speed compared to conventional methods. Field experiments further confirm the effectiveness of the proposed algorithm. Therefore, after training, the algorithm is able to accurately identify the directions of unknown signals received by sensor arrays.

Research on local path planning of unmanned vehicles based on improved driving risk field.

With the rapid development of the field of unmanned vehicles, motion planning based on field theory has become a research hotspot. A driving risk field is an effective means to evaluate driving safety in complex environments, and this method is frequently used in autonomous vehicle motion planning. However, existing risk field models are not sufficiently accurate for describing driving risks, often disregarding the size and driving direction restrictions of vehicles, amongst other aspects. Considering the aforementioned problems, this research improves and establishes a new risk field model, including a motor vehicle risk field, a road risk field and a pedestrian risk field. Simultaneously, it proposes a solution to the local minimum point problem caused by different scenarios and verifies the simulation in MATLAB. Finally, the Prescan and MATLAB/Simulink co-simulation platform is used to compare the traditional and improved field theory algorithms. Results show that the trajectory generated by the improved field theory algorithm is smoother, and the fluctuation amplitude and number of parameters, such as heading angle, yaw rate and roll angle during driving, are significantly reduced. These outcomes improve the stability of driving whilst smoothly reaching the target point, demonstrating high application potential for the proposed model.

Construct, merge, solve and adapt

AbstractThe CMSA algorithm for combinatorial optimization is a hybrid technique based on repeatedly solving sub-instances to the original problem instance. The incumbent sub-instance is extended at each iteration by the probabilistic generation of valid solutions to the original problem instance and by adding the components found in these solutions to the sub-instance. In addition, the incumbent sub-instance is reduced at each iteration by removing seemingly useless solution components. In recent years the usefulness of the CMSA algorithm has been shown by a range of applications to different combinatorial optimization problems. In this work, we provide a gentle introduction to CMSA by describing the application to the so-called minimum global domination problem as an example.

Generative adversarial network-based ultrasonic full waveform inversion for high-density polyethylene structures

Accurate detection and characterization of defects are crucial to safety assurance of high-density polyethylene (HDPE) pipes used in the nuclear industry. Ultrasonic non-destructive evaluation (NDE) has the advantages of deep defect detection and high sensitivity, which can play a crucial role in the structural integrity of HDPE pipes. However, the quantitative reconstruction of high-contrast defects remains a significant challenge. To address this, a generative adversarial network based full waveform inversion (GAN-FWI) method is proposed for quantitatively reconstructing hidden defects in HDPE materials. This unsupervised learning method employs an acoustic wave equation based generator to optimize modeled data based on the feedback from a critic, which is used to differentiate between modeled data and measured data by adjusting network parameters. Compared to conventional full waveform inversion, the incorporation of physically constrained learning in the proposed GAN-FWI method can effectively alleviate the local minimum problem and aid in reconstructing high-contrast defects by reducing the sensitivity to the initial model and noise. Numerical and experimental results demonstrate the effectiveness of the proposed method in accurately and quantitatively reconstructing high-contrast defects in HDPE materials.

A pursuit-evasion game robot controller design based on a neural network with an improved optimization algorithm

A pursuit-evasion game (PEG) is a type of game that utilizes one or several cooperative pursuers to capture one or several evaders. The PEG game concept has been used in different multi-robot applications such as transportation or navigation applications, search and rescue, surveillance applications such as collision avoidance and air traffic control systems, multi-defense applications such as missile guidance systems, and medical applications such as analyzing biological behaviors. Regardless of the benefits of PEG, one of the main drawbacks of such systems is the computational burden and the immense time required to learn such systems. For this reason, this work proposes a neural network game based on the pursuit-evasion game, where the leader (evader) robot tries to eat several particles/apples distributed inside a closed game environment with boundary and inner obstacles. In contrast, a follower (pursuer) robot tries to capture the leader robot and stop the particle-eating process. The leader and follower robots were designed based on a differential two-wheel robot (DTWR). The neural network is presented to control and learn the leader and follower robot directions with respect to the boundary and inside obstacles in the game environment. The neural network weights are learned using an improved sine cosine algorithm based on chaotic theory (ISCACT). The ISCACT is proposed to solve and avoid the proposed game of being trapped in the local minimum problem. The ISCACT is tested based on five multimodal benchmark functions. The ISCACT has been used in two cases, the first case arises when ISCACT is used in the follower robot’s learning process. In the second case, the ISCACT has been used in the leader robot’s learning process. The results for the first and second cases prove the superiority of the ISCACT compared with other existing works in enhancing the PEG performance time and reducing the computational burden for multi-robot applications.

Codesign of transmit waveform and reflective beamforming for active reconfigurable intelligent surface-aided MIMO ISAC system

This article discusses the active reconfigurable intelligent surface (ARIS)-aided integrated sensing and communication (ISAC) system for non-line-of-sight (NLoS) target sensing in cluttered environments while performing multi-user communication. To optimize sensing and communication performance simultaneously, we jointly design the shared transmit waveform, ARIS reflection coefficients and radar receive filter by using the multi-user interference and the reciprocal of radar output signal-to-interference-plus-noise ratio as metrics. Limited by practical requirements, the transmit waveform suffers from constant modulus or total energy constraints and the ARIS is subject to both maximum power and amplification gain constraints. Based on these considerations, the proposed codesign is formulated into a nonconvex constrained fractional function minimization problem. To tackle it effectively, we first translate the fractional objective into an integral form by employing Dinkelbach transform and then propose an alternating optimization-based algorithm, where the transmit waveform and ARIS reflection coefficients are respectively optimized by the customized algorithms based on the consensus alternating direction method of multipliers, and the receive filter has a closed-form optimal solution. Numerical results demonstrate that the ARIS-aided ISAC concurrently achieve superior NLoS sensing and communication performance to passive reconfigurable intelligent surface-aided and traditional ISACs in cluttered environments, regardless of waveform constraints and sensing-communication trade-off factor.

JointLIME: An interpretation method for machine learning survival models with endogenous time-varying covariates in credit scoring.

In this work, we introduce JointLIME, a novel interpretation method for explaining black-box survival (BBS) models with endogenous time-varying covariates (TVCs). Existing interpretation methods, like SurvLIME, are limited to BBS models only with time-invariant covariates. To fill this gap, JointLIME leverages the Local Interpretable Model-agnostic Explanations (LIME) framework to apply the joint model to approximate the survival functions predicted by the BBS model in a local area around a new individual. To achieve this, JointLIME minimizes the distances between survival functions predicted by the black-box survival model and those derived from the joint model. The outputs of this minimization problem are the coefficient values of each covariate in the joint model, serving as explanations to quantify their impact on survival predictions. JointLIME uniquely incorporates endogenous TVCs using a spline-based model coupled with the Monte Carlo method for precise estimations within any specified prediction period. These estimations are then integrated to formulate the joint model in the optimization problem. We illustrate the explanation results of JointLIME using a US mortgage data set and compare them with those ofSurvLIME.

Solving the minimum labeling global cut problem by mathematical programming

AbstractAn edge‐labeled graph (ELG) is a graph such that is the set of vertices, is the set of edges, is the set of labels (colors), and each edge has a label associated. Given an ELG , the goal of the minimum labeling global cut problem (MLGCP) is to find a subset such that the removal of all edges with labels in disconnects and is minimum. This work proposes three new mathematical formulations for the MLGCP, namely PART, VC, and TE as well as branch‐and‐cut algorithms to solve them. Additionally, a theoretical study was carried out on the MLGCP input graph, leading to the concept of chromatic closure, used in preprocessing algorithms for this model PART. Finally, a comprehensive polyhedral investigation of the model is performed. The computational experiments showed that the model, adopting the chromatic closure concept and its branch‐and‐cut algorithm, can solve small to average‐sized instances in reasonable times.