Alternating Direction Method Research Articles

Deep image prior and weighted anisotropic-isotropic total variation regularization for solving linear inverse problems

Deep learning, particularly unsupervised techniques, has been widely used to solve linear inverse problems due to its flexibility. A notable unsupervised approach is the deep image prior (DIP), which employs a predetermined deep neural network to regularize inverse problems by imposing constraints on the generated image. This article introduces an optimization technique (DIP-AITV) by combining the DIP with the weighted anisotropic-isotropic total variation (AITV) regularization. Furthermore, we utilize the alternating direction method of multipliers (ADMM), a highly flexible optimization technique, to solve the DIP-AITV minimization problem effectively. To demonstrate the benefits of the proposed DIP-AITV method over the state-of-the-art DIP, DIP-TV, DIP-WTV and CS-DIP, we solve two linear inverse problems, i.e., image denoising and compressed sensing. Computation examples on the MSE and PSNR values show that our method outperforms the existing DIP-based methods in both synthetic and real grayscale and color images.

Image reconstruction for compressed ultrafast photography based on manifold learning and the alternating direction method of multipliers

Compressed ultrafast photography (CUP) is a high-speed imaging technique with a frame rate of up to ten trillion frames per second (fps) and a sequence depth of hundreds of frames. This technique is a powerful tool for investigating ultrafast processes. However, since the reconstruction process is an ill-posed problem, the image reconstruction will be more difficult with the increase of the number of reconstruction frames and the number of pixels of each reconstruction frame. Recently, various deep-learning-based regularization terms have been used to improve the reconstruction quality of CUP, but most of them require extensive training and are not generalizable. In this paper, we propose a reconstruction algorithm for CUP based on the manifold learning and the alternating direction method of multipliers framework (ML-ADMM), which is an unsupervised learning algorithm. This algorithm improves the reconstruction stability and quality by initializing the iterative process with manifold modeling in embedded space (MMES) and processing the image obtained from each ADMM iterative with a nonlinear modeling based on manifold learning. The numerical simulation and experiment results indicate that most of the spatial details can be recovered and local noise can be eliminated. In addition, a high-spatiotemporal-resolution video sequence can be acquired. Therefore, this method can be applied for CUP with ultrafast imaging applications in the future.

Robust optimization for integrated energy systems based on multi-energy trading

Amid rising energy demands and environmental concerns, integrated energy systems (IESs) face conflicting interests. Conventional strategies of interest distribution face issues such as irrational resource allocation. Accordingly, establishing energy trading strategies with multiple stakeholders becomes essential. This paper proposes a robust optimization (RO) for IESs based on multi-energy trading to reduce energy trading cost. In this regard, a single-leader-multi-follower Stackelberg game is first modeled where IES acts as the leader, and the users and the electric vehicles (EVs) act as the followers. Secondly, this model is transformed into a single-layer linear model and integrated into the multi-stage RO. With this, the nonanticipativity in the two-stage RO can be effectively handled. Besides, by constructing multi-interval uncertainty sets for renewable energy, load, and electricity prices, the conservatism of the model is reduced, making the optimization results closer to actual condition. Noteworthy that the Nash bargaining method ensures a fair distribution of benefits among IESs and encourages them to participate in energy trading. Finally, the multi-energy trading model is solved using the prediction-correction-based alternating direction method with multipliers (PCB-ADMM) algorithm. The PCB-ADMM not only protects each IES’s privacy but also has less execution time than the ADMM algorithm. The effectiveness of the proposed strategy is validated through simulation using Matlab.

EfficientQ: An efficient and accurate post-training neural network quantization method for medical image segmentation

Model quantization is a promising technique that can simultaneously compress and accelerate a deep neural network by limiting its computation bit-width, which plays a crucial role in the fast-growing AI industry. Despite model quantization’s success in producing well-performing low-bit models, the quantization process itself can still be expensive, which may involve a long fine-tuning stage on a large, well-annotated training set. To make the quantization process more efficient in terms of both time and data requirements, this paper proposes a fast and accurate post-training quantization method, namely EfficientQ. We develop this new method with a layer-wise optimization strategy and leverage the powerful alternating direction method of multipliers (ADMM) algorithm to ensure fast convergence. Furthermore, a weight regularization scheme is incorporated to provide more guidance for the optimization of the discrete weights, and a self-adaptive attention mechanism is proposed to combat the class imbalance problem. Extensive comparison and ablation experiments are conducted on two publicly available medical image segmentation datasets, i.e., LiTS and BraTS2020, and the results demonstrate the superiority of the proposed method over various existing post-training quantization methods in terms of both accuracy and optimization speed. Remarkably, with EfficientQ, the quantization of a practical 3D UNet only requires less than 5 min on a single GPU and one data sample. The source code is available at https://github.com/rongzhao-zhang/EfficientQ.

An asynchronous distributed optimal wake control scheme for suppressing fatigue load and increasing power extraction in wind farms

The fatigue damage and power generation of wind farms (WFs) is related to their service lifetime and economic benefits. In this paper, an asynchronous distributed optimal wake control (OWC) scheme is proposed to suppress fatigue load and increase power extraction in doubly-fed induction generator (DFIG)-based wind farm (WF). To improve control flexibility, the generator torque and pitch angle of a wind turbine (WT) is simultaneously controlled. Then, considering the influence of wake effect, the wind speed sensitivity is used to combine with the states of WTs for formulating the model predictive control (MPC)-based optimal control objectives. To deal with the higher computational complexity of the aerodynamic interactions efficiently, an asynchronous distributed alternating direction method of multipliers (AD-ADMM) is designed for the control system. A layout of a 3 × 5 wake WF is considered in the case studies to validate the outstanding performance of the proposed scheme.

Two-Tier Efficient QoE Optimization for Partitioning and Resource Allocation in UAV-Assisted MEC.

Unmanned aerial vehicles (UAVs) have increasingly become integral to multi-access edge computing (MEC) due to their flexibility and cost-effectiveness, especially in the B5G and 6G eras. This paper aims to enhance the quality of experience (QoE) in large-scale UAV-MEC networks by minimizing the shrinkage ratio through optimal decision-making in computation mode selection for each user device (UD), UAV flight trajectory, bandwidth allocation, and computing resource allocation at edge servers. However, the interdependencies among UAV trajectory, binary task offloading mode, and computing/network resource allocation across numerous IoT nodes pose significant challenges. To address these challenges, we formulate the shrinkage ratio minimization problem as a mixed-integer nonlinear programming (MINLP) problem and propose a two-tier optimization strategy. To reduce the scale of the optimization problem, we first design a low-complexity UAV partition coverage algorithm based on the Welzl method and determine the UAV flight trajectory by solving a traveling salesman problem (TSP). Subsequently, we develop a coordinate descent (CD)-based method and an alternating direction method of multipliers (ADMM)-based method for network bandwidth and computing resource allocation in the MEC system. Extensive simulations demonstrate that the CD-based method is simple to implement and highly efficient in large-scale UAV-MEC networks, reducing the time complexity by three orders of magnitude compared to convex optimization methods. Meanwhile, the ADMM-based joint optimization method achieves approximately an 8% reduction in shrinkage ratio optimization compared to baseline methods.

TIHN: Tensor Improved Huber Norm for low-rank tensor recovery

Tensor Robust Principal Component Analysis (TRPCA) has received much attention in many real-world applications which aims to recover a low-rank tensor corrupted by sparse noise. Most existing TRPCA methods usually regularize the low-rank component by minimizing its Tensor Nuclear Norm (TNN). However, the original TNN shrinks all of the singular values with the same penalty which generally leads to suboptimal results. The reason is that the large singular values should be less penalized since they usually correspond to salient information of the real-world tensor. In this work, we develop a Tensor Improved Huber Norm (TIHN) which considers the difference between the singular values of tensor. The TIHN can achieve better performance by recovering the large and significant singular values exactly. We also establish the Huber TRPCA (HTRPCA) method by utilizing the proposed TIHN. In addition, an efficient optimization algorithm based on the half-quadratic theory and Alternating Direction Method of Multipliers (ADMM) framework is designed to implement the HTRPCA. Finally, experiments on color image recovery and video recovery validate the effectiveness of the proposed method.

Impact Force Localization and Reconstruction via ADMM-based Sparse Regularization Method

In practice, simultaneous impact localization and time history reconstruction can hardly be achieved, due to the ill-posed and under-determined problems induced by the constrained and harsh measuring conditions. Although ℓ1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell_{1}$$\\end{document} regularization can be used to obtain sparse solutions, it tends to underestimate solution amplitudes as a biased estimator. To address this issue, a novel impact force identification method with ℓp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell_{p}$$\\end{document} regularization is proposed in this paper, using the alternating direction method of multipliers (ADMM). By decomposing the complex primal problem into sub-problems solvable in parallel via proximal operators, ADMM can address the challenge effectively. To mitigate the sensitivity to regularization parameters, an adaptive regularization parameter is derived based on the K-sparsity strategy. Then, an ADMM-based sparse regularization method is developed, which is capable of handling ℓp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell_{p}$$\\end{document} regularization with arbitrary p values using adaptively-updated parameters. The effectiveness and performance of the proposed method are validated on an aircraft skin-like composite structure. Additionally, an investigation into the optimal p value for achieving high-accuracy solutions via ℓp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell_{p}$$\\end{document} regularization is conducted. It turns out that ℓ0.6\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell_{0.6}$$\\end{document} regularization consistently yields sparser and more accurate solutions for impact force identification compared to the classic ℓ1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell_{1}$$\\end{document} regularization method. The impact force identification method proposed in this paper can simultaneously reconstruct impact time history with high accuracy and accurately localize the impact using an under-determined sensor configuration.

Joint k-ω Space Image Reconstruction and Data Fitting for Chemical Exchange Saturation Transfer Magnetic Resonance Imaging.

Chemical exchange saturation transfer (CEST) magnetic resonance imaging (MRI) is a novel MRI technology to image certain compounds at extremely low concentrations. Long acquisition time to measure signals at a set of offset frequencies of the Z-spectra and to repeat measurements to reduce noise pose significant challenges to its applications. This study explores correlations of CEST MR images along the spatial and Z-spectral dimensions to improve MR image quality and robustness of magnetization transfer ratio (MTR) asymmetry estimation via a joint k-ω reconstruction model. The model was formulated as an optimization problem with respect to MR images at all frequencies ω, while incorporating regularizations along the spatial and spectral dimensions. The solution was subject to a self-consistency condition that the Z-spectrum of each pixel follows a multi-peak data fitting model corresponding to different CEST pools. The optimization problem was solved using the alternating direction method of multipliers. The proposed joint reconstruction method was evaluated on a simulated CEST MRI phantom and semi-experimentally on choline and iopamidol phantoms with added Gaussian noise of various levels. Results demonstrated that the joint reconstruction method was more tolerable to noise and reduction in number of offset frequencies by improving signal-to-noise ratio (SNR) of the reconstructed images and reducing uncertainty in MTR asymmetry estimation. In the choline and iopamidol phantom cases with 10.5% noise in the measurement data, our method achieved an averaged SNR of 31.0 dB and 32.2 dB compared to the SNR of 24.7 dB and 24.4 dB in the conventional reconstruction approach. It reduced uncertainty of the MTR asymmetry estimation over all regions of interest by 54.4% and 43.7%, from 1.71 and 2.38 to 0.78 and 1.71, respectively.

Imaging quality enhancement in photon-counting single-pixel imaging via an ADMM-based deep unfolding network in small animal fluorescence imaging.

Photon-counting single-pixel imaging (SPI) can image under low-light conditions with high-sensitivity detection. However, the imaging quality of these systems will degrade due to the undersampling and intrinsic photon-noise in practical applications. Here, we propose a deep unfolding network based on the Bayesian maximum a posterior (MAP) estimation and alternating direction method of multipliers (ADMM) algorithm. The reconstruction framework adopts a learnable denoiser by convolutional neural network (CNN) instead of explicit function with hand-crafted prior. Our method enhances the imaging quality compared to traditional methods and data-driven CNN under different photon-noise levels at a low sampling rate of 8%. Using our method, the sensitivity of photon-counting SPI prototype system for fluorescence imaging can reach 7.4 pmol/ml. In-vivo imaging of a mouse bearing tumor demonstrates an 8-times imaging efficiency improvement.

Crafting a decentralized framework for energy sharing among smart homes in a peer-to-peer market

In this paper, a new framework for the participation of smart homes in the peer-to-peer energy market is presented, in which smart homes share their surplus energy with each other to maintain the balance between generation and demand through bilateral trading. The seller smart homes, players who always sell their excess energy to the buyer smart homes in the peer-to-peer market, are equipped with distributed energy resources including electric vehicles, power-only units, and uncontrollable loads. The buyer smart homes always interact with sellers as energy consumers to supply their flexible loads. The presence of battery energy storage systems in the model of these players, as well as active participation in load response programs, increases the flexibility and profit obtained from energy transactions. All players regardless of their role in the peer-to-peer market, have bilateral trading with the retail market considering real-time energy pricing rates to gain more profit. A fully decentralized method called the Fast Alternating Direction Method of multipliers (FADMM) is used to clear the developed market. To prove the effectiveness of the proposed framework, numerical studies are considered for the energy transactions of several smart homes with each other and with the retail market.

Framework design of distributed P2P2G energy transaction considering photovoltaic power and dispatch of manufacturing facilities

In this paper, a distributed peer-to-peer-to-grid (P2P2G) energy transaction framework considering photovoltaic generation and dispatch of manufacturing facilities is designed to increase the penetration of renewable energy resource. The factory with typical manufacturing assembly production task, photovoltaic panels and battery is regarded as a micro-grid (MG) to participate in the energy transaction market. The designed framework contains two levels. In the upper level, each MG determines the scheduling of manufacturing facilities and battery and the quantities of P2P2G energy transactions based on the given P2P energy transaction price and distribution locational marginal price (DLMP), in which the complex typical manufacturing assembly process is formulated as a kind of deferable load and the DLMP is updated according to the optimal power flow (OPF) of the network. In the lower level, peer-to-peer (P2P) energy transaction prices are updated by non-cooperative games and the homologous Rosen-Nash equilibrium points are obtained by using Nikaido-Isoda relaxation and alternating direction method of multipliers (ADMM) algorithm. The efficacy of the designed framework is demonstrated by a case study, and the results indicate the designed framework can improve the utilisation rate of renewable energy.

Considering the Tiered Low-Carbon Optimal Dispatching of Multi-Integrated Energy Microgrid with P2G-CCS

The paper addresses the overlooked interaction between power-to-gas (P2G) devices and carbon capture and storage (CCS) equipment, along with the stepwise carbon trading mechanism in the context of current multi-park integrated energy microgrids (IEMGs). Additionally, it covers the economic and coordinated low-carbon operation issues in multi-park IEMGs under the carbon trading system. It proposes a multi-park IEMG low-carbon operation strategy based on the synchronous Alternating Direction Method of Multipliers (ADMM) algorithm. The algorithm first enables the distribution of cost relationships among multi-park IEMGs. Then, using a method that combines a CCS device with a P2G unit in line with the tiered carbon trading scheme, it expands on the model of single IEMGs managing thermal, electrical, and refrigeration energy. Finally, the comparison of simulation cases proves that the proposed strategy significantly reduces the external energy dependence while keeping the total cost of the users unchanged, and the cost of interaction with the external grid is reduced by 56.64%, the gas cost is reduced by 27.78%, and the carbon emission cost is reduced by 29.54% by joining the stepped carbon trading mechanism.

Human-safe and economic operation of renewable hydrogen-based microgrids under plateau conditions

The cold and hypoxic environment threatens the health and quality of people's lives in plateau areas. Appropriate air pressure, oxygen, and temperature are crucial for human-safe aspects in plateau microgrids. Facing extreme scenarios in these areas, we propose a two-stage robust operation strategy for renewable hydrogen-based microgrids considering the reliable supply of hydrogen and oxygen. To cope with rapid weather changes in these areas, the distributionally robust optimization (DRO) model that incorporates first-moment information is employed to depict renewable energy uncertainty. This study focuses on two weakly connected microgrids with distinct characteristics, showcasing their potential for mutual support. In the first stage, the plateau microgrid fulfills demands for electric power, hydrogen, and oxygen, with the power support from the another microgrid. In the second stage, reserves such as hydrogen fuel cells, diesel generators, and hydropower are utilized to mitigate renewable energy uncertainty. The operation optimization problem based on DRO is solved using the alternating direction method of multipliers. Case studies show that the strategy can coordinate complementary operation to enhance reliability, guarantee human safety, and save costs by 39.29 % and 38.73 % for the two microgrids.

Self-adaptive alternating direction method of multiplier for a fourth order variational inequality

We propose an alternating direction method of multiplier for approximation solution of the unilateral obstacle problem with the biharmonic operator. We introduce an auxiliary unknown and augmented Lagrangian functional to deal with the inequality constrained, and we deduce a constrained minimization problem that is equivalent to a saddle-point problem. Then the alternating direction method of multiplier is applied to the corresponding problem. By using iterative functions, a self-adaptive rule is used to adjust the penalty parameter automatically. We show the convergence of the method and give the penalty parameter approximation in detail. Finally, the numerical results are given to illustrate the efficiency of the proposed method.

Dynamic Consensus-Based ADMM Strategy for Economic Dispatch with Demand Response in Power Grids

This paper introduces a dynamic consensus-based economic dispatch (ED) algorithm utilizing the Alternating Direction Method of Multipliers (ADMM) to optimize real-time pricing and generation/demand decisions within a decentralized energy management framework. The increasing complexity of modern energy markets, driven by the proliferation of Distributed Energy Resources (DER) and variable demands from hybrid electric vehicles, necessitates a departure from traditional centralized dispatch methods. This research proposes a novel ADMM-based solution tailored for non-responsive and responsive demand units that integrates demand response mechanisms to adaptively manage real-time fluctuations while enhancing security and privacy through distributed data management. The testing of the algorithm on the IEEE 39 bus system under various load conditions over 24 h demonstrated the algorithm’s effectiveness in handling traditional and renewable energy sources, particularly highlighting the economic benefits of shifting controllable loads to periods of low-cost renewable availability. The findings underscore the algorithm’s potential to reduce energy costs, enhance energy efficiency, and offer a scalable solution across diverse grid systems, contributing significantly to advancing global energy policy and sustainable management practices.

A Semi-Supervised Adaptive Matrix Machine Approach for Fault Diagnosis in Railway Switch Machine.

The switch machine, an essential element of railway infrastructure, is crucial in maintaining the safety of railway operations. Traditional methods for fault diagnosis are constrained by their dependence on extensive labeled datasets. Semi-supervised learning (SSL), although a promising solution to the scarcity of samples, faces challenges such as the imbalance of pseudo-labels and inadequate data representation. In response, this paper presents the Semi-Supervised Adaptive Matrix Machine (SAMM) model, designed for the fault diagnosis of switch machine. SAMM amalgamates semi-supervised learning with adaptive technologies, leveraging adaptive low-rank regularizer to discern the fundamental links between the rows and columns of matrix data and applying adaptive penalty items to correct imbalances across sample categories. This model methodically enlarges its labeled dataset using probabilistic outputs and semi-supervised, automatically adjusting parameters to accommodate diverse data distributions and structural nuances. The SAMM model's optimization process employs the alternating direction method of multipliers (ADMM) to identify solutions efficiently. Experimental evidence from a dataset containing current signals from switch machines indicates that SAMM outperforms existing baseline models, demonstrating its exceptional status diagnostic capabilities in situations where labeled samples are scarce. Consequently, SAMM offers an innovative and effective approach to semi-supervised classification tasks involving matrix data.

Low-carbon optimal scheduling of park-integrated energy system based on bidirectional Stackelberg-Nash game theory

Multi-stakeholder participation is crucial in facilitating the development of park-integrated energy systems (PIES). Balancing the diverse interests of various stakeholders, each with its distinct requirements presents a notable challenge. Concurrently, the model's complexity increases due to the engagement of various stakeholders, posing challenges to its resolution through traditional methods. In this context, this paper aims to investigate an optimal scheduling model that incorporates shared energy storage (SES) system, microgrids operator (MGO), electric vehicles station (EVS), and user aggregator (UA) with multiple prosumers. To comprehensively address the interests of all stakeholders, this study introduces a tri-level optimization model. The proposed model integrates a bidirectional Stackelberg-Nash game framework, in which the SES acts as the leader, the MGO acts as the secondary leader, and the UA-EVS acts as the followers while allocating benefits based on the asymmetric Nash bargaining theory. The Stackelberg game model between MGO and UA-EVS is analyzed using the Karush-Kuhn-Tucker (KKT) condition, while the Stackelberg game model between SES and MGO is resolved using the bisection method. Meanwhile, the Nash bargaining method among users is solved using the alternating direction method of multipliers (ADMM) technique. The analysis indicates that the proposed strategy can reduce PIES's costs and carbon emissions, yielding a win-win situation for all stakeholders.

Low-complexity frequency-invariant beampattern synthesis using accurate response control for speech extraction

Beampattern synthesis generally designs weight parameters to form a main beam towards the direction of interest while suppressing interference and environmental noise simultaneously, playing a critical role in fixed beamforming techniques. In practical applications, when the position of the interference source changes, the filter design requirements will correspondingly change, thus necessitating accurate response control and rapid update capabilities. A low-complexity method is also highly desired in broadband beampattern synthesis, especially for time-domain beamformers with a relatively large number of weight parameters. To achieve rapid update and reduce the complexity for time-domain beamformers, this paper proposes a low-complexity frequency-invariant beampattern synthesis method using accurate response control. By introducing the null-forming scheme of adaptive beamformers, a virtual interference-plus-noise covariance matrix is constructed to precisely control the beampattern. Additionally, the spatial response variation is considered in the optimization problem for a constant mainlobe pattern, avoiding the complexity of determining a desired beampattern. Compared with the existing algorithms based on the interior-point method and the alternating direction method of multipliers, the proposed method reduces the computational complexity of each iteration, showing higher efficiency in broadband beampattern synthesis. Furthermore, the proposed method adjusts the beamformer flexibly on the basis of a pre-designed beampattern, achieving fast beampattern update when the acoustic environment changes. Numerical simulations on beam patterns and experimental results on speech extraction demonstrate better interference suppression performance of the beamformers designed by the proposed method.

A novel fast robust optimization algorithm for intensity-modulated proton therapy with minimum monitor unit constraint.

Intensity-modulated proton therapy (IMPT) optimizes spot intensities and position, providing better conformability. However, the successful application of IMPT is dependent upon addressing the challenges posed by range and setup uncertainties. In order to address the uncertainties in IMPT, robust optimization is essential. This study aims to develop a novel fast algorithm for robust optimization of IMPT with minimum monitor unit (MU) constraint. The study formulates a robust optimization problem and proposes a novel, fast algorithm based on the alternating direction method of multipliers (ADMM) framework. This algorithm enables distributed computation and parallel processing. Ten clinical cases were used as test scenarios to evaluate the performance of the proposed approach. The robust optimization method (RBO-NEW) was compared with plans that only consider nominal optimization using CTV (NMO-CTV) without handling uncertainties and PTV (NMO-PTV) to handle the uncertainties, as well as with conventional robust-optimized plans (RBO-CONV). Dosimetric metrics, including D95, homogeneity index, and Dmean, were used to evaluate the dose distribution quality. The area under the root-mean-square dose (RMSD)-volume histogram curves (AUC) and dose-volume histogram (DVH) bands were used to evaluate the robustness of the treatment plan. Optimization time cost was also assessed to measure computationalefficiency. The results demonstrated that the RBO plans exhibited better plan quality and robustness than the NMO plans, with RBO-NEW showing superior computational efficiency and plan quality compared to RBO-CONV. Specifically, statistical analysis results indicated that RBO-NEW was able to reduce the computational time from to s ( ) and reduce the mean organ-at-risk (OAR) dose from % of the prescription dose to % of the prescription dose ( ) compared toRBO-CONV. This study introduces a novel fast robust optimization algorithm for IMPT treatment planning with minimum MU constraint. Such an algorithm is not only able to enhance the plan's robustness and computational efficiency without compromising OAR sparing but also able to improve treatment plan quality andreliability.