Complicated Optimization Problem Research Articles

Multi-Agent Reinforcement Learning With Measured Difference Reward for Multi-Association in Ultra-Dense mmWave Network

Millimeter Wave (mmWave) communication technology is anticipated to play a vital role in meeting the growing demand for the scarce bandwidth in wireless communications. However, mmWave networks are highly susceptible to blockage. Thus, some mitigation techniques, such as multi-connectivity, need to be considered. Densely deploying mmWave base stations (mBSs) to form an ultra-dense network (UDN) also helps. With a mix of different technologies, optimally allocating resources becomes challenging. In this paper, we study mmWave user multi-association in a two-tier heterogeneous ultra-dense network (HetUDN) with a relatively large number of user equipments (UEs). We propose a framework of multi-agent reinforcement learning (MARL) to tackle the complicated optimization problem, leveraging its adaptivity to the communication environment. The proposed scheme considers mmWave beam-division based multi-connectivity and takes advantage of a macro base station (MBS) for indirect cooperation among agents (UEs). In particular, we borrow a credit-assignment technique called difference reward (DR) to deal with a relatively large MARL system with a large action space, which, to the best of our knowledge, is the first time to apply MARL with DR in user association. Furthermore, the proposed schemes are scalable mainly due to fixed observation dimensions and individual actions taken by UEs independently, ensuring that the operation is independent of the numbers of mBSs and UEs. Numerical results suggest that the two MARL schemes with measured DR could achieve a good balance between energy efficiency and QoS outage, and the one using extended DR (EDR) offers additional performance improvement.

The Multi-Objective Design of Laminated Structure with Non-Dominated Sorting Genetic Algorithm

Non-dominated sorting genetic algorithm has shown excellent advantages in solving complicated optimization problems with discrete variables in a variety of domains. In this paper, we implement a multi-objective genetic algorithm to guide the design of the laminated structure with two objectives: minimizing the mass and maximizing the strength of a specified structure simultaneously, classical lamination theory and failure theory are adopted to compute the strength of a laminate. The simulation results have shown that a non-dominated genetic algorithm has great advantages in the design of laminated composite material. Experiment results also suggest that optimal run times are from 16 to 32 for the design of glass-epoxy laminate with non-dominated sorting genetic algorithm. We also observed that two stages involve the optimization process in which the number of individuals in the first frontier first increases, and then decreases. These simulation results are helpful to decide the proper run times of genetic algorithms for glass-epoxy design and reduce computation costs.

Battery parameter identification strategy based on modified coot optimization algorithm

Coot algorithm (COOT) is a new metaheuristic algorithm (MA) based on the swarm of COOT birds' behavior. Like other MAs, the COOT algorithm may easily become trapped in local optima, tends to low diversity, slow convergence speed, and imperfect balance between exploitation/exploration. Thus, to overcome these shortcomings of the original COOT, seven effective strategies are injected into COOT, such as - best agent guide, control randomization, Transition Factor (TF), adjusting the position, leading the group towards the optimal area, phasor operator and Opposition Based Learning (OBL). To be specific, to alleviate these shortcomings, a new modified Coot algorithm (COOT) called mCOOT algorithm is proposed. To investigate the efficiency of mCOOT, it has been verified on complex and standard test suite given in IEEE CEC’2017 and its performance are compared with seven stable well-known MAs. In addition, the Wilcoxon sign rank test has been used to check the results’ significance. To ensure the robustness of mCOOT, it has been used to identify the optimal lithium-ion (Li-ion) battery model parameter. Battery parameters are unmeasured parameters that can only be estimated. The main advantage of this identification algorithm is the high accuracy where the root mean square error (RMSE) has been reduced to 7.11548 × 10–4 with an efficiency of 99.25% and STD of 0.59466 × 10–5. Thus, we observed that mCOOT algorithm can be used as an efficient algorithm to solve electrical and complicated optimization problems.

An Adaptive Covariance Scaling Estimation of Distribution Algorithm

Optimization problems are ubiquitous in every field, and they are becoming more and more complex, which greatly challenges the effectiveness of existing optimization methods. To solve the increasingly complicated optimization problems with high effectiveness, this paper proposes an adaptive covariance scaling estimation of distribution algorithm (ACSEDA) based on the Gaussian distribution model. Unlike traditional EDAs, which estimate the covariance and the mean vector, based on the same selected promising individuals, ACSEDA calculates the covariance according to an enlarged number of promising individuals (compared with those for the mean vector). To alleviate the sensitivity of the parameters in promising individual selections, this paper further devises an adaptive promising individual selection strategy for the estimation of the mean vector and an adaptive covariance scaling strategy for the covariance estimation. These two adaptive strategies dynamically adjust the associated numbers of promising individuals as the evolution continues. In addition, we further devise a cross-generation individual selection strategy for the parent population, used to estimate the probability distribution by combing the sampled offspring in the last generation and the one in the current generation. With the above mechanisms, ACSEDA is expected to compromise intensification and diversification of the search process to explore and exploit the solution space and thus could achieve promising performance. To verify the effectiveness of ACSEDA, extensive experiments are conducted on 30 widely used benchmark optimization problems with different dimension sizes. Experimental results demonstrate that the proposed ACSEDA presents significant superiority to several state-of-the-art EDA variants, and it preserves good scalability in solving optimization problems.

An Enhanced Slime Mould Algorithm and Its Application for Digital IIR Filter Design

In the past few decades, metaheuristic algorithms (MA) have been developed tremendously and have been successfully applied in many fields. In recent years, a large number of new MA have been proposed. Slime mould algorithm (SMA) is a novel swarm-based intelligence optimization algorithm. SMA solves the optimization problem by imitating the foraging and movement behavior of slime mould. It can effectively obtain a promising global optimal solution. However, it still suffers some shortcomings such as the unstable convergence speed, the imprecise search accuracy, and incapability of identifying a local optimal solution when faced with complicated optimization problems. With the purpose of overcoming the shortcomings of SMA, this paper proposed a multistrategy enhanced version of SMA called ESMA. The three enhanced strategies are chaotic initialization strategy (CIS), orthogonal learning strategy (OLS), and boundary reset strategy (BRS). The CIS is used to generate an initial population with diversity in the early stage of ESMA, which can increase the convergence speed of the algorithm and the quality of the final solution. Then, the OLS is used to discover the useful information of the best solutions and offer a potential search direction, which enhances the local search ability and raises the convergence rate. Finally, the BRS is used to correct individual positions, which ensures the population diversity and enhances the overall search capabilities of ESMA. The performance of ESMA was validated on the 30 IEEE CEC2014 functions and three IIR model identification problems, compared with other nine well-regarded and state-of-the-art algorithms. Simulation results and analysis prove that the ESMA has a superior performance. The three strategies involved in ESMA have significantly improved the performance of the basic SMA.

An Efficient Bi-Objective Optimization Workflow Using the Distributed Quasi-Newton Method and Its Application to Well-Location Optimization

SummaryAlthough it is possible to apply traditional optimization algorithms to determine the Pareto front of a multiobjective optimization problem, the computational cost is extremely high when the objective function evaluation requires solving a complex reservoir simulation problem and optimization cannot benefit from adjoint-based gradients. This paper proposes a novel workflow to solve bi-objective optimization problems using the distributed quasi-Newton (DQN) method, which is a well-parallelized and derivative-free optimization (DFO) method. Numerical tests confirm that the DQN method performs efficiently and robustly.The efficiency of the DQN optimizer stems from a distributed computing mechanism that effectively shares the available information discovered in prior iterations. Rather than performing multiple quasi-Newton optimization tasks in isolation, simulation results are shared among distinct DQN optimization tasks or threads. In this paper, the DQN method is applied to the optimization of a weighted average of two objectives, using different weighting factors for different optimization threads. In each iteration, the DQN optimizer generates an ensemble of search points (or simulation cases) in parallel, and a set of nondominated points is updated accordingly. Different DQN optimization threads, which use the same set of simulation results but different weighting factors in their objective functions, converge to different optima of the weighted average objective function. The nondominated points found in the last iteration form a set of Pareto-optimal solutions. Robustness as well as efficiency of the DQN optimizer originates from reliance on a large, shared set of intermediate search points. On the one hand, this set of searching points is (much) smaller than the combined sets needed if all optimizations with different weighting factors would be executed separately; on the other hand, the size of this set produces a high fault tolerance, which means even if some simulations fail at a given iteration, the DQN method’s distributed-parallel information-sharing protocol is designed and implemented such that the optimization process can still proceed to the next iteration.The proposed DQN optimization method is first validated on synthetic examples with analytical objective functions. Then, it is tested on well-location optimization (WLO) problems by maximizing the oil production and minimizing the water production. Furthermore, the proposed method is benchmarked against a bi-objective implementation of the mesh adaptive direct search (MADS) method, and the numerical results reinforce the auspicious computational attributes of DQN observed for the test problems.To the best of our knowledge, this is the first time that a well-parallelized and derivative-free DQN optimization method has been developed and tested on bi-objective optimization problems. The methodology proposed can help improve efficiency and robustness in solving complicated bi-objective optimization problems by taking advantage of model-based search algorithms with an effective information-sharing mechanism.NOTE: This paper is also published as part of the 2021 SPE Reservoir Simulation Conference Special Issue.

Integrated optimization algorithm: A metaheuristic approach for complicated optimization

This paper proposes an integrated optimization algorithm (IOA) designed for solving complicated optimization problems that are non-convex, non-differentiable, non-continuous, or computationally intensive. IOA is synthesized from 5 sub-algorithms: follower search, leader search, wanderer search, crossover search, and role learning. The follower search finds better solutions by tracing the leaders. The leader search refines current optimal solutions by approaching or deviating from the central point of the population and then executes a single-round coordinate descent. The wanderer search carries out comprehensive search space expansion. The crossover search generates offspring using solutions from superior parents. Role learning automates the process in which a search agent decides whether to become a follower or a wanderer. A global optima estimation framework (GOEF) is proposed to offer guidelines for designing an efficient optimization algorithm, and IOA is proved to attain global optima. A differentiable integrated optimization algorithm (DIOA) that extends gradient descent is put forward to train deep learning models. Empirical case studies conclude that IOA shows a much faster convergence speed and finds better solutions than the other 8 comparative algorithms based on 27 benchmark functions. IOA has also been applied to solve unit commitment problems in the power system and shows satisfactory results. A power line sub-image classification model based on a convolutional neural network (CNN) is optimized by DIOA. Compared with the pure gradient descent approach, DIOA converges significantly faster and obtains a high test set accuracy with much fewer training epochs.

Multi-Objective Optimization of Home Healthcare with Working-Time Balancing and Care Continuity

The ageing population in most parts of the world becomes a grand challenge for healthcare decision-makers. The care of elderly persons and general hygienic care at patients’ homes are two main reasons to motivate an optimization problem, namely, home healthcare (HHC). A robust plan for caregivers to have sustainable HHC operations management is to consider working-time balancing of caregivers, care continuity and uncertainties, e.g., the uncertainty of patients’ availability in addition to service and travel times as well as the regulations of companies to meet the standards of high-quality home care services. Based on these motivations and challenges to this field, this study firstly established a multi-objective robust optimization of the HHC which is multi-depot, multi-period and multi-service. The demand of each patient in each period may be different due to promptness of services. Each caregiver plays one of the roles of nurses, doctors, physiotherapists and nutritionists. The types of services are directly related to these roles. The objectives were optimizing the total cost of logistic activities as well as the total unemployment time of caregivers and care continuity. As a complicated optimization problem, this study innovated efficient heuristics and an enhanced nature-inspired metaheuristic. Finally, an extensive comparison with regards to the criteria of the multi-objective algorithms’ assessment was conducted. Some sensitivity analyses were conducted to conclude some practical insights.

Metaheuristics for pharmacometrics.

Metaheuristics is a powerful optimization tool that is increasingly used across disciplines to tackle general purpose optimization problems. Nature‐inspired metaheuristic algorithms is a subclass of metaheuristic algorithms and have been shown to be particularly flexible and useful in solving complicated optimization problems in computer science and engineering. A common practice with metaheuristics is to hybridize it with another suitably chosen algorithm for enhanced performance. This paper reviews metaheuristic algorithms and demonstrates some of its utility in tackling pharmacometric problems. Specifically, we provide three applications using one of its most celebrated members, particle swarm optimization (PSO), and show that PSO can effectively estimate parameters in complicated nonlinear mixed‐effects models and to gain insights into statistical identifiability issues in a complex compartment model. In the third application, we demonstrate how to hybridize PSO with sparse grid, which is an often‐used technique to evaluate high dimensional integrals, to search for D‐efficient designs for estimating parameters in nonlinear mixed‐effects models with a count outcome. We also show the proposed hybrid algorithm outperforms its competitors when sparse grid is replaced by its competitor, adaptive gaussian quadrature to approximate the integral, or when PSO is replaced by three notable nature‐inspired metaheuristic algorithms.

Blind image deblurring with Gaussian curvature of the image surface

This paper presents a blind image deblurring algorithm by utilizing Gaussian curvature (GC) of the image surface. GC is an intrinsic geometric feature and related to the developability of the surface. In recent years, numerous variational models based on GC for image denoising and image reconstruction have been proposed. In this paper, we show that GC regularization is also effective for blind image deblurring when combines it with L0-norm of image gradients. By minimizing the combined regularization, our algorithm gradually preserves sharp edges and removes detrimental structures and noises in intermediate latent images. The sharp latent images can then accurately guide the estimation of the blur kernel. However, a complicated optimization problem will occur once the proposed regularization has been involved. As we know, traditional diffusion methods for minimizing the GC regularization not only converges slowly but also requires the differentiability of signals. Moreover, the L0-norm of image gradients is non-convex. Consequently, Gaussian curvature filter (GCF) and the half-quadratic splitting strategy are adopted to solve the optimization problem. Extensive experimental results show that the proposed deblurring method achieves state-of-the-art results on benchmark datasets and performs favorably on real-world blurry images.

PCA-ANN integrated NSGA-III framework for dormitory building design optimization: Energy efficiency, daylight, and thermal comfort

“Framework” and “case-study” are the two most prominent features in the optimization of architectural building design. The first can improve the speed of the process, the accuracy and integration of the outcome. The second can evolve the design knowledge of the various types of buildings in practice. This paper presents the “PCA-ANN integrated NSGA-III” framework that is a fast, accurate, and integrated solution for practical, erosive, and complicated building performance optimization problems, especially when the case study is a multi-type building such as a dormitory. The novelty and contribution of this paper exist in all three phases of its research framework. First, the dormitory building was parametrically modeled as a case study in two conventional dormitory building types. Second, an artificial model called PCA-ANN is presented in which problems that have a more complicated and practical case study with a multi-type building can be solved simply, integratively, and accurately. Third, the capabilities of NSGA-III are introduced and showed that it could be more accurate in comparison with NSGA-II, and then NSGA-III was integrated with the PCA-ANN model to solve the many-objective problem posed in this study. R2 score for PCA-ANN in the best-presented structure reached 0.99. In the overall best building alternative, energy efficiency and daylight performance for the whole year and thermal comfort performance in the free running period improved by 41.27%, 42.24%, and 15.57%, respectively, compared to its reference model. The presented framework can be applied and generalized to building energy optimization problems in practical and complex cases.

Spare simple MKKM with semi‐infinite linear program optimization

Multiple kernel clustering (MKC) optimally combines a group of predefined kernel matrices to improve clustering performance. Although demonstrating promising performance in various applications, most of existing approaches adopt the min–min formulation, which could be sensitive to perturbation with adversarial samples. Moreover, existing MKC algorithms often involve several hypermeters preventing them into further real applications. To address these issues, we propose a parameter-free effective sparse simple multiple kernel k-means algorithm with max–min optimization formulation in this paper. To be specific, we propose to optimize the widely used unsupervised kernel alignment criterion by minimizing the kernel coefficient and maximizing the clustering partition matrix. Unlike traditional min–min formulation, the max–min kernel alignment is robust to adversarial sample perturbation and free of hyper-parameters. An optimization method based on semi-infinite linear program is designed to solve the complicated optimization problem. Extensive experiments on six multiple kernel benchmark data sets demonstrate the effectiveness of the proposed method.

A novel hybrid particle swarm optimization using adaptive strategy

Particle swarm optimization (PSO) has been employed to solve numerous real-world problems because of its strong optimization ability and easy implementation. However, PSO still has some shortcomings in solving complicated optimization problems, such as premature convergence and poor balance between global exploration and local exploitation. A novel hybrid particle swarm optimization using adaptive strategy (ASPSO) is developed to address associated difficulties. The contribution of ASPSO is threefold: (1) a chaotic map and an adaptive position updating strategy to balance exploration behavior and exploitation nature in the search progress; (2) elite and dimensional learning strategies to enhance the diversity of the population effectively; (3) a competitive substitution mechanism to improve the accuracy of solutions. Based on various functions from CEC 2017, the numerical experiment results demonstrate that ASPSO is significantly better than the other 16 optimization algorithms. Furthermore, we apply ASPSO to a typical industrial problem, the optimization of melt spinning progress, where the results indicate that ASPSO performs better than other algorithms.

Buffer-Aided Relay Selection for Cooperative Hybrid NOMA/OMA Networks With Asynchronous Deep Reinforcement Learning

This paper investigates asynchronous reinforcement learning algorithms for joint buffer-aided relay selection and power allocation in the non-orthogonal-multiple-access (NOMA) relay network. With the hybrid NOMA/OMA transmission, we investigate joint relay selection and power allocation to maximize the throughput with the delay constraint. To solve this complicated high-dimensional optimization problem, we propose two asynchronous reinforcement learning-based schemes: the asynchronous deep Q-Learning network (ADQN)-based scheme and the asynchronous advantage actor-critic (A3C)-based scheme, respectively. The A3C-based scheme achieves better performance and robustness when the action space is large, while the ADQN-based scheme converges faster with a small action space. Moreover, a-prior information is exploited to improve the convergence of the proposed schemes. The simulation results show that the proposed asynchronous learning-based schemes can learn from the environment and achieve good convergence.

Swarm-Based 4D Path Planning For Drone Operations in Urban Environments

Drones have a wide range of applications in urban environments as they can both enhance people's daily activities and commercial activities through various operations and deployments. With the increasing number of drones, flight safety and efficiency become the main concern, and effective drone operations can make a difference. Accordingly, 4D path planning for drone operations is the focus of this paper, and the swarm-based method is proposed to solve this complicated optimization problem. Under the framework of ‘AirMatrix’, the problem is solved in two levels, i.e., 3D path planning for a single drone and conflict resolution among drones. In the multi-path planning level, multiple alternative flight paths for each drone are generated to increase the acceptance rate of a flight request. The constraints on a single flight path and two different flight paths are considered. The goal is to obtain several different short flight paths as alternatives. A clustering improved ant colony optimization CIACO) algorithm is employed to solve the multi-path planning problem. The crowding mechanism is used in clustering, and some improvements are made to strengthen the global and local search ability in the early and later phases of iterations. In the task scheduling level, the conflicts between two drones are defined in two circumstances. One is for the time interval of passing the same path point, another one is for the right-angle collision between two drones. A three-layer fitness function is proposed to maximize the number of permitted flights according to the safety requirement, in which the airspace utilization and the operators’ requests are both considered. A ‘cross-off’ strategy is developed to calculate the fitness value, and a ‘distributed-centralized’ strategy is applied considering the task priorities of drones. A genetic algorithm GA)-based task scheduling algorithm is also developed according to the characteristic of the established model. Simulation results demonstrate that 4D flight path of each drone can be generated by the proposed swarmed-based algorithms, and safe and efficient drone operations in a specific airspace can be ensured.

IGS-CMAES: A Two-Stage Optimization for Ground Deformation and DEM Error Estimation in Time Series InSAR Data

Interferometric synthetic aperture radar (InSAR) has become an increasingly recognized remote sensing technology for earth surface monitoring. Slow and subtle terrain displacements can be estimated using time-series InSAR (TSInSAR) data. However, a substantial increase in the availability of exclusive time series data necessitates the development of more efficient and effective algorithms. Research in these areas is usually carried out by solving complicated optimization problems, which is very computationally expensive and time-consuming. This work proposes a two-stage black-box optimization framework to jointly estimate the average ground deformation rate and terrain digital elevation model (DEM) error. The method performs an iterative grid search (IGS) to acquire coarse candidate solutions, and then a covariance matrix adaptive evolution strategy (CMAES) is adopted to obtain the final local results. The performance of our method is evaluated using both simulated and real datasets. Both quantitative and qualitative comparisons using different optimizers support the reliability and effectiveness of our work. The proposed IGS-CMAES achieves higher accuracy with a significantly fewer number of objective function evaluations than other established algorithms. It offers the possibility for wide-area monitoring, where high precision and real-time processing is essential.

WOA-TLBO: Whale optimization algorithm with Teaching-learning-based optimization for global optimization and facial emotion recognition

The Whale Optimization Algorithm (WOA) is a recently developed algorithm that is based on the chasing mechanism of humpback whales. Benefiting from the unique structure, WOA has virtuous global search capability. One of the drawbacks of this algorithm is the slow convergence rate that limits its real-world application. In resolving complicated global optimization problems, without any exertion for adequate fine-tuning preliminary constraints, Teaching-learning-based optimization (TLBO) is smooth to plunge into local optimal, but it has a fast convergence speed. By given the features of WOA and TLBO, an active hybrid WOA-TLBO algorithm is proposed for resolving optimization difficulties. To explore the enactment of the proposed WOA-TLBO algorithm, several experimentations are accompanied by regular benchmark test functions and compared with six other algorithms. The investigational outcomes indicate the more magnificent concert of the proposed WOA-TLBO algorithm for the benchmark function results. The proposed method has also been applied to the Facial Emotion Recognition (FER) functional problem. FER is the thought-provoking investigation zone that empowers us to classify the expression of the human face in everyday life. Centered on the portions’ actions in the human face, the maximum of the standard approaches fail to distinguish the expressions precisely as the expressions. In this paper, we have proposed FER’s productive process using WOA-TLBO based MultiSVNN (Multi-Support Vector Neural Network). Investigational outcomes deliver an indication of the virtuous enactment of the proposed technique resolutions in terms of accurateness.

A FRAMEWORK TO MANAGE UNCERTAINTY IN THE COMPUTATION OF WASTE COLLECTION ROUTES AFTER A FLOOD

Abstract. In this paper, we describe a framework to find a good quality waste collection tour after a flood, without having to solve a complicated optimization problem from scratch in limited time. We model the computation of a waste collection tour as a capacitated routing problem, on the vertices or on the edges of a graph, with uncertain waste quantities and uncertain road availability. Multiple models have been conceived to manage uncertainty in routing problems, and we build on the ideas of discretizing the uncertain parameters and computing master solutions that can be adapted to propose an original method to compute efficient solutions. We first introduce our model for the progressive removal of the uncertainty, then outline our method to compute solutions: our method first considers a low-dimensional set of random variables that govern the behaviour of the problem parameters, discretizes these variables and computes a solution for each discrete point before the flood, and then uses these solutions as a basis to build operational solutions when there are enough information about the parameters of the routing problem. We then give computational tools to implement this method. We give a framework to compute the basis of solutions in an efficient way, by computing all the solutions simultaneously and sharing information (that can lead to good quality solutions) between the different problems based on how close their parameters are, and we also describe how real solutions can be derived from this basis. Our main contributions are our model for the progressive removal of uncertainty, our multi-step method to compute efficient solutions, and our intrusive framework to compute solutions on the discrete grid of parameters.

Protein Structure Prediction: Conventional and Deep Learning Perspectives.

Protein structure prediction is a way to bridge the sequence-structure gap, one of the main challenges in computational biology and chemistry. Predicting any protein's accurate structure is of paramount importance for the scientific community, as these structures govern their function. Moreover, this is one of the complicated optimization problems that computational biologists have ever faced. Experimental protein structure determination methods include X-ray crystallography, Nuclear Magnetic Resonance Spectroscopy and Electron Microscopy. All of these are tedious and time-consuming procedures that require expertise. To make the process less cumbersome, scientists use predictive tools as part of computational methods, using data consolidated in the protein repositories. In recent years, machine learning approaches have raised the interest of the structure prediction community. Most of the machine learning approaches for protein structure prediction are centred on co-evolution based methods. The accuracy of these approaches depends on the number of homologous protein sequences available in the databases. The prediction problem becomes challenging for many proteins, especially those without enough sequence homologs. Deep learning methods allow for the extraction of intricate features from protein sequence data without making any intuitions. Accurately predicted protein structures are employed for drug discovery, antibody designs, understanding protein-protein interactions, and interactions with other molecules. This article provides a review of conventional and deep learning approaches in protein structure prediction. We conclude this review by outlining a few publicly available datasets and deep learning architectures currently employed for protein structure prediction tasks.

Long-step path-following algorithm for quantum information theory: Some numerical aspects and applications

<p style='text-indent:20px;'>We consider some important computational aspects of the long-step path-following algorithm developed in our previous work and show that a broad class of complicated optimization problems arising in quantum information theory can be solved using this approach. In particular, we consider one difficult optimization problem involving the quantum relative entropy in quantum key distribution and show that our method can solve problems of this type much faster in comparison with (very few) available options.</p>