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Multiobjective optimization is a ubiquitous problem that arises naturally in many scientific and industrial areas. Network routing optimization with multiobjective performance demands falls into this problem class, and finding good quality solutions at large scales is generally challenging. In this work, we develop a scheme with which near-term quantum computers can be applied to solve multiobjective combinatorial optimization problems. We study the application of this scheme to the network routing problem in detail, by first mapping it to the multiobjective shortest-path problem. Focusing on an implementation based on the quantum approximate optimization algorithm (QAOA)—the go-to approach for tackling optimization problems on near-term quantum computers—we examine the Pareto plot that results from the scheme and qualitatively analyze its ability to produce Pareto-optimal solutions. We further provide theoretical and numerical scaling analyses of the resource requirements and performance of QAOA and identify key challenges associated with this approach. Finally, through Amazon Braket, we execute small-scale implementations of our scheme on the IonQ Harmony 11-qubit quantum computer.

How can we efficiently find very large numbers of clusters C in very large datasets N of potentially high dimensionality D? Here we address the question by using a novel variational approach to optimize Gaussian mixture models (GMMs) with diagonal covariance matrices. The variational method approximates expectation maximization (EM) by applying truncated posteriors as variational distributions and partial E-steps in combination with coresets. Run time complexity to optimize the clustering objective then reduces from O(NCD) per conventional EM iteration to O(N′G2D) for a variational EM iteration on coresets (with coreset size N ′ ≤ N and truncation parameter G ≪ C). Based on the strongly reduced run time complexity per iteration, which scales sublinearly with NC, we then provide a concrete, practically applicable, parallelized and highly efficient clustering algorithm. In numerical experiments on standard large-scale benchmarks we (A) show that also overall clustering times scale sublinearly with NC, and (B) observe substantial wall-clock speedups compared to already highly efficient recently reported results. The algorithm’s sublinear scaling allows for applications at scales where alternative methods cease to be applicable. We demonstrate such very large-scale applicability using the YFCC100M benchmark, for which we realize with a GMM of up to 50.000 clusters an optimization of a data density model with up to 150 M parameters. [ABSTRACT FROM AUTHOR]

Adversarial attacks have been extensively studied in recent years since they can identify the vulnerability of deep learning models before deployed. In this paper, we consider the black-box adversarial setting, where the adversary needs to craft adversarial examples without access to the gradients of a target model. Previous methods attempted to approximate the true gradient either by using the transfer gradient of a surrogate white-box model or based on the feedback of model queries. However, the existing methods inevitably suffer from low attack success rates or poor query efficiency since it is difficult to estimate the gradient in a high-dimensional input space with limited information. To address these problems and improve black-box attacks, we propose two prior-guided random gradient-free (PRGF) algorithms based on biased sampling and gradient averaging, respectively. Our methods can take the advantage of a transfer-based prior given by the gradient of a surrogate model and the query information simultaneously. Through theoretical analyses, the transfer-based prior is appropriately integrated with model queries by an optimal coefficient in each method. Extensive experiments demonstrate that, in comparison with the alternative state-of-the-arts, both of our methods require much fewer queries to attack black-box models with higher success rates. [ABSTRACT FROM AUTHOR]

This article presents a rough-to-fine evolutionary multiobjective optimization algorithm based on the decomposition for solving problems in which the solutions are initially far from the Pareto-optimal set. Subsequently, a tree is constructed by a modified $k$ -means algorithm on $N$ uniform weight vectors, and each node of the tree contains a weight vector. Each node is associated with a subproblem with the help of its weight vector. Consequently, a subproblem tree can be established. It is easy to find that the descendant subproblems are refinements of their ancestor subproblems. The proposed algorithm approaches the Pareto front (PF) by solving a few subproblems in the first few levels to obtain a rough PF and gradually refining the PF by involving the subproblems level-by-level. This strategy is highly favorable for solving problems in which the solutions are initially far from the Pareto set. Moreover, the proposed algorithm has lower time complexity. Theoretical analysis shows the complexity of dealing with a new candidate solution is $\mathcal {O}(M \log N)$ , where $M$ is the number of objectives. Empirical studies demonstrate the efficacy of the proposed algorithm. [ABSTRACT FROM AUTHOR]

Although cell-free massive multiple-input–multiple-output (mMIMO) Internet-of-Things (IoT) systems can improve spectral efficiency (SE) significantly compared with conventional cellular mMIMO IoT systems due to high coverage and no intercell interference, the pilot contamination and active sensors (ASs) interference can still seriously limit the further improvement of SE. In this article, we first develop a model of the cell-free mMIMO IoT system and then derive a mathematical expression of the SE to formulate a maximized sum SE optimization problem for jointly controlling pilot and data power. This problem is nonconvex; thus, a successive convex approximation (SCA) iteration algorithm is proposed, which, in fact, approximates the objective and constraint function in each iteration, and converts the initial optimization problem into an easy-to-solve geometric programming (GP) problem. Finally, simulation results show that our proposed algorithm outperforms the existing algorithms related to power control optimization in terms of the sum SE performance. [ABSTRACT FROM AUTHOR]

This paper revisits the temporal difference (TD) learning algorithm for the policy evaluation tasks in reinforcement learning. Typically, the performance of TD(0) and TD($\lambda$ λ ) is very sensitive to the choice of stepsizes. Oftentimes, TD(0) suffers from slow convergence. Motivated by the tight link between the TD(0) learning algorithm and the stochastic gradient methods, we develop a provably convergent adaptive projected variant of the TD(0) learning algorithm with linear function approximation that we term AdaTD(0). In contrast to the TD(0), AdaTD(0) is robust or less sensitive to the choice of stepsizes. Analytically, we establish that to reach an $\epsilon$ ε accuracy, the number of iterations needed is $\tilde{O}(\epsilon ^{-2}\ln ^4\frac{1}{\epsilon }/\ln ^4\frac{1}{\rho })$ O ˜ (ε - 2 ln 4 1 ε / ln 4 1 ρ) in the general case, where $\rho$ ρ represents the speed of the underlying Markov chain converges to the stationary distribution. This implies that the iteration complexity of AdaTD(0) is no worse than that of TD(0) in the worst case. When the stochastic semi-gradients are sparse, we provide theoretical acceleration of AdaTD(0). Going beyond TD(0), we develop an adaptive variant of TD($\lambda$ λ ), which is referred to as AdaTD($\lambda$ λ ). Empirically, we evaluate the performance of AdaTD(0) and AdaTD($\lambda$ λ ) on several standard reinforcement learning tasks, which demonstrate the effectiveness of our new approaches. [ABSTRACT FROM AUTHOR]

Multi-relational learning on knowledge graphs infers high-order relations among the entities across the graphs. This learning task can be solved by label propagation on the tensor product of the knowledge graphs to learn the high-order relations as a tensor. In this paper, we generalize a widely used label propagation model to the normalized tensor product graph, and propose an optimization formulation and the scalable Low-rank Tensor-based Label Propagation algorithm (LowrankTLP) to infer multi-relations for two learning tasks, hyperlink prediction and multiple graph alignment. The optimization formulation minimizes the upper bound of the noisy-tensor estimating error for multiple graph alignment, by learning with a subset of the eigen-pairs in the spectrum of the normalized tensor product graph. We also provide a data-dependent transductive Rademacher bound for binary hyperlink prediction. We accelerate LowrankTLP with parallel tensor computation which enables label propagation on a tensor product of 100 graphs each of size 1000 in less than half hour in the simulation. LowrankTLP was also applied to predicting the author-paper-venue hyperlinks in publication records, alignment of segmented regions across up to 26 CT-scan images and alignment of protein-protein interaction networks across multiple species. The experiments demonstrate that LowrankTLP indeed well approximates the original label propagation with better scalability and accuracy. [ABSTRACT FROM AUTHOR]

One of the most extensively studied combinatorial optimization problems is the Travelling Salesman Problem (TSP). Considerable research efforts in the past have resulted in exact solvers. However, the runtime of such hand-crafted solutions increases exponentially with problem size. Ising model based solvers have also gained prominence due to their abilities to find fast and approximate solutions for combinatorial optimization problems. However, such Ising based heuristics also suffer from scalability as the solution quality becomes increasingly sub-optimal with increase in problem size. In this work, we propose Neuro-Ising – a machine learning framework which uses Ising models to find clusters of near-optimal partial solutions of large scale TSPs and combines those solutions by employing a supervised data driven mechanism, which we model as a Graph Neural Network (GNN). The GNN is trained from solution instances obtained through exact solvers and hence, the proposed approach generalizes to unseen problems while avoiding the run-time complexity otherwise required, if the solution is built from scratch. Using standard computing resources, our proposed framework rapidly converges to near-optimal solutions for 15 TSPs (upto $\sim 5k$ cities) from the TSPLib benchmark suite. We report $\sim 10.66\times $ speedup over Tabu Search for 8 problems. Furthermore, compared to two state-of-the-art clustering-based TSP solvers, Neuro-Ising achieves $\sim 38 \times $ faster convergence along with $\sim 8.9\%$ better quality of solution, on average. [ABSTRACT FROM AUTHOR]

Owing to its good compression ability, discrete cosine transform (DCT) is widely used in signal processing, including high-efficiency video coding (HEVC). For efficient implementation, approximate DCTs and integer DCTs were proposed, but the challenge in the tradeoff between coding performance and implementation cost always exists. To solve this problem, this article proposes a new algorithm to derive Int-DCT with good coding performance and low hardware cost. To address the multiobjective optimization problem, we extract information from the initially shortlisted transform matrices to derive the weighting factors for the different matrices properties. Those properties having a higher impact to the performance are assigned with higher weights and vice versa. Subsequently, an efficient search algorithm is proposed to shortlist those candidates which can lead to better coding performance evaluated by the proposed measure. Last but not the least, a splitting method is proposed to search the efficient extended double-base number system (EDBNS) representation of the coefficients for hardware implementation and select the solution which encounters the lowest cost. The experimental results in ASIC demonstrate that the implementation area and power cost of designs by the new algorithm are reduced by at least 11.1% and 17.5% for 32-point transform, respectively, over other competing algorithms. Meanwhile, the transforms generated by the proposed algorithm causes negligible impact to coding performance compared to the original transforms in HEVC. [ABSTRACT FROM AUTHOR]

This article deals with nonconvex stochastic optimization problems in deep learning. Appropriate learning rates, based on theory, for adaptive-learning-rate optimization algorithms (e.g., Adam and AMSGrad) to approximate the stationary points of such problems are provided. These rates are shown to allow faster convergence than previously reported for these algorithms. Specifically, the algorithms are examined in numerical experiments on text and image classification and are shown in experiments to perform better with constant learning rates than algorithms using diminishing learning rates. [ABSTRACT FROM AUTHOR]

This article introduces a new deep learning approach to approximately solve the covering salesman problem (CSP). In this approach, given the city locations of a CSP as input, a deep neural network model is designed to directly output the solution. It is trained using the deep reinforcement learning without supervision. Specifically, in the model, we apply the multihead attention (MHA) to capture the structural patterns, and design a dynamic embedding to handle the dynamic patterns of the problem. Once the model is trained, it can generalize to various types of CSP tasks (different sizes and topologies) without the need of retraining. Through controlled experiments, the proposed approach shows desirable time complexity: it runs more than 20 times faster than the traditional heuristic solvers with a tiny gap of optimality. Moreover, it significantly outperforms the current state-of-the-art deep learning approaches for combinatorial optimization in the aspect of both training and inference. In comparison with traditional solvers, this approach is highly desirable for most of the challenging tasks in practice that are usually large scale and require quick decisions. [ABSTRACT FROM AUTHOR]

Automated hyperparameter optimization (HPO) has gained great popularity and is an important component of most automated machine learning frameworks. However, the process of designing HPO algorithms is still an unsystematic and manual process: new algorithms are often built on top of prior work, where limitations are identified and improvements are proposed. Even though this approach is guided by expert knowledge, it is still somewhat arbitrary. The process rarely allows for gaining a holistic understanding of which algorithmic components drive performance and carries the risk of overlooking good algorithmic design choices. We present a principled approach to automated benchmark-driven algorithm design applied to multifidelity HPO (MF-HPO). First, we formalize a rich space of MF-HPO candidates that includes, but is not limited to, common existing HPO algorithms and then present a configurable framework covering this space. To find the best candidate automatically and systematically, we follow a programming-by-optimization approach and search over the space of algorithm candidates via Bayesian optimization. We challenge whether the found design choices are necessary or could be replaced by more naive and simpler ones by performing an ablation analysis. We observe that using a relatively simple configuration (in some ways, simpler than established methods) performs very well as long as some critical configuration parameters are set to the right value. [ABSTRACT FROM AUTHOR]

In a large cloud data center HPC system, a critical problem is how to allocate the submitted tasks to heterogeneous servers that will achieve the goal of maximizing the system’s gain defined as the value of completed tasks minus system operation costs. We consider this problem in the online setting that tasks arrive in batches and propose a novel deep reinforcement learning (DRL) enhanced greedy optimization algorithm of two-stage scheduling interacting task sequencing and task allocation. For task sequencing, we deploy a DRL module to predict the best allocation sequence for each arriving batch of tasks based on the knowledge (allocation strategies) learnt from previous batches. For task allocation, we propose a greedy strategy that allocates tasks to servers one by one online following the allocation sequence to maximize the total gain increase. We show that our greedy strategy has a performance guarantee of competitive ratio $\frac{1}{1+\kappa }$ 1 1 + κ to the optimal offline solution, which improves the existing result for the same problem, where $\kappa$ κ is upper bounded by the maximum cost-to-gain ratio of each task. While our DRL module enhances the greedy algorithm by providing the likely-optimal allocation sequence for each batch of arriving tasks, our greedy strategy bounds DRL’s prediction error within a proven worst-case performance guarantee for any allocation sequence. It enables a better solution quality than that obtainable from both DRL and greedy optimization alone. Extensive experiment evaluation results in both simulation and real application environments demonstrate the effectiveness and efficiency of our proposed algorithm. Compared with the state-of-the-art baselines, our algorithm increases the system gain by about 10% to 30%. Our algorithm provides an interesting example of combining machine learning (ML) and greedy optimization techniques to improve ML-based solutions with a worst-case performance guarantee for solving hard optimization problems. [ABSTRACT FROM AUTHOR]

This brief addresses the problem of implementing very large constant multiplications by a single variable under the shift-adds architecture using a minimum number of adders/subtractors. Due to the intrinsic complexity of the problem, we introduce an approximate algorithm, called TÕLL, which partitions the very large constants into smaller ones. To reduce the number of operations, TÕLL incorporates graph-based and common subexpression elimination methods proposed for the shift-adds design of constant multiplications. It can also consider the delay of a multiplierless design defined in terms of the maximum number of operations in series, i.e., the number of adder-steps, while reducing the number of operations. High-level experimental results show that the adder-steps of a shift-adds design can be reduced significantly with a little overhead in the number of operations. Gate-level experimental results indicate that while the shift-adds design can lead to a 36.6% reduction in gate-level area with respect to a design using a multiplier, the delay-aware optimization can yield a 48.3% reduction in minimum achievable delay of the shift-adds design when compared to the area-aware optimization. [ABSTRACT FROM AUTHOR]

Convolutional transform learning (CTL), learning filters by minimizing the data fidelity loss function in an unsupervised way, is becoming very pervasive, resulting from keeping the best of both worlds: the benefit of unsupervised learning and the success of the convolutional neural network. There have been growing interests in developing efficient CTL algorithms. However, developing a convergent and accelerated CTL algorithm with accurate representations simultaneously with proper sparsity is an open problem. This article presents a new CTL framework with a $log$ regularizer that can not only obtain accurate representations but also yield strong sparsity. To efficiently address our nonconvex composite optimization, we propose to employ the proximal difference of the convex algorithm (PDCA) which relies on decomposing the nonconvex regularizer into the difference of two convex parts and then optimizes the convex subproblems. Furthermore, we introduce the extrapolation technology to accelerate the algorithm, leading to a fast and efficient CTL algorithm. In particular, we provide a rigorous convergence analysis for the proposed algorithm under the accelerated PDCA. The experimental results demonstrate that the proposed algorithm can converge more stably to desirable solutions with lower approximation error and simultaneously with stronger sparsity and, thus, learn filters efficiently. Meanwhile, the convergence speed is faster than the existing CTL algorithms. [ABSTRACT FROM AUTHOR]

Bilevel optimization involves two levels of optimization, where one optimization problem is nested within the other. The structure of the problem often requires solving a large number of inner optimization problems that make these kinds of optimization problems expensive to solve. The reaction set mapping and the lower level optimal value function mapping are often used to reduce bilevel optimization problems to a single level; however, the mappings are not known a priori, and the need is to be estimated. Though there exist a few studies that rely on the estimation of these mappings, they are often applied to problems where one of these mappings has a known form, that is, piecewise linear, convex, etc. In this article, we utilize both these mappings together to solve general bilevel optimization problems without any assumptions on the structure of these mappings. Kriging approximations are created during the generations of an evolutionary algorithm, where the population members serve as the samples for creating the approximations. One of the important features of the proposed algorithm is the creation of an auxiliary optimization problem using the Kriging-based metamodel of the lower level optimal value function that solves an approximate relaxation of the bilevel optimization problem. The auxiliary problem when used for local search is able to accelerate the evolutionary algorithm toward the bilevel optimal solution. We perform experiments on two sets of test problems and a problem from the domain of control theory. Our experiments suggest that the approach is quite promising and can lead to substantial savings when solving bilevel optimization problems. The approach is able to outperform state-of-the-art methods that are available for solving bilevel problems, in particular, the savings in function evaluations for the lower level problem are substantial with the proposed approach. [ABSTRACT FROM AUTHOR]

By employing time-varying proximal functions, adaptive subgradient methods (ADAGRAD) have improved the regret bound and been widely used in online learning and optimization. However, ADAGRAD with full matrix proximal functions (ADA-FULL) cannot handle large-scale problems due to the impractical $O(d^3)$ O (d 3) time and $O(d^2)$ O (d 2) space complexities, though it has better performance when gradients are correlated. In this paper, we propose two efficient variants of ADA-FULL via a matrix sketching technique called frequent directions (FD). The first variant named as ADA-FD directly utilizes FD to maintain and manipulate low-rank matrices, which reduces the space and time complexities to $O(\tau d)$ O (τ d) and $O(\tau ^2d)$ O (τ 2 d) respectively, where $d$ d is the dimensionality and $\tau \ll d$ τ ≪ d is the sketching size. The second variant named as ADA-FFD further adopts a doubling trick to accelerate FD used in ADA-FD, which reduces the average time complexity to $O(\tau d)$ O (τ d) while only doubles the space complexity of ADA-FD. Theoretical analysis reveals that the regret of ADA-FD and ADA-FFD is close to that of ADA-FULL as long as the outer product matrix of gradients is approximately low-rank. Experimental results demonstrate the efficiency and effectiveness of our algorithms. [ABSTRACT FROM AUTHOR]

This article investigates the finite-time distributed approximate optimization problem of higher order multiagent systems, where the local cost functions are considered to be quadratic functions. This problem is solved via penalty-function-based method. First, by the penalty-function method, a global approximate cost function is constructed. Second, nonlinear distributed optimization algorithms are proposed for higher order multiagent systems by the tool of adding a power integrator technique. In the optimization algorithms design, the gradients of the approximate cost function are utilized. Under the proposed optimization algorithms, the agents approach the approximate optimal solution in finite time. Although there exist errors (they may be called approximation errors) between the approximate minimizers and the global accurate minimizer, the approximation errors can be regulated by penalty parameter and the relationship between the bound of the approximation errors and the penalty parameter is given explicitly. Furthermore, the proposed distributed approximate optimization algorithms are applied to the optimal rendezvous problem of wheeled multimobile robots, making the mobile robots achieve approximate optimization rendezvous in finite time. The effectiveness of the proposed distributed optimization algorithms and their applications to optimal rendezvous problem are validated by simulations. [ABSTRACT FROM AUTHOR]

In this article, we discuss a sequential algorithm for the computation of a minimum-time speed profile over a given path, under velocity, acceleration, and jerk constraints. Such a problem arises in industrial contexts, such as automated warehouses, where LGVs need to perform assigned tasks as fast as possible in order to increase productivity. It can be reformulated as an optimization problem with a convex objective function, linear velocity and acceleration constraints, and nonconvex jerk constraints, which, thus, represent the main source of the difficulty. While existing nonlinear programming (NLP) solvers can be employed for the solution of this problem, it turns out that the performance and robustness of such solvers can be enhanced by the sequential line-search algorithm proposed in this article. At each iteration, a feasible direction, with respect to the current feasible solution, is computed, and a step along such direction is taken in order to compute the next iterate. The computation of the feasible direction is based on the solution of a linearized version of the problem, and the solution of the linearized problem, through an approach that strongly exploits its special structure, represents the main contribution of this work. The efficiency of the proposed approach with respect to existing NLP solvers is proven through different computational experiments. Note to Practitioners—This article was motivated by the needs of LGV manufacturers. In particular, it presents an algorithm for computing the minimum-time speed law for an LGV along a preassigned path, respecting assigned velocity, acceleration, and jerk constraints. The solution algorithm should be: 1) fast, since speed planning is made continuously throughout the workday, not only when an LGV receives a new task but also during the execution of the task itself, since conditions may change, e.g., if the LGV has to be halted for security reasons and 2) reliable, i.e., it should return solutions of high quality, because a better speed profile allows to save time and even small percentage improvements, say a 5% improvement, has a considerable impact on the productivity of the warehouse, and, thus, determines a significant economic gain. The algorithm that we propose meets these two requirements, and we believe that it can be a useful tool for LGV manufacturers and users. It is obvious that the proposed method also applies to the speed planning problem for vehicles other than LGVs, e.g., road vehicles. [ABSTRACT FROM AUTHOR]

We study a class of generic charging path optimization problems arising from emerging networking applications, where mobile chargers are dispatched to deliver energy to mobile agents (e.g., robots, drones, vehicles), which have specified tasks and mobility patterns. We instantiate our work by focusing on finding the charging path maximizing the number of nodes charged within a fixed time horizon. We show that this problem is APX-hard. By recursively decomposing the problem into sub-problems of searching sub-paths, we design quasi-polynomial-time algorithms achieving logarithmic approximation to the optimum charging path. Our approximation algorithms can be further adapted and extended to solve a variety of charging path optimization and scheduling problems with realistic constraints, such as limited time and energy budget. [ABSTRACT FROM AUTHOR]

This article is the second part of a two-part survey series on large-scale global optimization. The first part covered two major algorithmic approaches to large-scale optimization, namely, decomposition methods and hybridization methods, such as memetic algorithms and local search. In this part, we focus on sampling and variation operators, approximation and surrogate modeling, initialization methods, and parallelization. We also cover a range of problem areas in relation to large-scale global optimization, such as multiobjective optimization, constraint handling, overlapping components, the component imbalance issue and benchmarks, and applications. The article also includes a discussion on pitfalls and challenges of the current research and identifies several potential areas of future research. [ABSTRACT FROM AUTHOR]

This article focuses on developing a static output feedback (SOF) control scheme for Takagi–Sugeno fuzzy systems. The proposed SOF controller does not share the same premise membership functions with the model, which permits enhancing the flexibility in controller design and implementation. By contrast with the state feedback case, the SOF control generally results in nonconvex design conditions. To circumvent this problem, we develop a successive convex optimization algorithm, which is based on solving a sequence of more tractable convex optimization problems obtained by approximating the nonconvex constraints with some convex ones. As a heuristic algorithm, the validity of the developed successive convex optimization algorithm is highly affected by initial conditions, and, recognizing this, we put forward an iterative procedure for determining the feasible initial condition. Finally, two illustrative examples are presented to validate the efficiency of the proposed algorithms. [ABSTRACT FROM AUTHOR]

Rough clustering has attracted increasing attention due to well dealing with the fuzziness and uncertainty of data. It is well known that it needs to manually set the threshold to determine the upper and lower approximations of rough clusters, which may bring a great effect on the clustering performance. When applied to image segmentation, rough clustering is always sensitive to the initialized cluster centers and image noise. Furthermore, only one clustering criterion is considered in rough clustering, which cannot satisfy diverse practical requirements. To handle these issues, a particle competitive mechanism based multiobjective rough clustering algorithm (PCM-MORCA) for image segmentation is proposed. First, a rough intraclass compactness function considering the nonlocal spatial information derived from an image is constructed to overcome the sensitivity to image noise. Next, the constructed rough intraclass compactness function and an interclass separation function are optimized simultaneously to make cluster centers meet diverse segmentation requirements. Then, an adaptive threshold determination mechanism by which the threshold adaptively varies with the clustered data is presented to well determine the upper and lower approximations of rough clusters. After that, to effectively search appropriate cluster centers, a novel pair competition-based particle weight updating strategy is designed for multiobjective particle swarm optimization by improving the elite particle selection and particle update. Finally, a rough clustering index with the nonlocal spatial information is constructed for selecting the optimal solution for PCM-MORCA. Segmentation experiments on Berkeley and magnetic resonance images reveal that PCM-MORCA behaves well on the segmentation accuracy and noise robustness. [ABSTRACT FROM AUTHOR]

With an increasing penetration of distributed energy resources (DERs), it has become a public interest regarding how to efficiently manage large-scale DERs into wholesale markets. The existing literature has focused on various coordination methods including iteration-based (e.g., Lagrangian relaxation) and projection-based (e.g., vertex enumeration) ones, which may not work for the cases with complicated constraints due to curse of dimensionality. In this article, we propose an efficient projection-based method that adapts to characterizing the feasible region of active distribution networks with temporal-coupled constraints. The problem of finding the temporal-coupled feasible region is formulated as a max–min optimization program, aiming to determine the largest set allowing zero-valued slack variables. Based on Duality Principle, the max–min model is transformed into a solvable mixed-integer linear programming that can be efficiently optimized by off-the-shelf solvers. In contrast to the existing methods searching for a vertex from interior points, we develop an outer progressive approximation algorithm that generates feasibility cuts for each iteration. Case studies based on a modified IEEE 33-bus feeder and Caracas 141-bus system demonstrate the effectiveness and efficiency of our proposed method in characterizing the feasible region with large-scale temporal-coupled constraints. [ABSTRACT FROM AUTHOR]

This paper explores the non-convex composition optimization consisting of inner and outer finite-sum functions with a large number of component functions. This problem arises in important applications such as nonlinear embedding and reinforcement learning. Although existing approaches such as stochastic gradient descent (SGD) and stochastic variance reduced gradient (SVRG) descent can be applied to solve this problem, their query complexities tend to be high, especially when the number of inner component functions is large. Therefore, to significantly improve the query complexity of current approaches, we have devised the stochastic composition via variance reduction (SCVR). What's more, we analyze the query complexity under different numbers of inner function and outer function. Based on different kinds of estimation of inner component function, we also present the SCVRII algorithm, though the order of query complexities are the same with SCVR. Additionally, we propose an extension to handle the mini-batch cases, which improve the query complexity under the optimal mini-batch size. The experimental results validate our proposed algorithms and theoretical analyses. [ABSTRACT FROM AUTHOR]

Reorganizing bus frequencies to cater for the actual travel demands can significantly save the cost of the public transport system. Many, if not all, previous studies formulate this as a bus frequency optimization problem that tries to minimize passengers’ average waiting time. On the other hand, many investigations have confirmed that the user satisfaction drops faster as the waiting time increases. Consequently, this paper studies the bus frequency optimization problem considering the user satisfaction. Specifically, for the first time to our best knowledge, we study how to schedule the buses such that the total number of passengers who could receive their bus services within the waiting time threshold can be maximized. We propose two variants of the problem, FAST and FASTCO, to cater for different application needs and prove that both are NP-hard. To solve FAST effectively and efficiently, we first present an index-based $(1-1/e)$ (1 - 1 / e) -approximation algorithm. By exploiting the locality property of routes in a bus network, we further propose a partition-based greedy method that achieves a $(1-\rho)(1-1/e)$ (1 - ρ) (1 - 1 / e) approximation ratio. Then we propose a progressive partition-based greedy method to further boost the efficiency while achieving a $(1-\rho)(1-1/e-\varepsilon)$ (1 - ρ) (1 - 1 / e - ɛ) approximation ratio. For the FASTCO problem, two greedy-based heuristic methods are proposed. Experiments on a real city-wide bus dataset in Singapore have been conducted to verify the efficiency, effectiveness, and scalability of our methods in addressing FAST and FASTCO respectively. [ABSTRACT FROM AUTHOR]

Revenue maximization (RM) is one of the most important problems in social networks, which attempts to find a small subset of users that make the expected revenue maximized. It has been studied in depth before. However, most of the existing literature was based on nonadaptive seeding strategies and simple information diffusion models. It considered the number of influenced users as a measurement unit to quantify the revenue. Until the emergence of the collaborate game model, it considered the activity as a basic object to compute the revenue. An activity initiated by a user can only influence those users whose distances are within ${k}$ -hop from the initiator. Based on that, we adopt an adaptive seed strategy and formulate an RM under the size budget (RMSB) problem. If taking into account the product’s promotion, we extend it to an RM under the community budget problem, where the influence can be distributed over the whole network uniformly. We can prove that our objective function is adaptive monotone and not adaptive submodular, but it is adaptive submodular in some special cases. We study these two problems under both the special submodular cases and general nonsubmodular cases, and propose RMSBSolver and RMCBSolver to solve them with strong theoretical guarantees, respectively. In particular, we give a data-dependent approximation ratio by adaptive primal curvature for the RMSB in general nonsubmodular cases. Finally, we evaluate our proposed algorithms by conducting experiments on real datasets, and show the effectiveness and accuracy of our solutions. [ABSTRACT FROM AUTHOR]

The design of a fuzzy neural network (FNN) has long been a challenging problem since most methods rely on approximation error to train an FNN, which may easily result in overfitting phenomenon to degrade the generalization performance. To improve the generalization performance, an FNN with a multiobjective optimization algorithm (MOO-FNN) is proposed in this article. First, the multilevel learning objectives are designed around the generalization performance to guide the training process of an FNN. Then, the method utilizes the approximation error, the structure complexity, and the output smoothness indicators instead of a single indicator to improve the evaluation accuracy of generalization performance. Second, an MOO algorithm with continuous–discrete variables is developed to optimize the FNN. Then, MOO is able to use a novel particle update method to adjust both the structure and parameters rather than adjusting them separately, thereby achieving suitable generalization performance of the FNN. Third, the convergence of MOO-FNN is analyzed in detail to guarantee its successful applications. Finally, the experimental studies of MOO-FNN have been performed on model identification of nonlinear systems to verify the effectiveness. The results illustrate that MOO-FNN has a significant improvement over some state-of-the-art algorithms. [ABSTRACT FROM AUTHOR]

Considering multi-user downlink ultra-high reliability and low latency communications (URLLC), this paper employs the artificial noise (AN) technique to establish a secure and covert broadcast communication paradigm for the first time. Specifically, a multi-antenna transmitter (Alice) broadcasts the confidential information to multiple legitimate users in the presence of a multi-antenna malicious warden (Willie) and a multi-antenna eavesdropper (Eve). It is well known that AN is an effective technique for securing the physical layer security (PLS) of signal transmissions. Nevertheless, AN emission also exposes the signal transmission and decreases the signal covertness. Taking into account the impact of short-packet URLLC transmissions, we investigate the joint optimization of the precoder and AN to maximize the secrecy rate under the covertness constraint. Although the considered problem is nonconvex, we propose a branch-reduce-and-bound (BRB)-based algorithm to solve it optimally. However, the nested-loop structure of the BRB-based algorithm incurs a high computational complexity. To strike a balance between the performance and computational complexity, we also propose a low-complexity penalty successive convex approximation (SCA)-based algorithm, whose performance approaches that of the optimal BRB-based algorithm, particularly in the low to medium transmit power regime. Simulation results demonstrate the excellent performance of our proposed optimization algorithms compared with various benchmark algorithms and unveil the importance of exploiting AN for secrecy provisioning. [ABSTRACT FROM AUTHOR]

With the integration of Mobile Edge Computing (MEC) and Network Function Virtualization (NFV), service providers are able to provide low-latency services to mobile users for profit. In this paper, we study the online service placement and request assignment problem in an MEC network, where service requests arrive one by one without the knowledge of future arrivals, and each arrived request demands a specific service with a tolerable service delay requirement with the aim to maximize the profit of the service provider, through admitting as many service requests as possible for a given monitoring period. This optimization objective is achieved by assigning service requests to appropriate cloudlets in the MEC network, pre-installing service instances into cloudlets to shorten service delays, and accommodating new services by revoking some idle service instances from cloudlets due to limited computing resources in MEC networks. In this paper, we first show that the problem is NP-hard. We then devise an efficient deep reinforcement learning algorithm for the online service placement and request assignment problem that consists of a deep reinforcement learning-based prediction mechanism for dynamic service placement, followed by a dynamic request assignment procedure to assign requests to cloudlets. We finally evaluate the performance of the proposed algorithms by conducting experiments through simulations. Simulation results demonstrate that the proposed algorithm is promising, improving performance by 46.8% compared with that of the comparison algorithms. [ABSTRACT FROM AUTHOR]

In this article, we consider the optimal coordination of automated vehicles at intersections under fixed crossing orders. We formulate the problem using direct optimal control and exploit the structure to construct a semidistributed primal-dual interior-point algorithm to solve it by parallelizing most of the computations. Differently from standard distributed optimization algorithms, where the optimization problem is split, in our approach we split the linear algebra steps, such that the algorithm takes the same steps as a fully centralized one, while still performing computations in a distributed fashion. We analyze the communication requirements of the algorithm, and propose an approximation scheme which can significantly reduce the data exchange. We demonstrate the effectiveness of the algorithm in hard but realistic scenarios, which show that the approximation leads to reductions in communicated data of almost 99% of the exact formulation, at the expense of less than 1% suboptimality. [ABSTRACT FROM AUTHOR]

A subset ${\mathcal{ S}}\subseteq V$ of vertices of an undirected graph $G=(V,E)$ is a hub cover when for each edge $(u,v) \in E$ , at least one of its endpoints belongs to ${\mathcal{ S}}$ , or there exists a vertex $r \in {\mathcal{ S}}$ that is a neighbor of both $u$ and $v$ . The problem of computing a minimum hub cover set in arbitrary graphs is NP-hard. This problem has applications for indexing large databases. This paper proposes $\Psi $ -MHC, the first approximation algorithm for the minimum hub cover set in arbitrary graphs to the best of our knowledge. The approximation ratio of this algorithm is $\ln \mu $ , where $\mu $ is upper bounded by $\min \left\{{\frac {1}{2}(\Delta +1)^{2}, \vert E\vert }\right\}$ and $\Delta $ is the degree of $G$ . The execution time of $\Psi $ -MHC is $O((\Delta + 1) \vert E \vert + \vert {\mathcal{ S}} \vert \vert V \vert)$ . Experimental results show that $\Psi $ -MHC far outperforms the theoretical approximation ratio for the input graph instances.

Recent advances in quantum technologies pave the way for noisy intermediate-scale quantum (NISQ) devices, where the quantum approximation optimization algorithm (QAOA) constitutes a promising candidate for demonstrating tangible quantum advantages based on NISQ devices. In this paper, we consider the maximum likelihood (ML) detection problem of binary symbols transmitted over a multiple-input and multiple-output (MIMO) channel, where finding the optimal solution is exponentially hard using classical computers. Here, we apply the QAOA for the ML detection by encoding the problem of interest into a level- $p$ QAOA circuit having $2p$ variational parameters, which can be optimized by classical optimizers. This level- $p$ QAOA circuit is constructed by applying the prepared Hamiltonian to our problem and the initial Hamiltonian alternately in $p$ consecutive rounds. More explicitly, we first encode the optimal solution of the ML detection problem into the ground state of a problem Hamiltonian. Using the quantum adiabatic evolution technique, we provide both analytical and numerical results for characterizing the evolution of the eigenvalues of the quantum system used for ML detection. Then, for level-1 QAOA circuits, we derive the analytical expressions of the expectation values of the QAOA and discuss the complexity of the QAOA based ML detector. Explicitly, we evaluate the computational complexity of the classical optimizer used and the storage requirement of simulating the QAOA. Finally, we evaluate the bit error rate (BER) of the QAOA based ML detector and compare it both to the classical ML detector and to the classical minimum mean squared error (MMSE) detector, demonstrating that the QAOA based ML detector is capable of approaching the performance of the classical ML detector. [ABSTRACT FROM AUTHOR]

Toward better approximation for the minimum-weighted vertex cover (MWVC) problem in multiagent systems, we present a distributed algorithm from the perspective of learning in games. For self-organized coordination and optimization, we see each vertex as a potential game player who makes decisions using local information of its own and the immediate neighbors. The resulting Nash equilibrium is classified into two categories, i.e., the inferior Nash equilibrium (INE) and the dominant Nash equilibrium (DNE). We show that the optimal solution must be a DNE. To achieve better approximation ratios, local rules of perturbation and weighted memory are designed, with the former destroying the stability of an INE and the latter facilitating the refinement of a DNE. By showing the existence of an improvement path from any INE to a DNE, we prove that when the memory length is larger than 1, our algorithm converges in finite time to DNEs, which could not be improved by exchanging the action of a selected node with all its unselected neighbors. Moreover, additional freedom for solution efficiency refinement is provided by increasing the memory length. Finally, intensive comparison experiments demonstrate the superiority of the presented methodology to the state of the art, both in solution efficiency and computation speed. [ABSTRACT FROM AUTHOR]

We consider decentralized large-scale continuous submodular constrained optimization problems over networks, where the goal is to maximize a sum of nonconvex functions with diminishing returns property. However, the computations of the projection step and the whole gradient can become prohibitive in high-dimensional constrained optimization problems. For this reason, a decentralized randomized block-coordinate Frank-Wolfe algorithm is proposed for submoduar maximization over networks by local communication and computation, which adopts the randomized block-coordinate descent and the Frank-Wolfe technique. We also show that the proposed algorithm converges to an approximation fact $(1-e^{-p_{\max }/p_{\min }})$ of the global maximal points at a rate of $\mathcal {O}(1/T)$ by choosing a suitable stepsize, where $T$ is the number of iterations. In addition, we confirm the theoretical results by experiments. [ABSTRACT FROM AUTHOR]

Subset selection plays an important role in the field of evolutionary multiobjective optimization (EMO). Especially, in an EMO algorithm with an unbounded external archive (UEA), subset selection is an essential post-processing procedure to select a prespecified number of solutions as the final result. In this article, we discuss the efficiency of greedy subset selection for the hypervolume, inverted generational distance (IGD), and IGD plus (IGD+) indicators. Greedy algorithms usually efficiently handle the subset selection. However, when a large number of solutions are given (e.g., subset selection from tens of thousands of solutions in a UEA), they often become time consuming. Our idea is to use the submodular property, which is known for the hypervolume indicator, to improve their efficiency. First, we prove that the IGD and IGD+ indicators are also submodular. Next, based on the submodular property, we propose an efficient greedy inclusion algorithm for each indicator. We demonstrate through computational experiments that the proposed algorithms are much faster than the standard greedy subset selection algorithms. The proposed algorithms also help the research on performance indicators. [ABSTRACT FROM AUTHOR]

There is an increasing need for charging station recommendation to minimize the overall charging time for electric vehicles and balance load for the charging stations. To grant this need, we model the recommendation problem as a Markov Decision Process (MDP) problem. However, the traditional MDP model has the issue of ‘curse of dimensionality’. To address this issue, we propose an extension of MDP: multiple-phase MDP, in which the state transition of MDP is decomposing into several phases, so as to reduce the state space and state transition complexities. This is done by introducing two states other than the normal state defined in MDP: post decision state and intermediate decision state. Then, we propose an online learning based algorithm to solve the formulated multiple-phase MDP model. Thanks to the reduced complexities of the state space and state transition, the proposed online algorithm can converge fast. By comparing to other recommendation mechanisms, such as game theory based recommendation and Q-learning based recommendation, our simulation evaluation demonstrates that our proposition can bring good performance. [ABSTRACT FROM AUTHOR]

Many problems in industrial smart manufacturing, such as process operational optimization and decision-making, can be regarded as distributed nonconvex optimization problems, whose goal is to utilize distributed nodes to cooperatively search for the minimal value of the global objective function. With the consideration of data transmission mode, transmission condition, and communication waste in industrial applications, it is meaningful to study the distributed nonconvex optimization problem with an event-triggered strategy over time-varying directed networks. To solve such a problem, a distributed nonconvex event-triggered algorithm is proposed in this article. Under some assumptions on local objective functions, gradients, and step sizes, the convergence of the proposed event-triggered algorithm to the local minimum is established theoretically. Moreover, it is obtained that the proposed distributed event-triggered algorithm has a convergence rate of $O(1/\ln (t))$. Finally, two examples of industrial systems are provided to validate the effectiveness of the proposed algorithm. [ABSTRACT FROM AUTHOR]

This letter investigates reconfigurable intelligent surface (RIS)-assisted full-duplex multiple-input single-output wireless system, where the beamforming and RIS phase shifts are optimized to maximize the sum-rate for both single and distributed RIS deployment schemes. The preference of using the single or distributed RIS deployment scheme is investigated through three practical scenarios based on the links’ quality. The closed-form solution is derived to optimize the beamforming vectors and a novel deep reinforcement learning (DRL) algorithm is proposed to optimize the RIS phase shifts. Simulation results illustrate that the choice of the deployment scheme depends on the scenario and the links’ quality. It is further shown that the proposed algorithm significantly improves the sum-rate compared to the non-optimized scenario in both single and distributed RIS deployment schemes. Besides, the proposed beamforming derivation achieves a remarkable improvement compared to the approximated derivation in previous works. Finally, the complexity analysis confirms that the proposed DRL algorithm reduces the computation complexity compared to the DRL algorithm in the literature. [ABSTRACT FROM AUTHOR]

This paper proposes a real-time schedule model of a microgrid (MG) for maximizing battery energy storage (BES) utilization. To this end, a BES life model is linearized using piece-wise linearization and big-M method to assess the BES life loss (BLL) in a real-time manner. The cost-effective schedule model of the MG with multiple energy resources aims to maximize BES utilization while ensuring its sufficient lifespan. Corresponding to the optimization model, approximate dynamic programming (ADP) for maximizing BES utilization (ADP-MBU) in the real-time schedule is proposed to optimize the system under stochastic environments. In ADP-MBU, a new value function approximation method employing the BES cumulative life loss (BCLL) is developed to improve the optimality and applicability. The proposed ADP-MBU algorithm can achieve satisfactory approximate optimality while reflecting the variation of real-time BLL. Case studies validate the applicability of the proposed MG schedule model and the advantages of the proposed ADP-MBU algorithm. [ABSTRACT FROM AUTHOR]

Nonlinear precoding and pulse shaping are jointly considered in multi-user massive multiple-input multiple-output (MIMO) systems with low resolution D/A-converters (DACs) in terms of algorithmic approach as well as large system performance. Two design criteria are investigated: the mean squared error (MSE) with active constellation extension (ACE) and the symbol error rate (SER). Both formulations are solved based on a modified version of the generalized approximate message passing (GAMP) algorithm. Furthermore, theoretical performance results are derived based on the state evolution analysis of the GAMP algorithm. The MSE based technique is extended to jointly perform over-the-air (OTA) spectral shaping and precoding for frequency-selective channels, in which the spectral performance is characterized at the transmitter and at the receiver. Simulation and analytical results demonstrate that the MSE based approach yields the same performance as the SER based formulation in terms of uncoded SER. The analytical results provide good performance predictions up to medium SNR. Substantial improvements in detection, as well as spectral performance, are obtained from the proposed combined pulse shaping and precoding approach compared to standard linear methods. [ABSTRACT FROM AUTHOR]

In this paper, we investigate an intelligent reflecting surface (IRS) assisted multi-user multiple-input multiple-output (MIMO) full-duplex (FD) system. We jointly optimize the active beamforming matrices at the access point (AP) and uplink users, and the passive beamforming matrix at the IRS to maximize the weighted sum-rate of the system. Since it is practically difficult to acquire the channel state information (CSI) for IRS-related links due to its passive operation and large number of elements, we conceive a mixed-timescale beamforming scheme. Specifically, the high-dimensional passive beamforming matrix at the IRS is updated based on the channel statistics while the active beamforming matrices are optimized relied on the low-dimensional real-time effective CSI at each time slot. We propose an efficient stochastic successive convex approximation (SSCA)-based algorithm for jointly designing the active and passive beamforming matrices. Moreover, due to the high computational complexity caused by the matrix inversion computation in the SSCA-based optimization algorithm, we further develop a deep-unfolding neural network (NN) to address this issue. The proposed deep-unfolding NN maintains the structure of the SSCA-based algorithm but introduces a novel non-linear activation function and some learnable parameters induced by the first-order Taylor expansion to approximate the matrix inversion. In addition, we develop a black-box NN as a benchmark. Simulation results show that the proposed mixed-timescale algorithm outperforms the existing single-timescale algorithm and the proposed deep-unfolding NN approaches the performance of the SSCA-based algorithm with much reduced computational complexity when deployed online. [ABSTRACT FROM AUTHOR]

Most existing random access schemes for machine-type communications (MTC) simply adopt a uniform preamble selection distribution, irrespective of the underlying device activity distributions. Hence, they may yield unsatisfactory access efficiency. In this paper, we model device activities for MTC as multiple Bernoulli random variables following an arbitrary multivariate Bernoulli distribution which can reflect both dependent and independent device activities. Then, we optimize preamble selection and access barring for random access in MTC according to the underlying joint device activity distribution. Specifically, we investigate three cases of the joint device activity distribution, i.e., the cases of perfect, imperfect, and unknown joint device activity distributions, and formulate the average, worst-case average, and sample average throughput maximization problems, respectively. The problems in the three cases are challenging nonconvex problems. In the case of perfect joint device activity distribution, we develop an iterative algorithm and a low-complexity iterative algorithm to obtain stationary points of the original problem and an approximate problem, respectively. In the case of imperfect joint device activity distribution, we develop an iterative algorithm and a low-complexity iterative algorithm to obtain a Karush-Kuhn-Tucker (KKT) point of an equivalent problem and a stationary point of an approximate problem, respectively. Finally, in the case of unknown joint device activity distribution, we develop an iterative algorithm to obtain a stationary point. The proposed solutions are widely applicable and outperform existing solutions for dependent and independent device activities. [ABSTRACT FROM AUTHOR]

Hammerstein model with a static nonlinearity followed by a linear filter is commonly used in numerous applications. This paper focuses on adaptive filtering techniques for parameter identification of Hammerstein systems and output prediction of nonlinear systems. By formulating the underlying filtering problem as a recursive bilinear least-squares optimization with the non-convex feasible region constraint, we develop a recursive non-convex projected least-squares (RncPLS) algorithm based on alternating direction method of multipliers (ADMM). The RncPLS algorithm alternates between implementing ridge regression and projecting on the non-convex feasible set, which successively refines the system parameters. The convergence and accuracy properties of the proposed RncPLS algorithm are theoretically investigated. Moreover, extensive simulation results in the context of system identification, nonlinear predication, and acoustic echo cancellation, are also included to demonstrate the performance characteristics of the proposed algorithm. [ABSTRACT FROM AUTHOR]

Numerical optimization has been playing an increasingly important role in the design of contemporary antenna systems. Due to the shortage of design-ready theoretical models, optimization is mainly based on electromagnetic (EM) analysis, which tends to be costly. Numerous techniques have evolved to abate this cost, including surrogate-assisted frameworks for global optimization, or sparse sensitivity updates for speeding up local search. In the latter, CPU-heavy updates of the system response sensitivity through finite differentiation are suppressed based on, e.g., the magnitude of design variability during the optimization run. Another approach is to incorporate variable-resolution simulations. Recently, a technique exploiting a continuous spectrum of admissible model fidelity levels has been reported, thereby allowing for a considerable reduction of computational expenditures. Seeking further savings, this work introduces an accelerated gradient-based algorithm with sparse sensitivity updates and variable-resolution EM simulations. Our technique is validated using four broadband antennas, and demonstrated to offer substantial (around 80%) savings over the benchmark while maintaining acceptable design quality. [ABSTRACT FROM AUTHOR]

Due to the rapid development and wide application of mobile devices, mobile edge computing (MEC) technology improves system performance by offloading vehicle computing tasks to the edges of the network. In order to save device power, shorten transmission delay, save network bandwidth, or shorten execution time, many related works have been proposed. Different from these works under the constraints of system resources, we adopt another method to maximize the admission of mobile users' tasks to meet the service requirements. Compared with these works, this method can allocate and utilize all system resources more effectively and admit as many tasks as possible. Further, we prove that the approximate ratio of its solution to the optimal solution of the problem of maximizing task admission in MEC is bounded. The above claim is verified by the conducted simulations and the simulation results show that there is a small difference between its solution and the optimal one. On the other hand, if the system resources can be increased at a bounded rate, another algorithm is proposed to control the compromise between the overall system performance and the robustness of its obtained solution. [ABSTRACT FROM AUTHOR]

We consider a class of non-convex complex modulus constrained problems (CMCPs), which has many important applications in signal processing for wireless communications, including multicast beamforming and multi-input multi-output detection. Due to the non-convex constraints, traditional optimization-based algorithms either obtain sub-optimal solutions or take exponential complexity to reach the optimum. In this paper, we propose a learning based branch-and-bound (LBB) algorithm for solving the considered CMCPs. LBB regards the search procedure of the global optimal argument-cut based branch-and-bound (AC-BB) algorithm as a sequential decision problem in a binary tree and learns the optimal pruning policy via supervised learning. We first propose to apply ensemble learning to train multiple classifiers and then combine them to achieve better performance. To tackle the imbalanced issue, we propose an undersampling-supervised learning method to sample several balanced subsets and train a classifier on each of them. We also show that the computational complexity of LBB is determined by the depth of the binary tree and give its expression in the worst case. Finally, we validate the proposed LBB with two classic problems in wireless communications, i.e., multicast beamforming and MIMO detection. Numerical results show that LBB runs significantly faster than AC-BB while achieving nearly the same optimal performance. [ABSTRACT FROM AUTHOR]