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Due to the advantages such as high flexibility, low cost and easy implementation offered by unmanned aerial vehicles (UAVs), a UAV-assisted network is regard as an appealing solution to a seamless coverage, high disaster-tolerant and on-demand wireless system. In this paper, we focus on the downlink transmission in a cache-enabled UAV-assisted wireless communication network, where UAVs cache popular content from a macro base station in advance and cooperatively transfer the content to users. We aim to minimize the average transmission latency of the system and to formulate an optimization problem that jointly optimizes the UAV location, beamforming and caching strategy. However, the formulated problem is very challenging because of its non-convexity and the highly coupled optimization variables. To solve this resulting problem efficiently, we decompose it into two subproblems, namely UAV location and beamforming optimization, and UAV caching strategy optimization. The first subproblem is an NP-hard joint optimization problem, while the second one is a linear programing problem. By adopting the first-order Taylor expansion, we propose a convex optimization algorithm based on the difference-of-convex (DC) method. Specifically, we bring out a method to apply linear approximation in the DC-based algorithm, which is particularly suitable to the problems involving complicated summations. The numerical results demonstrate that the proposed DC-based iterative optimization algorithm can efficiently reduce the average transmission latency of the system. [ABSTRACT FROM AUTHOR]

The Unsplittable Flow on a Path (UFP) problem has garnered considerable attention as a challenging combinatorial optimization problem with notable practical implications. Steered by its pivotal applications in power engineering, the present work formulates a novel generalization of UFP, wherein demands and capacities in the input instance are monotone step functions over the set of edges. As an initial step towards tackling this generalization, we draw on and extend ideas from prior research to devise a quasi-polynomial time approximation scheme under the premise that the demands and capacities lie in a quasi-polynomial range. Second, retaining the same assumption, an efficient logarithmic approximation is introduced for the single-source variant of the problem. Finally, we round up the contributions by designing a (kind of) black-box reduction that, under some mild conditions, allows to translate LP-based approximation algorithms for the studied problem into their counterparts for the Alternating Current Optimal Power Flow problem—a fundamental workflow in operation and control of power systems. [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]

There are problems of interference between Devices-to-Devices (D2D) communication users and cellular users, mutu¬al interference between D2D communication users, and environmental interference caused by the existence of environmental obstacles in the D2D communication system assisted by Reconfigurable Intelligent Surface (RIS). In order to solve the problems, an optimization algorithm aiming at maximizing the total rate of the system is proposed to maintain the high quality of services among cellular users and D2D communication. Simulation experiments show that the total system speed under the tabu search- successive convex approximation algorithm is improved by 56% compared with the phase shift optimization algorithm. The to¬tal system speed under the tabu search-successive convex approximation algorithm is improved by 21. 2% compared with the power allocation algorithm. [ABSTRACT FROM AUTHOR]

The variational quantum algorithm (VQA) is a hybrid classical–quantum algorithm. It can actually run in an intermediate-scale quantum device where the number of available qubits is too limited to perform quantum error correction, so it is one of the most promising quantum algorithms in the noisy intermediate-scale quantum era. In this paper, two ideas for solving the learning with errors problem (LWE) using VQA are proposed. First, after reducing the LWE problem into the bounded distance decoding problem, the quantum approximation optimization algorithm (QAOA) is introduced to improve classical methods. Second, after the LWE problem is reduced into the unique shortest vector problem, the variational quantum eigensolver (VQE) is used to solve it, and the number of qubits required is calculated in detail. Small-scale experiments are carried out for the two LWE variational quantum algorithms, and the experiments show that VQA improves the quality of the classical solutions. [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 paper, the problem we study is how to maximize a monotone non-submodular function with cardinality constraint. Different from the previous streaming algorithms, this paper mainly considers the sliding window model. Based on the concept of diminishing-return ratio γ , we propose a (1 3 γ 2 −) -approximation algorithm with the memory O (k log 2 (k Φ 1 γ) 2 ) , where Φ is the ratio between maximum and minimum values of any singleton element of function f. Then, we improve the approximation ratio to (1 2 γ −) through the sub-windows at the expense of losing some memory. Our results generalize the corresponding results for the submodular case. [ABSTRACT FROM AUTHOR]

We consider the problem of fairly allocating indivisible goods, focusing on a recently introduced notion of fairness called maximin share guarantee: each player's value for his allocation should be at least as high as what he can guarantee by dividing the items into as many bundles as there are players and receiving his least desirable bundle. Assuming additive valuation functions, we show that such allocations may not exist, but allocations guaranteeing each player 2/3 of the above value always exist. These theoretical results have direct practical implications. [ABSTRACT FROM AUTHOR]

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]

Quantum Approximate Optimization algorithm (QAOA) aims to search for approximate solutions to discrete optimization problems with near-term quantum computers. As there are no algorithmic guarantee possible for QAOA to outperform classical computers, without a proof that bounded-error quantum polynomial time (BQP) ≠ nondeterministic polynomial time (NP), it is necessary to investigate the empirical advantages of QAOA. We identify a computational phase transition of QAOA when solving hard problems such as SAT—random instances are most difficult to train at a critical problem density. We connect the transition to the controllability and the complexity of QAOA circuits. Moreover, we find that the critical problem density in general deviates from the SAT-UNSAT phase transition, where the hardest instances for classical algorithms lies. Then, we show that the high problem density region, which limits QAOA's performance in hard optimization problems (reachability deficits), is actually a good place to utilize QAOA: its approximation ratio has a much slower decay with the problem density, compared to classical approximate algorithms. Indeed, it is exactly in this region that quantum advantages of QAOA over classical approximate algorithms can be identified. [ABSTRACT FROM AUTHOR]

The manta ray foraging optimization algorithm (MRFO) is one of the promised meta-heuristic optimization algorithms. However, it can stick to a local minimum, consuming iterations without reaching the optimum solution. So, this paper proposes a hybridization between MRFO, and the genetic algorithm (GA) based on a pseudo parameter; where the GA can help MRFO to escape from falling into the local minimum. It is called a pseudo genetic algorithm with manta-ray foraging optimization (PGA-MRFO). The proposed algorithm is not a classical hybridization between MRFO and GA, wherein the classical hybridization consumes time in the search process as each algorithm is applied to all system variables. In addition, the classical hybridization results in an extended search algorithm, especially in systems with many variables. The PGA-MRFO hybridizes the pseudo-parameter-based GA and the MRFO algorithm to produce a more efficient algorithm that combines the advantages of both algorithms without getting stuck in a local minimum or taking a long time in the calculations. The pseudo parameter enables the GA to be applied to a specific number of variables and not to all system variables leading to reduce the computation time and burden. Also, the proposed algorithm used an approximation for the gradient of the objective function, which leads to dispensing derivatives calculations. Besides, PGA-MRFO depends on the pseudo inverse of non-square matrices, which saves calculations time; where the dependence on the pseudo inverse gives the algorithm more flexibility to deal with square and non-square systems. The proposed algorithm will be tested on the test functions that the main MRFO failed to find their optimum solution to prove its capability and efficiency. In addition, it will be applied to solve the unit commitment (UC) problem as one of the vital power system problems to show the validity of the proposed algorithm in practical applications. Finally, several analyses will be applied to the proposed algorithm to illustrate its effectiveness and reliability. [ABSTRACT FROM AUTHOR]

For a continuous multi-objective optimization problem, it is usually not a practical approach to compute all its nondominated points because there are infinitely many of them. For this reason, a typical approach is to compute an approximation of the nondominated set. A common technique for this approach is to generate a polyhedron which contains the nondominated set. However, often these approximations are used for further evaluations. For those applications a polyhedron is a structure that is not easy to handle. In this paper, we introduce an approximation with a simpler structure respecting the natural ordering. In particular, we compute a box-coverage of the nondominated set. To do so, we use an approach that, in general, allows us to update not only one but several boxes whenever a new nondominated point is found. The algorithm is guaranteed to stop with a finite number of boxes, each being sufficiently thin. [ABSTRACT FROM AUTHOR]

Existing wideband beampattern synthesis methods usually require multiple beamformers, each corresponding to one frequency bin, which results in excessive resource consumption in engineering practice. To alleviate the resource consumption of wideband array beamforming, we propose a novel time-domain wideband digital beamformer that can control the beam shape over the whole bandwidth for interference mitigation by using only one set of amplitude–phase weights and integer time delay filters. To tackle the resultant optimization problem, two effective algorithms are introduced, and both can solve the problem within polynomial time. One is an alternating optimization algorithm that decomposes the original problem into multiple tractable subproblems iteratively. The other is the convex approximation algorithm that transforms the original problem into a series of second-order cone programming problems iteratively. The two algorithms exhibit their respective advantages in terms of convergence and computational complexity. Extensive numerical examples are presented to evaluate the proposed time-domain wideband beamforming strategy, highlighting that it can form a deep notch in any interested spatial–spectral region to suppress the strong interferences. [ABSTRACT FROM AUTHOR]

We study the problem of maximizing a monotone submodular function subject to a Multiple Knapsack constraint. The input is a set I of items, each has a non-negative weight, and a set of bins of arbitrary capacities. Also, we are given a submodular, monotone and non-negative function f over subsets of the items. The objective is to find a packing of a subset of items A ⊆ I in the bins such that f (A) is maximized. Our main result is an almost optimal polynomial time (1 − e − 1 − ε) -approximation algorithm for the problem, for any ε > 0. The algorithm relies on a structuring technique which converts a general multiple knapsack constraint to a constraint in which the bins are partitioned into groups of exponentially increasing cardinalities, each consisting of bins of uniform capacity. We derive the result by combining structuring with a refined analysis of techniques for submodular optimization subject to knapsack constraints. [ABSTRACT FROM AUTHOR]

Non-orthogonal multiple access (NOMA) and backscatter communication are two emerging technologies for low-power communication. In this paper, we consider a NOMA backscatter system, where signals from two backscatter devices are multiplexed on a frequency resource block using NOMA in each time slot. Our objective is to maximize the average energy efficiency by optimizing backscatter device pairing, reflection coefficients of backscatter devices, and the transmit power of the reader. We formulate the average energy efficiency maximization problem subject to the minimum circuit power and the minimum data rate requirements of the backscatter devices, and the transmit power constraint of the reader. The formulated problem is nonconvex. To obtain a suboptimal solution for this problem, we use alternating optimization technique and decompose the problem into two subproblems. The subproblems are solved by using fractional programming, Dinkelbach’s algorithm, and successive convex approximation method. Simulation results show that our proposed algorithm converges quickly to a suboptimal solution. Our proposed algorithm outperforms several baseline algorithms, including the genetic algorithm, fixed device pairing scheme, conventional device pairing scheme, maximum transmit power allocation scheme, and random reflection coefficient selection scheme, in terms of the average energy efficiency. The optimality gap of our proposed algorithm is investigated by comparing with the optimal scheme, in which the optimal device pairing is obtained based on exhaustive search. [ABSTRACT FROM AUTHOR]

Quantum state tomography is a process for estimating an unknown quantum state; which is innately probabilistic. The exponential growth of unknown parameters to be estimated is a fundamental difficulty in realizing quantum state tomography for higher dimensions. Iterative optimization algorithms like self-guided quantum tomography have been effective in robust and accurate ascertaining a quantum state even with exponential growth in Hilbert space. We propose a faster convergent simultaneous perturbation stochastic approximation algorithm which is more practical in a resource-deprived situation for determining the underlying quantum states by incorporating the Barzilai–Borwein two-point step size gradient method with minimal loss of accuracy. [ABSTRACT FROM AUTHOR]

We present a simple deterministic distributed (2 + ε)-approximation algorithm for minimum-weight vertex cover, which completes in O(logΔ/ε log logΔ) rounds, where Δ is the maximum degree in the graph, for any ε > 0 that is at most O(1). For a constant Δ, this implies a constant approximation in O(logΔ/log logΔ) rounds, which contradicts the lower bound of [KMW10]. [ABSTRACT FROM AUTHOR]

We suggest a new optimization technique for minimizing the sum $\sum _{i=1}^n g_i(x)$ ∑ i = 1 n g i (x) of $n$ n non-convex real functions that satisfy a property that we call piecewise log-Lipschitz. This is by forging links between techniques in computational geometry, combinatorics and convex optimization. As an example application, we provide the first constant-factor approximation algorithms whose running-times are polynomial in $n$ n for the fundamental problem of Points-to-Lines alignment: Given $n$ n points $p_1,\ldots,p_n$ p 1 ,... , p n and $n$ n lines $\ell _1,\ldots,\ell _n$ ℓ 1 ,... , ℓ n on the plane and $z>0$ z > 0 , compute the matching $\pi :[n]\to [n]$ π : [ n ] → [ n ] and alignment (rotation matrix $R$ R and translation vector $t$ t ) that minimize the sum of euclidean distances $\sum _{i=1}^n \mathrm{dist}(Rp_i-t,\ell _{\pi (i)})^z$ ∑ i = 1 n dist (R p i - t , ℓ π (i)) z between each point to its corresponding line. This problem is non-trivial even if $z=1$ z = 1 and the matching $\pi$ π is given. If $\pi$ π is given, our algorithms run in $O(n^3)$ O (n 3) time, and even near-linear in $n$ n using core-sets that support: streaming, dynamic, and distributed parallel computations in poly-logarithmic update time. Generalizations for handling e.g., outliers or pseudo-distances such as $M$ M -estimators for the problem are also provided. Experimental results and open source code show that our algorithms improve existing heuristics also in practice. A companion demonstration video in the context of Augmented Reality shows how such algorithms may be used in real-time systems. [ABSTRACT FROM AUTHOR]

Recent research has revealed new and unexpected applications of network control science within biomedicine, pharmacology, and medical therapeutics. These new insights and new applications generated in turn a rediscovery of some old, unresolved algorithmic problems. One of these problems is the Structural Target Control optimization problem, known in previous literature also as Structural Output Controllability problem, which is defined as follows. Given a directed network and a target subset of nodes, the task is to select a small (or the smallest) set of nodes from which the target can be independently controlled, i.e., there exists a set of paths from the selected set of nodes (called driver nodes) to the target nodes such that no two paths intersect at the same distance from their targets. Recently, Structural Target Control optimization problem has been shown to be NP-hard, and several heuristic algorithms were introduced and analyzed, both on randomly generated networks, and on biomedical ones. In this paper, we show that the Structural Target Controllability problem is fixed parameter tractable when parameterized by the number of target nodes. We also prove that the problem is hard to approximate at a factor better than O(log n). Taking into consideration the real case formulations of this problem we identify two more parameters which are naturally constrained by smaller bounds: the maximal length of a controlling path and the size of the set of nodes from which the control can start. With these new parameters we provide an approximation algorithm which is of exponential complexity in the size of the set of nodes from which the control can start and polynomial in all the other parameters. [ABSTRACT FROM AUTHOR]

• The placement of data and the scheduling of request are studied in edge caching. • Network unreliability and limited storage and bandwidth resources are considered. • Two algorithms with different computation complexity and cache efficiency are developed. • The experiments are conducted to evaluate the performance of proposed caching strategies. Edge caching is a prevailing media delivery technology where data is hosted at the edge nodes with computing and storage capability in close proximity to users, in order to expand the backhaul network capacity and enhance users' quality of experience (QoE). The existing work in this area often neglects the fact that large-scale distributed cache networks are not particularly reliable and many edge nodes are prone to failure. In this paper we investigate and develop a novel, cooperative caching mechanism for content placement and request routing. We aim to minimize the content access delay and achieve the optimization in polynomial time, taking into account failures in an unreliable network environment with limited edge storage and bandwidth. We introduce two optimization algorithms: 1) a primal-dual algorithm that is based on the Lagrangian dual decomposition and subgradient method, and 2) a greedy-based approximation algorithm with a proven approximation ratio. Numerical results show that the proposed algorithms outperform other comparative approaches in synthetic and real network environments, and the approximation algorithm is particularly suitable for networking scenarios with sparse node connectivity and resources in short supply. [ABSTRACT FROM AUTHOR]

Algorithms for approximating the nondominated set of multiobjective optimization problems are reviewed. The approaches are categorized into general methods that are applicable under mild assumptions and, thus, to a wide range of problems, and into algorithms that are specifically tailored to structured problems. All in all, this survey covers 52 articles published within the last 41 years, that is, between 1979 and 2020. Summary of Contribution: In many problems in operations research, several conflicting objective functions have to be optimized simultaneously, and one is interested in finding Pareto optimal solutions. Because of the high complexity of finding Pareto optimal solutions and their usually very large number, however, the exact solution of such multiobjective problems is often very difficult, which motivates the study of approximation algorithms for multiobjective optimization problems. This research area uses techniques and methods from algorithmics and computing in order to efficiently determine approximate solutions to many well-known multiobjective problems from operations research. Even though approximation algorithms for multiobjective optimization problems have been investigated for more than 40 years and more than 50 research articles have been published on this topic, this paper provides the first survey of this important area at the intersection of computing and operations research. [ABSTRACT FROM AUTHOR]

Unsourced random-access (U-RA) is a type of grant-free random access with a virtually unlimited number of users, of which only a certain number $K_{a}$ are active on the same time slot. Users employ exactly the same codebook, and the task of the receiver is to decode the list of transmitted messages. We present a concatenated coding construction for U-RA on the AWGN channel, in which a sparse regression code (SPARC) is used as an inner code to create an effective outer OR-channel. Then an outer code is used to resolve the multiple-access interference in the OR-MAC. We propose a modified version of the approximate message passing (AMP) algorithm as an inner decoder and give a precise asymptotic analysis of the error probabilities of the AMP decoder and of a hypothetical optimal inner MAP decoder. This analysis shows that the concatenated construction under optimal decoding can achieve a vanishing per-user error probability in the limit of large blocklength and a large number of active users at sum-rates up to the symmetric Shannon capacity, i.e. as long as $K_{a}R < 0.5\log _{2}(1+K_{a} { \mathsf {SNR}})$. This extends previous point-to-point optimality results about SPARCs to the unsourced multiuser scenario. Furthermore, we give an optimization algorithm to find the power allocation for the inner SPARC code that minimizes the ${\mathsf {SNR}}$ required to achieve a given target per-user error probability with the AMP decoder. [ABSTRACT FROM AUTHOR]

We consider a distributed stochastic optimization problem in networks with finite number of nodes. Each node adjusts its action to optimize the global utility of the network, which is defined as the sum of local utilities of all nodes. While Gradient descent method is a common technique to solve such optimization problem, the computation of the gradient may require much information exchange. In this paper, we consider that each node can only have a noisy numerical observation of its local utility, of which the closed-form expression is not available. This assumption is quite realistic, especially when the system is either too complex or constantly changing. Nodes may exchange partially the observation of their local utilities to estimate the global utility at each timeslot. We propose a distributed algorithm based on stochastic perturbation, under the assumption that each node has only part of the local utilities of the other nodes. We use stochastic approximation tools to prove that our algorithm converges almost surely to the optimum, given that the objective function is smooth and strictly concave. The convergence rate is also derived, under the additional assumption of strongly concave objective function. It is shown that the convergence rate scales as O (K−0.5) after a sufficient number of iterations K > K0, which is the optimal rate order in terms of K for our problem. Although the proposed algorithm can be applied to general optimization problems, we perform simulations for a typical power control problem in wireless networks and present numerical results to corroborate our claims. [ABSTRACT FROM AUTHOR]

In this work, we consider a logarithmic nonlinear viscoelastic wave equation with a delay term in a bounded domain. We obtain the local existence of the solution by using the Faedo-Galerkin approximation. Then, under suitable conditions, we prove the blow up of solutions in finite time. [ABSTRACT FROM AUTHOR]

We revisit the problem of online linear optimization in the case where the set of feasible actions is accessible through an approximated linear optimization oracle with a factor α multiplicative approximation guarantee. This setting in particular is interesting because it captures natural online extensions of well-studied offline linear optimization problems that are NP-hard yet admit efficient approximation algorithms. The goal here is to minimize the α-regret, which is the natural extension to this setting of the standard regret in online learning. We present new algorithms with significantly improved oracle complexity for both the full-information and bandit variants of the problem. Mainly, for both variants, we present α-regret bounds of O (T − 1 / 3 ) , were T is the number of prediction rounds, using only O (l o g T) calls to the approximation oracle per iteration, on average. These are the first results to obtain both the average oracle complexity of O (l o g T) (or even polylogarithmic in T) and α -regret bound O (T − c ) for a constant c > 0 for both variants. [ABSTRACT FROM AUTHOR]

The emergence of big data has caused a dramatic shift in the operating regime for optimization algorithms. The performance bottleneck, which used to be computations, is now often communications. Several gradient compression techniques have been proposed to reduce the communication load at the price of a loss in solution accuracy. Recently, it has been shown how compression errors can be compensated for in the optimization algorithm to improve the solution accuracy. Even though convergence guarantees for error-compensated algorithms have been established, there is very limited theoretical support for quantifying the observed improvements in solution accuracy. In this paper, we show that Hessian-aided error compensation, unlike other existing schemes, avoids accumulation of compression errors on quadratic problems. We also present strong convergence guarantees of Hessian-based error compensation for stochastic gradient descent. Our numerical experiments highlight the benefits of Hessian-based error compensation, and demonstrate that similar convergence improvements are attained when only a diagonal Hessian approximation is used. [ABSTRACT FROM AUTHOR]

This paper considers the networked system optimization problem by cooperatively finding an approximately optimal solution to the overall network convex cost function, which is the sum of the individual cost functions of nodes (agents) in a networked system. A new distributed gradient descent algorithm is proposed based on the distributed estimation of the sum of gradients of individual cost functions using a consensus-type coordination algorithm. The proposed algorithm can address unknown directed communication topologies and only requires limited communications and information exchanges among nodes in the networked system. Under the assumption that the communication graph is strongly connected, the convergence of the proposed algorithm is rigorously analyzed. Simulation examples are presented to demonstrate the applicability and effectiveness of the proposed algorithm. [ABSTRACT FROM AUTHOR]

An important tool in the study of the complexity of Constraint Satisfaction Problems (CSPs) is the notion of a relational clone, which is the set of all relations expressible using primitive positive formulas over a particular set of base relations. Post's lattice gives a complete classification of all Boolean relational clones, and this has been used to classify the computational difficulty of CSPs. Motivated by a desire to understand the computational complexity of (weighted) counting CSPs, we develop an analogous notion of functional clones and study the landscape of these clones. One of these clones is the collection of log-supermodular (lsm) functions, which turns out to play a significant role in classifying counting CSPs. In the conservative case (where all nonnegative unary functions are available), we show that there are no functional clones lying strictly between the clone of lsm functions and the total clone (containing all functions). Thus, any counting CSP that contains a single nontrivial non-lsm function is computationally as hard to approximate as any problem in #P. Furthermore, we show that any nontrivial functional clone (in a sense that will be made precise) contains the binary function "implies". As a consequence, in the conservative case, all nontrivial counting CSPs are as hard to approximate as #BIS, the problem of counting independent sets in a bipartite graph. Given the complexity-theoretic results, it is natural to ask whether the "implies" clone is equivalent to the clone of lsm functions. We use the Möbius transform and the Fourier transform to show that these clones coincide precisely up to arity 3. It is an intriguing open question whether the lsm clone is finitely generated. Finally, we investigate functional clones in which only restricted classes of unary functions are available. [ABSTRACT FROM AUTHOR]

A common approach to clustering data is to view data objects as points in a metric space, and then to optimize a natural distance-based objective such as the k-median, k-means, or min-sum score. For applications such as clustering proteins by function or clustering images by subject, the implicit hope in taking this approach is that the optimal solution for the chosen objective will closely match the desired "target" clustering (e.g., a correct clustering of proteins by function or of images by who is in them). However, most distance-based objectives, including those mentioned here, are NP-hard to optimize. So, this assumption by itself is not sufficient, assuming P ≠ NP, to achieve clusterings of low-error via polynomial time algorithms. In this article, we show that we can bypass this barrier if we slightly extend this assumption to ask that for some small constant c, not only the optimal solution, but also all c-approximations to the optimal solution, differ from the target on at most some ∈ fraction of points--we call this (c, ∈)-approximation-stability. We show that under this condition, it is possible to efficiently obtain low-error clusterings even if the property holds only for values c for which the objective is known to be NP-hard to approximate. Specifically, for any constant c > 1, (c, ∈)-approximation-stability of k-median or k-means objectives can be used to efficiently produce a clustering of error O(∈) with respect to the target clustering, as can stability of the min-sum objective if the target clusters are sufficiently large. Thus, we can perform nearly as well in terms of agreement with the target clustering as if we could approximate these objectives to this NP-hard value. [ABSTRACT FROM AUTHOR]

We investigate variants of Lloyd's heuristic for clustering high-dimensional data in an attempt to explain its popularity (a half century after its introduction) among practitioners, and in order to suggest improvements in its application. We propose and justify a clusterability criterion for data sets.We present variants of Lloyd's heuristic that quickly lead to provably near-optimal clustering solutions when applied to well-clusterable instances. This is the first performance guarantee for a variant of Lloyd's heuristic. The provision of a guarantee on output quality does not come at the expense of speed: some of our algorithms are candidates for being faster in practice than currently used variants of Lloyd's method. In addition, our other algorithms are faster on well-clusterable instances than recently proposed approximation algorithms, while maintaining similar guarantees on clustering quality. Our main algorithmic contribution is a novel probabilistic seeding process for the starting configuration of a Lloyd-type iteration. [ABSTRACT FROM AUTHOR]

Stochastic optimization problems attempt to model uncertainty in the data by assuming that the input is specified by a probability distribution. We consider the well-studied paradigm of 2-stage models with recourse: first, given only distributional information about (some of) the data one commits on initial actions, and then once the actual data is realized (according to the distribution), further (recourse) actions can be taken. We show that for a broad class of 2-stage linear models with recourse, one can, for any ∈ > 0, in time polynomial in 1/∈ and the size of the input, compute a solution of value within a factor (1 + ∈) of the optimum, in spite of the fact that exponentially many second-stage scenarios may occur. In conjunction with a suitable rounding scheme, this yields the first approximation algorithms for 2-stage stochastic integer optimization problems where the underlying random data is given by a "black box" and no restrictions are placed on the costs in the two stages. Our rounding approach for stochastic integer programs shows that an approximation algorithm for a deterministic analogue yields, with a small constant-factor loss, provably near-optimal solutions for the stochastic generalization. Among the range of applications, we consider are stochastic versions of the multicommodity flow, set cover, vertex cover, and facility location problems. [ABSTRACT FROM AUTHOR]

We study a novel genre of optimization problems, which we call segmentation problems, motivated in part by certain aspects of clustering and data mining. For any classical optimization problem, the corresponding segmentation problem seeks to partition a set of cost vectors into several segments, so that the overall cost is optimized. We focus on two natural and interesting (but MAXSNP-complete) problems in this class, the HYPERCUBE SEGMENTATION PROBLEM and the CATALOG SEGMENTATION PROBLEM, and present approximation algorithms for them. We also present a general greedy scheme, which can be specialized to approximate any segmentation problem. [ABSTRACT FROM AUTHOR]

This paper proposes robust reformulations of ambiguous chance constraints when the underlying family of distributions is discrete and supported in a so-called "p-box" or "p-ellipsoidal" uncertainty set. Using the robust optimization paradigm, the deterministic counterparts of the ambiguous chance constraints are reformulated as mixed-integer programming problems which can be tackled by commercial solvers for moderate sized instances. For larger sized instances, we propose a safe approximation algorithm that is computationally efficient and yields high quality solutions. The associated approach and the algorithm can be easily extended to joint chance constraints, nonlinear inequalities, and dependent data without introducing additional mathematical optimization complexity to that of the original robust reformulation. In numerical experiments, we first present our approach over a toy-sized chance constrained knapsack problem. Then, we compare optimality and computational performances of the safe approximation algorithm with those of the exact and the randomized approaches for larger sized instances via Monte Carlo simulation. [ABSTRACT FROM AUTHOR]

Abstract It is well-known that every non-negative univariate real polynomial can be written as the sum of two polynomial squares with real coefficients. When one allows a (non-negatively) weighted sum of finitely many squares instead of a sum of two squares, then one can choose all coefficients in the representation to lie in the field generated by the coefficients of the polynomial. In particular, this allows for an effective treatment of polynomials with rational coefficients. In this article, we describe, analyze and compare, from both the theoretical and practical points of view, two algorithms computing such a weighted sum of squares decomposition for univariate polynomials with rational coefficients. The first algorithm, due to the third author, relies on real root isolation, quadratic approximations of positive polynomials and square-free decomposition, but its complexity was not analyzed. We provide bit complexity estimates, both on the runtime and the output size of this algorithm. They are exponential in the degree of the input univariate polynomial and linear in the maximum bitsize of its complexity. This analysis is obtained using quantifier elimination and root isolation bounds. The second algorithm, due to Chevillard, Harrison, Joldes and Lauter, relies on complex root isolation and square-free decomposition, and was introduced for certifying positiveness of polynomials in the context of computer arithmetic. Again, its complexity was not analyzed. We provide bit complexity estimates, both on the runtime and the output size of this algorithm, which are polynomial in the degree of the input polynomial and linear in the maximum bitsize of its complexity. This analysis is obtained using Vieta's formula and root isolation bounds. Finally, we report on our implementations of both algorithms and compare them in practice on several application benchmarks. While the second algorithm is, as expected from the complexity result, more efficient on most of examples, we exhibit families of non-negative polynomials for which the first algorithm is better. [ABSTRACT FROM AUTHOR]

This paper introduces an efficient algorithm for finding the dominant generalized eigenvectors of a pair of symmetric matrices. Combining tools from approximation theory and convex optimization, we develop a simple scalable algorithm with strong theoretical performance guarantees. More precisely, the algorithm retains the simplicity of the well-known power method but enjoys the asymptotic iteration complexity of the powerful Lanczos method. Unlike these classic techniques, our algorithm is designed to decompose the overall problem into a series of subproblems that only need to be solved approximately. The combination of good initializations, fast iterative solvers, and appropriate error control in solving the subproblems lead to a linear running time in the input sizes compared to the superlinear time for the traditional methods. The improved running time immediately offers acceleration for several applications. As an example, we demonstrate how the proposed algorithm can be used to accelerate canonical correlation analysis, which is a fundamental statistical tool for learning of a low-dimensional representation of high-dimensional objects. Numerical experiments on real-world datasets confirm that our approach yields significant improvements over the current state of the art. [ABSTRACT FROM AUTHOR]

This paper introduces a novel methodology for residential building contouring from large-scale airborne point clouds. Unlike other methods that handle linearization and regularization of the linear primitives separately by imposing rigid constraints, we propose an optimization-based linearization and global regularization to form accurate, topologically error-free, and lightweight polygons. To this end, we enhance the classic density-based spatial clustering of applications with noise algorithm to segment individual building entities at the instance level. The initial contours of each individual building are then delineated and further decomposed by a novel topologically aware propagation process and a global optimization technique. The decomposed linear primitives are fed into the global regularization step, from which the regular shapes are learned and enforced hierarchically by imposing constraints, such as parallelism, homogeneity, orthogonality, and collinearity. Based on the concept of hybrid representation, the regularized and unaltered linear primitives are jointly connected in an esthetic way. Various experiments using representative buildings and large-scale residential scenes from the Dutch AHN3 data set have shown that the proposed methodology generates meaningful building contouring representation in terms of accuracy, compactness, topology, and levels of detail abstraction while being robust and scalable. [ABSTRACT FROM AUTHOR]

We consider an $\ell _{2}$ -regularized non-convex optimization problem for recovering signals from their noisy phaseless observations. We design and study the performance of a message passing algorithm that aims to solve this optimization problem. We consider the asymptotic setting $m,n \rightarrow \infty $ , $m/n \rightarrow \delta $ and obtain sharp performance bounds, where $m$ is the number of measurements and $n$ is the signal dimension. We show that for complex signals, the algorithm can perform accurate recovery with only $m = (({64}/{\pi ^{2}})-4)n \approx 2.5n$ measurements. Also, we provide a sharp analysis on the sensitivity of the algorithm to noise. We highlight the following facts about our message passing algorithm: 1) adding $\ell _{2}$ regularization to the non-convex loss function can be beneficial and 2) spectral initialization has a marginal impact on the performance of the algorithm. The sharp analyses, in this paper, not only enable us to compare the performance of our method with other phase recovery schemes but also shed light on designing better iterative algorithms for other non-convex optimization problems. [ABSTRACT FROM AUTHOR]

Contactors are typical electromagnetic devices whose optimization is multi-objective optimization problems. Contactors’ static and dynamic characteristics obtained by the finite-element method (FEM) are typical objectives. Due to the low efficiency of FEM, the intelligent optimization algorithms that require multiple computations cannot be applied well here. In this paper, a fast algorithm for characteristics of contactors based on radial basis function neural network approximate model is proposed. The orthogonal least squares, data pre-processing and subsection modeling methods are adopted here to make the model more accurate. Meanwhile, a hybrid multi-objective optimization algorithm which combines the differential evolution and the particle swarm optimization is also proposed to speed up convergence and obtain the limit optimal solutions. The feasibility of the above-mentioned methods has been verified by solving a multi-objective optimization problem of the contactor with solenoid structure. [ABSTRACT FROM AUTHOR]

Approximate Bayesian computation (ABC) is a method for Bayesian inference when the likelihood is unavailable but simulating from the model is possible. However, many ABC algorithms require a large number of simulations, which can be costly. To reduce the computational cost, Bayesian optimisation (BO) and surrogate models such as Gaussian processes have been proposed. Bayesian optimisation enables one to intelligently decide where to evaluate the model next but common BO strategies are not designed for the goal of estimating the posterior distribution. Our paper addresses this gap in the literature. We propose to compute the uncertainty in the ABC posterior density, which is due to a lack of simulations to estimate this quantity accurately, and define a loss function that measures this uncertainty. We then propose to select the next evaluation location to minimise the expected loss. Experiments show that the proposed method often produces the most accurate approximations as compared to common BO strategies. [ABSTRACT FROM AUTHOR]

In this paper, we devise a communication-efficient decentralized algorithm, named as communication-censored alternating direction method of multipliers (ADMM) (COCA), to solve a convex consensus optimization problem defined over a network. Similar to popular decentralized consensus optimization algorithms such as ADMM, at every iteration of COCA, a node exchanges its local variable with neighbors, and then updates its local variable according to the received neighboring variables and its local cost function. A different feature of COCA is that a node is not allowed to transmit its local variable to neighbors, if this variable is not sufficiently different to the previously transmitted one. The sufficiency of the difference is evaluated by a properly designed censoring function. Though this censoring strategy may slow down the optimization process, it effectively reduces the communication cost. We prove that when the censoring function is properly chosen, COCA converges to an optimal solution of the convex consensus optimization problem. Furthermore, if the local cost functions are strongly convex, COCA has a fast linear convergence rate. Numerical experiments demonstrate that, given a target solution accuracy, COCA is able to significantly reduce the overall communication cost compared to existing algorithms including ADMM, and hence fits for applications where network communication is a bottleneck. [ABSTRACT FROM AUTHOR]

Multi-access edge computing (MEC) has formed a major improvement in the existing mobile cloud computing paradigm, due to its ability in addressing the rising number of latency-sensitive services. However, bearing in mind the limited capacity that edge servers possess which offsets their benefits in the periods of high load, a hierarchical arrangement of the edge cloudlets has been studied and has shown to be successful in expanding their capabilities. Yet, considering the emerging business models in 5G networks, the cost disparity between the edge tiers has been until now ignored, leading to cost-inefficient solutions with respect to the network operators (NOs). In this paper, we consider an NO that is leasing resources of a high-tier central cloudlet for task offload, where we jointly minimize the NO’s computational cost and devices’ energy consumption in a multi-tier MEC system, by optimizing the offloading decision, the allocated transmission power and radio resources on the uplink channel, and the assigned servers’ computation, while respecting the devices’ latency requirement. We mathematically formulate our mixed-integer non-convex program and propose a Branch-and-Bound (BnB) algorithm for obtaining the optimal solution. Due to the BnB complexity, we propose a low-complexity algorithm based on the successive convex approximation method to solve and obtain a high-quality solution and also present an inflation-based algorithm for obtaining a polynomial-time and efficient solution. The numerical results show the performance and scalability of the algorithms, demonstrate their efficiency, and uncover insights for helping NOs to better manage their resources following various configurations. [ABSTRACT FROM AUTHOR]

Shared mesh protection (SMP) is a protection mechanism that recovers traffic delivery against failures on a working path as rapidly as linear protection. In addition, it allows for resource sharing among the protection paths with different endpoints in a mesh network by coordinating the use of shared resources when multiple protection paths compete for these resources. Owing to limited protection resources and the preemption in consequence of priority comparison, the SMP introduces a new optimization problem whose objective is to find the optimal priority assignment that maximizes the number of protected services against any failure case for a given SMP network. In this paper, we recognize that the problem of finding the optimal solution is NP-hard and propose min-degree and capacity-density approximation algorithms, whose approximation ratios are presented with the maximum degree of services and the maximum capacity of segments of an SMP network. Moreover, we propose a depth-based algorithm that always finds the optimal solution for a hypertree-shaped SMP network. We compare the performance of the proposed algorithms in terms of the protection ratio, which we define in this paper. [ABSTRACT FROM AUTHOR]

This paper focuses on developing a decentralized framework for coordinated minimum power beamforming wherein $L$ base stations (BSs), each equipped with $N$ antennas, serve $K$ single-antenna users with specific rate constraints. This is realized by considering user specific intercell interference (ICI) strength as the principal coupling parameter among BSs. First, explicit deterministic expressions for transmit powers are derived for spatially correlated channels in the asymptotic regime in which $N$ and $K$ grow large with a non-trivial ratio $K/N$. These asymptotic expressions are then used to compute approximations of the optimal ICI values that depend only on the channel statistics. By relying on the approximate ICI values as coordination parameters, a distributed non-iterative coordination algorithm, suitable for large networks with limited backhaul, is proposed. A heuristic algorithm is also proposed relaxing coordination requirements even further as it only needs pathloss values for non-local channels. The proposed algorithms satisfy the target rates for all users even when $N$ and $K$ are relatively small. Finally, the potential benefits of grouping users with similar statistics are investigated to further reduce the overhead and computational effort of the proposed solutions. Simulation results show that the proposed methods yield near-optimal performance. [ABSTRACT FROM AUTHOR]