IA Scholar Query: Improved Approximation Algorithms for Maximum Cut and Satisfiability Problems Using Semidefinite Programming.
https://scholar.archive.org/
Internet Archive Scholar query results feedeninfo@archive.orgTue, 27 Sep 2022 00:00:00 GMTfatcat-scholarhttps://scholar.archive.org/help1440Survey Descent: A Multipoint Generalization of Gradient Descent for Nonsmooth Optimization
https://scholar.archive.org/work/73uii2nbh5gypmqocm3omqz7re
For strongly convex objectives that are smooth, the classical theory of gradient descent ensures linear convergence relative to the number of gradient evaluations. An analogous nonsmooth theory is challenging. Even when the objective is smooth at every iterate, the corresponding local models are unstable and the number of cutting planes invoked by traditional remedies is difficult to bound, leading to convergences guarantees that are sublinear relative to the cumulative number of gradient evaluations. We instead propose a multipoint generalization of the gradient descent iteration for local optimization. While designed with general objectives in mind, we are motivated by a "max-of-smooth" model that captures the subdifferential dimension at optimality. We prove linear convergence when the objective is itself max-of-smooth, and experiments suggest a more general phenomenon.X.Y. Han, Adrian S. Lewiswork_73uii2nbh5gypmqocm3omqz7reTue, 27 Sep 2022 00:00:00 GMTCalculating the Moore–Penrose Generalized Inverse on Massively Parallel Systems
https://scholar.archive.org/work/xza5krlk2ngyhc3hwxuiikbs2q
In this work, we consider the problem of calculating the generalized Moore–Penrose inverse, which is essential in many applications of graph theory. We propose an algorithm for the massively parallel systems based on the recursive algorithm for the generalized Moore–Penrose inverse, the generalized Cholesky factorization, and Strassen's matrix inversion algorithm. Computational experiments with our new algorithm based on a parallel computing architecture known as the Compute Unified Device Architecture (CUDA) on a graphic processing unit (GPU) show the significant advantages of using GPU for large matrices (with millions of elements) in comparison with the CPU implementation from the OpenCV library (Intel, Santa Clara, CA, USA).Vukašin Stanojević, Lev Kazakovtsev, Predrag S. Stanimirović, Natalya Rezova, Guzel Shkaberinawork_xza5krlk2ngyhc3hwxuiikbs2qTue, 27 Sep 2022 00:00:00 GMTWarm-Started QAOA with Custom Mixers Provably Converges and Computationally Beats Goemans-Williamson's Max-Cut at Low Circuit Depths
https://scholar.archive.org/work/3zxknzjybfcuxatqk5xxwipzja
We generalize the Quantum Approximate Optimization Algorithm (QAOA) of Farhi et al. (2014) to allow for arbitrary separable initial states and corresponding mixers such that the starting state is the most excited state of the mixing Hamiltonian. We demonstrate this version of QAOA,which we call QAOA-warmest, by simulating Max-Cut on weighted graphs. We initialize the starting state as a warm-start using rank-2 and rank-3 approximations obtained using randomized projections of solutions to Max-Cut's semi-definite program, and define a warm-start dependent custom mixer. We show that these warm-starts initialize the QAOA circuit with constant-factor approximations of 0.658 for rank-2 and 0.585 for rank-3 warm-starts for graphs with non-negative edge weights, improving upon previously known trivial (i.e., 0.5 for standard initialization) worst-case bounds. We further show that QAOA-warmest with any separable initial state converges to Max-Cut under the adiabatic limit as p→∞. Our numerical simulations with this generalization yield higher quality cuts (compared to standard QAOA, the classical Goemans-Williamson algorithm, and a warm-started QAOA without custom mixers) for an instance library of 1148 graphs (upto 11 nodes) and depth p=8. We further show that QAOA-warmest outperforms the standard QAOA of Farhi et al. in experiments on current IBM-Q and Quantinuum hardware.Reuben Tate and Jai Moondra and Bryan Gard and Greg Mohler and Swati Guptawork_3zxknzjybfcuxatqk5xxwipzjaTue, 27 Sep 2022 00:00:00 GMTGlobal Optimization for Cardinality-constrained Minimum Sum-of-Squares Clustering via Semidefinite Programming
https://scholar.archive.org/work/4ndqu6umsjehxbkevjw4ylddoi
The minimum sum-of-squares clustering (MSSC), or k-means type clustering, has been recently extended to exploit prior knowledge on the cardinality of each cluster. Such knowledge is used to increase performance as well as solution quality. In this paper, we propose an exact approach based on the branch-and-cut technique to solve the cardinality-constrained MSSC. For the lower bound routine, we use the semidefinite programming (SDP) relaxation recently proposed by Rujeerapaiboon et al. [SIAM J. Optim. 29(2), 1211-1239, (2019)]. However, this relaxation can be used in a branch-and-cut method only for small-size instances. Therefore, we derive a new SDP relaxation that scales better with the instance size and the number of clusters. In both cases, we strengthen the bound by adding polyhedral cuts. Benefiting from a tailored branching strategy which enforces pairwise constraints, we reduce the complexity of the problems arising in the children nodes. For the upper bound, instead, we present a local search procedure that exploits the solution of the SDP relaxation solved at each node. Computational results show that the proposed algorithm globally solves, for the first time, real-world instances of size 10 times larger than those solved by state-of-the-art exact methods.Veronica Piccialli, Antonio M. Sudosowork_4ndqu6umsjehxbkevjw4ylddoiSun, 25 Sep 2022 00:00:00 GMTLearning-Augmented Algorithms for Online Linear and Semidefinite Programming
https://scholar.archive.org/work/spz7ak4xpfgp3cgt53ingvgdj4
Semidefinite programming (SDP) is a unifying framework that generalizes both linear programming and quadratically-constrained quadratic programming, while also yielding efficient solvers, both in theory and in practice. However, there exist known impossibility results for approximating the optimal solution when constraints for covering SDPs arrive in an online fashion. In this paper, we study online covering linear and semidefinite programs in which the algorithm is augmented with advice from a possibly erroneous predictor. We show that if the predictor is accurate, we can efficiently bypass these impossibility results and achieve a constant-factor approximation to the optimal solution, i.e., consistency. On the other hand, if the predictor is inaccurate, under some technical conditions, we achieve results that match both the classical optimal upper bounds and the tight lower bounds up to constant factors, i.e., robustness. More broadly, we introduce a framework that extends both (1) the online set cover problem augmented with machine-learning predictors, studied by Bamas, Maggiori, and Svensson (NeurIPS 2020), and (2) the online covering SDP problem, initiated by Elad, Kale, and Naor (ICALP 2016). Specifically, we obtain general online learning-augmented algorithms for covering linear programs with fractional advice and constraints, and initiate the study of learning-augmented algorithms for covering SDP problems. Our techniques are based on the primal-dual framework of Buchbinder and Naor (Mathematics of Operations Research, 34, 2009) and can be further adjusted to handle constraints where the variables lie in a bounded region, i.e., box constraints.Elena Grigorescu, Young-San Lin, Sandeep Silwal, Maoyuan Song, Samson Zhouwork_spz7ak4xpfgp3cgt53ingvgdj4Wed, 21 Sep 2022 00:00:00 GMTConvex integer optimization with Frank-Wolfe methods
https://scholar.archive.org/work/r6o27gjwavbh3ixtlmppj2atxy
Mixed-integer nonlinear optimization is a broad class of problems that feature combinatorial structures and nonlinearities. Typical exact methods combine a branch-and-bound scheme with relaxation and separation subroutines. We investigate the properties and advantages of error-adaptive first-order methods based on the Frank-Wolfe algorithm for this setting, requiring only a gradient oracle for the objective function and linear optimization over the feasible set. In particular, we will study the algorithmic consequences of optimizing with a branch-and-bound approach where the subproblem is solved over the convex hull of the mixed-integer feasible set thanks to linear oracle calls, compared to solving the subproblems over the continuous relaxation of the same set. This novel approach computes feasible solutions while working on a single representation of the polyhedral constraints, leveraging the full extent of Mixed-Integer Programming (MIP) solvers without an outer approximation scheme.Deborah Hendrych and Hannah Troppens and Mathieu Besançon and Sebastian Pokuttawork_r6o27gjwavbh3ixtlmppj2atxyWed, 21 Sep 2022 00:00:00 GMTThe Quantum and Classical Streaming Complexity of Quantum and Classical Max-Cut
https://scholar.archive.org/work/bovqx5yajzblbgglr3pfsneamy
We investigate the space complexity of two graph streaming problems: Max-Cut and its quantum analogue, Quantum Max-Cut. Previous work by Kapralov and Krachun [STOC '19] resolved the classical complexity of the classical problem, showing that any (2 - ε)-approximation requires Ω(n) space (a 2-approximation is trivial with O(log n) space). We generalize both of these qualifiers, demonstrating Ω(n) space lower bounds for (2 - ε)-approximating Max-Cut and Quantum Max-Cut, even if the algorithm is allowed to maintain a quantum state. As the trivial approximation algorithm for Quantum Max-Cut only gives a 4-approximation, we show tightness with an algorithm that returns a (2 + ε)-approximation to the Quantum Max-Cut value of a graph in O(log n) space. Our work resolves the quantum and classical approximability of quantum and classical Max-Cut using o(n) space. We prove our lower bounds through the techniques of Boolean Fourier analysis. We give the first application of these methods to sequential one-way quantum communication, in which each player receives a quantum message from the previous player, and can then perform arbitrary quantum operations on it before sending it to the next. To this end, we show how Fourier-analytic techniques may be used to understand the application of a quantum channel.John Kallaugher, Ojas Parekhwork_bovqx5yajzblbgglr3pfsneamyTue, 20 Sep 2022 00:00:00 GMTStrong Parallel Repetition for Unique Games on Small Set Expanders
https://scholar.archive.org/work/3unjou3bjvhlblbcx4rdf75mvu
Strong Parallel Repetition for Unique Games on Small Set Expanders The strong parallel repetition problem for unique games is to efficiently reduce the 1-delta vs. 1-C*delta gap problem of Boolean unique games (where C>1 is a sufficiently large constant) to the 1-epsilon vs. epsilon gap problem of unique games over large alphabet. Due to its importance to the Unique Games Conjecture, this problem garnered a great deal of interest from the research community. There are positive results for certain easy unique games (e.g., unique games on expanders), and an impossibility result for hard unique games. In this paper we show how to bypass the impossibility result by enlarging the alphabet sufficiently before repetition. We consider the case of unique games on small set expanders for two setups: (i) Strong small set expanders that yield easy unique games. (ii) Weaker small set expanders underlying possibly hard unique games as long as the game is mildly fortified. We show how to fortify unique games in both cases, i.e., how to transform the game so sufficiently large induced sub-games have bounded value. We then prove strong parallel repetition for the fortified games. Prior to this work fortification was known for projection games but seemed hopeless for unique games.Dana Moshkovitzwork_3unjou3bjvhlblbcx4rdf75mvuTue, 20 Sep 2022 00:00:00 GMTPartitioning through projections: strong SDP bounds for large graph partition problems
https://scholar.archive.org/work/lhqip24omfelpgpipj5w22a5ca
The graph partition problem (GPP) aims at clustering the vertex set of a graph into a fixed number of disjoint subsets of given sizes such that the sum of weights of edges joining different sets is minimized. This paper investigates the quality of doubly nonnegative (DNN) relaxations, i.e., relaxations having matrix variables that are both positive semidefinite and nonnegative, strengthened by additional polyhedral cuts for two variations of the GPP: the k-equipartition and the graph bisection problem. After reducing the size of the relaxations by facial reduction, we solve them by a cutting-plane algorithm that combines an augmented Lagrangian method with Dykstra's projection algorithm. Since many components of our algorithm are general, the algorithm is suitable for solving various DNN relaxations with a large number of cutting planes. We are the first to show the power of DNN relaxations with additional cutting planes for the GPP on large benchmark instances up to 1,024 vertices. Computational results show impressive improvements in strengthened DNN bounds.Frank de Meijer, Renata Sotirov, Angelika Wiegele, Shudian Zhaowork_lhqip24omfelpgpipj5w22a5caMon, 19 Sep 2022 00:00:00 GMTDrone-Delivery Network for Opioid Overdose – Nonlinear Integer Queueing-Optimization Models and Methods
https://scholar.archive.org/work/fmmryesnkbbotivyhxeqns5s6e
We propose a new stochastic emergency network design model that uses a fleet of drones to quickly deliver naxolone in response to opioid overdoses. The network is represented as a collection of M/G/K queuing systems in which the capacity K of each system is a decision variable and the service time is modelled as a decision-dependent random variable. The model is an optimization-based queuing problem which locates fixed (drone bases) and mobile (drones) servers and determines the drone dispatching decisions, and takes the form of a nonlinear integer problem, which is intractable in its original form. We develop an efficient reformulation and algorithmic framework. Our approach reformulates the multiple nonlinearities (fractional, polynomial, exponential, factorial terms) to give a mixed-integer linear programming (MILP) formulation. We demonstrate its generalizablity and show that the problem of minimizing the average response time of a network of M/G/K queuing systems with unknown capacity K is always MILP-representable. We design two algorithms and demonstrate that the outer approximation branch-and-cut method is the most efficient and scales well. The analysis based on real-life overdose data reveals that drones can in Virginia Beach: 1) decrease the response time by 78%, 2) increase the survival chance by 432%, 3) save up to 34 additional lives per year, and 4) provide annually up to 287 additional quality-adjusted life years.Miguel Lejeune, Wenbo Mawork_fmmryesnkbbotivyhxeqns5s6eMon, 19 Sep 2022 00:00:00 GMTCheeger Inequalities for Vertex Expansion and Reweighted Eigenvalues
https://scholar.archive.org/work/hy36onlp7zaipmieqgjrb4wosq
The classical Cheeger's inequality relates the edge conductance ϕ of a graph and the second smallest eigenvalue λ_2 of the Laplacian matrix. Recently, Olesker-Taylor and Zanetti discovered a Cheeger-type inequality ψ^2 / log |V| ≲λ_2^* ≲ψ connecting the vertex expansion ψ of a graph G=(V,E) and the maximum reweighted second smallest eigenvalue λ_2^* of the Laplacian matrix. In this work, we first improve their result to ψ^2 / log d ≲λ_2^* ≲ψ where d is the maximum degree in G, which is optimal assuming the small-set expansion conjecture. Also, the improved result holds for weighted vertex expansion, answering an open question by Olesker-Taylor and Zanetti. Building on this connection, we then develop a new spectral theory for vertex expansion. We discover that several interesting generalizations of Cheeger inequalities relating edge conductances and eigenvalues have a close analog in relating vertex expansions and reweighted eigenvalues. These include an analog of Trevisan's result on bipartiteness, an analog of higher order Cheeger's inequality, and an analog of improved Cheeger's inequality. Finally, inspired by this connection, we present negative evidence to the 0/1-polytope edge expansion conjecture by Mihail and Vazirani. We construct 0/1-polytopes whose graphs have very poor vertex expansion. This implies that the fastest mixing time to the uniform distribution on the vertices of these 0/1-polytopes is almost linear in the graph size. This does not provide a counterexample to the conjecture, but this is in contrast with known positive results which proved poly-logarithmic mixing time to the uniform distribution on the vertices of subclasses of 0/1-polytopes.Tsz Chiu Kwok, Lap Chi Lau, Kam Chuen Tungwork_hy36onlp7zaipmieqgjrb4wosqMon, 19 Sep 2022 00:00:00 GMTQuadratic Regularization of Unit-Demand Envy-Free Pricing Problems and Application to Electricity Markets
https://scholar.archive.org/work/5sg5regmzrh2fcuvsqhorc33wq
We consider a profit-maximizing model for pricing contracts as an extension of the unit-demand envy-free pricing problem: customers aim to choose a contract maximizing their utility based on a reservation bill and multiple price coefficients (attributes). A classical approach supposes that the customers have deterministic utilities; then, the response of each customer is highly sensitive to price since it concentrates on the best offer. A second approach is to consider logit model to add a probabilistic behavior in the customers' choices. To circumvent the intrinsic instability of the former and the resolution difficulties of the latter, we introduce a quadratically regularized model of customer's response, which leads to a quadratic program under complementarity constraints (QPCC). This allows to robustify the deterministic model, while keeping a strong geometrical structure. In particular, we show that the customer's response is governed by a polyhedral complex, in which every polyhedral cell determines a set of contracts which is effectively chosen. Moreover, the deterministic model is recovered as a limit case of the regularized one. We exploit these geometrical properties to develop an efficient pivoting heuristic, which we compare with implicit or non-linear methods from bilevel programming. These results are illustrated by an application to the optimal pricing of electricity contracts on the French market.Quentin Jacquet, Wim van Ackooij, Clémence Alasseur, Stéphane Gaubertwork_5sg5regmzrh2fcuvsqhorc33wqMon, 19 Sep 2022 00:00:00 GMTThe Use of Computational Geometry Techniques to Resolve the Issues of Coverage and Connectivity in Wireless Sensor Networks
https://scholar.archive.org/work/am5bqiughfa5hkcowa2dpv6txm
Wireless Sensor Networks (WSNs) enhance the ability to sense and control the physical environment in various applications. The functionality of WSNs depends on various aspects like the localization of nodes, the strategies of node deployment, and a lifetime of nodes and routing techniques, etc. Coverage is an essential part of WSNs wherein the targeted area is covered by at least one node. Computational Geometry (CG) -based techniques significantly improve the coverage and connectivity of WSNs. This paper is a step towards employing some of the popular techniques in WSNs in a productive manner. Furthermore, this paper attempts to survey the existing research conducted using Computational Geometry-based methods in WSNs. In order to address coverage and connectivity issues in WSNs, the use of the Voronoi Diagram, Delaunay Triangulation, Voronoi Tessellation, and the Convex Hull have played a prominent role. Finally, the paper concludes by discussing various research challenges and proposed solutions using Computational Geometry-based techniques.Sharmila Devi, Anju Sangwan, Anupma Sangwan, Mazin Abed Mohammed, Krishna Kumar, Jan Nedoma, Radek Martinek, Petr Zmijwork_am5bqiughfa5hkcowa2dpv6txmFri, 16 Sep 2022 00:00:00 GMTCompF2: Theoretical Calculations and Simulation Topical Group Report
https://scholar.archive.org/work/gu3mrawvqva5nld52fu7e7ni2a
This report summarizes the work of the Computational Frontier topical group on theoretical calculations and simulation for Snowmass 2021. We discuss the challenges, potential solutions, and needs facing six diverse but related topical areas that span the subject of theoretical calculations and simulation in high energy physics (HEP): cosmic calculations, particle accelerator modeling, detector simulation, event generators, perturbative calculations, and lattice QCD (quantum chromodynamics). The challenges arise from the next generations of HEP experiments, which will include more complex instruments, provide larger data volumes, and perform more precise measurements. Calculations and simulations will need to keep up with these increased requirements. The other aspect of the challenge is the evolution of computing landscape away from general-purpose computing on CPUs and toward special-purpose accelerators and coprocessors such as GPUs and FPGAs. These newer devices can provide substantial improvements for certain categories of algorithms, at the expense of more specialized programming and memory and data access patterns.Peter Boyle, Kevin Pedro, Ji Qiangwork_gu3mrawvqva5nld52fu7e7ni2aFri, 16 Sep 2022 00:00:00 GMTCombinatorial geometry of neural codes, neural data analysis, and neural networks
https://scholar.archive.org/work/4hudx3fjozfltjck6hl5tww4pm
This dissertation explores applications of discrete geometry in mathematical neuroscience. We begin with convex neural codes, which model the activity of hippocampal place cells and other neurons with convex receptive fields. In Chapter 4, we introduce order-forcing, a tool for constraining convex realizations of codes, and use it to construct new examples of non-convex codes with no local obstructions. In Chapter 5, we relate oriented matroids to convex neural codes, showing that a code has a realization with convex polytopes iff it is the image of a representable oriented matroid under a neural code morphism. We also show that determining whether a code is convex is at least as difficult as determining whether an oriented matroid is representable, implying that the problem of determining whether a code is convex is NP-hard. Next, we turn to the problem of the underlying rank of a matrix. This problem is motivated by the problem of determining the dimensionality of (neural) data which has been corrupted by an unknown monotone transformation. In Chapter 6, we introduce two tools for computing underlying rank, the minimal nodes and the Radon rank. We apply these to analyze calcium imaging data from a larval zebrafish. In Chapter 7, we explore the underlying rank in more detail, establish connections to oriented matroid theory, and show that computing underlying rank is also NP-hard. Finally, we study the dynamics of threshold-linear networks (TLNs), a simple model of the activity of neural circuits. In Chapter 9, we describe the nullcline arrangement of a threshold linear network, and show that a subset of its chambers are an attracting set. In Chapter 10, we focus on combinatorial threshold linear networks (CTLNs), which are TLNs defined from a directed graph. We prove that if the graph of a CTLN is a directed acyclic graph, then all trajectories of the CTLN approach a fixed point.Caitlin Lienkaemperwork_4hudx3fjozfltjck6hl5tww4pmThu, 15 Sep 2022 00:00:00 GMTSome Results on Approximability of Minimum Sum Vertex Cover
https://scholar.archive.org/work/gzjl7gtzrvbrldczjdrjiexxmm
We study the Minimum Sum Vertex Cover problem, which asks for an ordering of vertices in a graph that minimizes the total cover time of edges. In particular, n vertices of the graph are visited according to an ordering, and for each edge this induces the first time it is covered. The goal of the problem is to find the ordering which minimizes the sum of the cover times over all edges in the graph. In this work we give the first explicit hardness of approximation result for Minimum Sum Vertex Cover. In particular, assuming the Unique Games Conjecture, we show that the Minimum Sum Vertex Cover problem cannot be approximated within 1.014. The best approximation ratio for Minimum Sum Vertex Cover as of now is 16/9, due to a recent work of Bansal, Batra, Farhadi, and Tetali. We also revisit an approximation algorithm for regular graphs outlined in the work of Feige, Lovász, and Tetali, and show that Minimum Sum Vertex Cover can be approximated within 1.225 on regular graphs.Aleksa Stanković, Amit Chakrabarti, Chaitanya Swamywork_gzjl7gtzrvbrldczjdrjiexxmmThu, 15 Sep 2022 00:00:00 GMTWasserstein K-means for clustering probability distributions
https://scholar.archive.org/work/ix4wgbckobhsflcrolztzyo2pm
Clustering is an important exploratory data analysis technique to group objects based on their similarity. The widely used K-means clustering method relies on some notion of distance to partition data into a fewer number of groups. In the Euclidean space, centroid-based and distance-based formulations of the K-means are equivalent. In modern machine learning applications, data often arise as probability distributions and a natural generalization to handle measure-valued data is to use the optimal transport metric. Due to non-negative Alexandrov curvature of the Wasserstein space, barycenters suffer from regularity and non-robustness issues. The peculiar behaviors of Wasserstein barycenters may make the centroid-based formulation fail to represent the within-cluster data points, while the more direct distance-based K-means approach and its semidefinite program (SDP) relaxation are capable of recovering the true cluster labels. In the special case of clustering Gaussian distributions, we show that the SDP relaxed Wasserstein K-means can achieve exact recovery given the clusters are well-separated under the 2-Wasserstein metric. Our simulation and real data examples also demonstrate that distance-based K-means can achieve better classification performance over the standard centroid-based K-means for clustering probability distributions and images.Yubo Zhuang, Xiaohui Chen, Yun Yangwork_ix4wgbckobhsflcrolztzyo2pmWed, 14 Sep 2022 00:00:00 GMTRiemannian Langevin Algorithm for Solving Semidefinite Programs
https://scholar.archive.org/work/sycwajipazbpvbvc5kaifjtatq
We propose a Langevin diffusion-based algorithm for non-convex optimization and sampling on a product manifold of spheres. Under a logarithmic Sobolev inequality, we establish a guarantee for finite iteration convergence to the Gibbs distribution in terms of Kullback–Leibler divergence. We show that with an appropriate temperature choice, the suboptimality gap to the global minimum is guaranteed to be arbitrarily small with high probability. As an application, we consider the Burer–Monteiro approach for solving a semidefinite program (SDP) with diagonal constraints, and analyze the proposed Langevin algorithm for optimizing the non-convex objective. In particular, we establish a logarithmic Sobolev inequality for the Burer–Monteiro problem when there are no spurious local minima, but under the presence saddle points. Combining the results, we then provide a global optimality guarantee for the SDP and the Max-Cut problem. More precisely, we show that the Langevin algorithm achieves ϵ accuracy with high probability in Ω( ϵ^-5 ) iterations.Mufan Bill Li, Murat A. Erdogduwork_sycwajipazbpvbvc5kaifjtatqWed, 14 Sep 2022 00:00:00 GMTFast quantum subroutines for the simplex method
https://scholar.archive.org/work/n5imgfb6nrfwtmiur2rii75vre
We propose quantum subroutines for the simplex method that avoid classical computation of the basis inverse. We show how to quantize all steps of the simplex algorithm, including checking optimality, unboundedness, and identifying a pivot (i.e., pricing the columns and performing the ratio test) according to Dantzig's rule or the steepest edge rule. The quantized subroutines obtain a polynomial speedup in the dimension of the problem, but have worse dependence on other numerical parameters. For example, for a problem with m constraints, n variables, at most d_c nonzero elements per column of the costraint matrix, at most d nonzero elements per column or row of the basis, basis condition number κ, and optimality tolerance ϵ, pricing can be performed in Õ(1/ϵκ d √(n)(d_c n + d m)) time, where the Õ notation hides polylogarithmic factors; classically, pricing requires O(d_c^0.7 m^1.9 + m^2 + o(1) + d_c n) time in the worst case using the fastest known algorithm for sparse matrix multiplication. For well-conditioned sparse problems the quantum subroutines scale better in m and n, and may therefore have an advantage for very large problems. The running time of the quantum subroutines can be improved if the constraint matrix admits an efficient algorithmic description, or if quantum RAM is available.Giacomo Nanniciniwork_n5imgfb6nrfwtmiur2rii75vreMon, 12 Sep 2022 00:00:00 GMTGlobally Optimal Spectrum- and Energy-Efficient Beamforming for Rate Splitting Multiple Access
https://scholar.archive.org/work/r5rw5i5z4feczmgmcpkn62pnly
Rate splitting multiple access (RSMA) is a promising non-orthogonal transmission strategy for next-generation wireless networks. It has been shown to outperform existing multiple access schemes in terms of spectral and energy efficiency when suboptimal beamforming schemes are employed. In this work, we fill the gap between suboptimal and truly optimal beamforming schemes and conclusively establish the superior spectral and energy efficiency of RSMA. To this end, we propose a successive incumbent transcending (SIT) branch and bound (BB) algorithm to find globally optimal beamforming solutions that maximize the weighted sum rate or energy efficiency of RSMA in Gaussian multiple-input single-output (MISO) broadcast channels. Numerical results show that RSMA exhibits an explicit globally optimal spectral and energy efficiency gain over conventional multi-user linear precoding (MU-LP) and power-domain non-orthogonal multiple access (NOMA). Compared to existing globally optimal beamforming algorithms for MU-LP, the proposed SIT BB not only improves the numerical stability but also achieves faster convergence. Moreover, for the first time, we show that the spectral/energy efficiency of RSMA achieved by suboptimal beamforming schemes (including weighted minimum mean squared error (WMMSE) and successive convex approximation) almost coincides with the corresponding globally optimal performance, making it a valid choice for performance comparisons. The globally optimal results provided in this work are imperative to the ongoing research on RSMA as they serve as benchmarks for existing suboptimal beamforming strategies and those to be developed in multi-antenna broadcast channels.Bho Matthiesen, Yijie Mao, Armin Dekorsy, Petar Popovski, Bruno Clerckxwork_r5rw5i5z4feczmgmcpkn62pnlyMon, 12 Sep 2022 00:00:00 GMT