Discrete-continuous Optimization Research Articles

Regularized Zero-Forcing Aided Hybrid Beamforming for Millimeter-Wave Multiuser MIMO Systems

This paper considers hybrid beamforming consisting of analog beamforming (ABF) coupled with digital baseband beamforming (DBF) which is designed for multi-user (MU) multiple input multiple output (MIMO) millimeter-wave (mmWave) communications. ABF uses a limited number of radio frequency (RF) chains and finite-resolution phase-shifters to alleviate the power consumption at the base station (BS), while DBF uses either zero-forcing beamforming (ZFB) or regularized zero forcing beamforming (RZFB) to restrain MU interference. The joint design of ABF and DBF constitutes a computationally challenging mixed discrete continuous optimization problem. The paper develops efficient algorithms for its solution, which iterate scalable-complex expressions. Furthermore, we conceive a new class of MU RZFB for attaining higher rates. Simulations are provided to demonstrate the viability of the proposed algorithms and the advantages of the conceived RZFB.

Data-driven computational framework for snap-through problems

The distance-minimizing data-driven computing is an emerging field of computational mechanics, which reformulates the classical boundary-value problem as a discrete-continuous optimization problem, seeking a minimum distance between the discrete material data space and the physical constraint space. However, the optimization problem becomes non-convex due to the limit equilibrium instability in snap-through phenomenon, which makes it challenging to obtain a desired convergent solution, especially near critical points. Towards this end, we propose a data-driven computational framework by virtue of the structural stability theory for analyzing snap-through problems. First, an auxiliary linearized perturbed system is established at each equilibrium state (i.e., convergent data-driven solution), which can be solved with the stiffness-based method and data-based method. Then, a stability indicator is employed to detect the critical point and the corresponding buckling mode, which is utilized to construct a proper start point to trace the equilibrium path in the neighborhood of critical points. Several numerical examples are performed to evaluate the reliability of the proposed framework. It is proved that even for the complex problem with nonlinear constitutive behavior, the proposed framework can correctly trace the entire equilibrium path of snap-through behavior.

Low-Resolution RIS-Aided Multiuser MIMO Signaling

A multi-antenna aided base station (BS) supporting several multi-antenna downlink users with the aid of a reconfigurable intelligent surface (RIS) of programmable reflecting elements (PREs) is considered. Low-resolution PREs constrained by a set of sparse discrete values are used for reasons of cost-efficiency. Our challenging objective is to jointly design the beamformers at the BS and the RIS’s PREs for improving the throughput of all users by maximizing their geometric-mean, under a variety of different access schemes. This constitutes a computationally challenging problem of mixed continuous-discrete optimization, because each user’s throughput is a complicated function of both the continuous-valued beamformer weights and of the discrete-valued PREs. We develop low-complexity algorithms, which iterate by directly evaluating low-complexity closed-form expressions. Our simulation results show the advantages of non-orthogonal multiple access-aided signaling, which allows the users to decode a part of the multi-user interference for enhancing their throughput.

Searching for Fast Demosaicking Algorithms

We present a method to automatically synthesize efficient, high-quality demosaicking algorithms, across a range of computational budgets, given a loss function and training data. It performs a multi-objective, discrete-continuous optimization which simultaneously solves for the program structure and parameters that best tradeoff computational cost and image quality. We design the method to exploit domain-specific structure for search efficiency. We apply it to several tasks, including demosaicking both Bayer and Fuji X-Trans color filter patterns, as well as joint demosaicking and super-resolution. In a few days on 8 GPUs, it produces a family of algorithms that significantly improves image quality relative to the prior state-of-the-art across a range of computational budgets from 10 s to 1000 s of operations per pixel (1 dB–3 dB higher quality at the same cost, or 8.5–200× higher throughput at same or better quality). The resulting programs combine features of both classical and deep learning-based demosaicking algorithms into more efficient hybrid combinations, which are bandwidth-efficient and vectorizable by construction. Finally, our method automatically schedules and compiles all generated programs into optimized SIMD code for modern processors.

Method for assessing the characteristics of the quality of service of networks

Variational quantum algorithms are at the forefront of modeling fault-tolerant quantum devices of the near future. While most variational quantum algorithms use only continuous optimization variables, their representative power can sometimes be greatly increased by adding certain discrete optimization variables, as illustrated by the example of the generalized quantum approximation optimization algorithm. However, the hybrid discrete-continuous optimization problem poses an optimization problem. The paper proposes a new algorithm that combines the Monte Carlo tree search method with an improved gradient solver to optimize discrete and continuous variables in a quantum circuit, respectively. The algorithm has been found to have excellent noise tolerance properties and is superior to prior algorithms in the field.

Computational Object-Wrapping Rope Nets

Wrapping objects using ropes is a common practice in our daily life. However, it is difficult to design and tie ropes on a 3D object with complex topology and geometry features while ensuring wrapping security and easy operation. In this article, we propose to compute a rope net that can tightly wrap around various 3D shapes. Our computed rope net not only immobilizes the object but also maintains the load balance during lifting. Based on the key observation that if every knot of the net has four adjacent curve edges, then only a single rope is needed to construct the entire net. We reformulate the rope net computation problem into a constrained curve network optimization. We propose a discrete-continuous optimization approach, where the topological constraints are satisfied in the discrete phase and the geometrical goals are achieved in the continuous stage. We also develop a hoist planning to pick anchor points so that the rope net equally distributes the load during hoisting. Furthermore, we simulate the wrapping process and use it to guide the physical rope net construction process. We demonstrate the effectiveness of our method on 3D objects with varying geometric and topological complexity. In addition, we conduct physical experiments to demonstrate the practicability of our method.

StrokeStrip

When creating freeform drawings, artists routinely employ clusters of overdrawn strokes to convey intended, aggregate curves. The ability to algorithmically fit these intended curves to their corresponding clusters is central to many applications that use artist drawings as inputs. However, while human observers effortlessly envision the intended curves given stroke clusters as input, existing fitting algorithms lack robustness and frequently fail when presented with input stroke clusters with non-trivial geometry or topology. We present StrokeStrip , a new and robust method for fitting intended curves to vector-format stroke clusters. Our method generates fitting outputs consistent with viewer expectations across a vast range of input stroke cluster configurations. We observe that viewers perceive stroke clusters as continuous, varying-width strips whose paths are described by the intended curves. An arc length parameterization of these strips defines a natural mapping from a strip to its path. We recast the curve fitting problem as one of parameterizing the cluster strokes using a joint 1D parameterization that is the restriction of the natural arc length parameterization of this strip to the strokes in the cluster. We simultaneously compute the joint cluster parameterization and implicitly reconstruct the a priori unknown strip geometry by solving a variational problem using a discrete-continuous optimization framework. We use this parameterization to compute parametric aggregate curves whose shape reflects the geometric properties of the cluster strokes at the corresponding isovalues. We demonstrate StrokeStrip outputs to be significantly better aligned with observer preferences compared to those of prior art; in a perceptual study, viewers preferred our fitting outputs by a factor of 12:1 compared to alternatives. We further validate our algorithmic choices via a range of ablation studies; extend our framework to raster data; and illustrate applications that benefit from the parameterizations produced.

A novel discrete–continuous material orientation optimization model for stiffness-based concurrent design of fiber composite

The cooperation of tailoring fiber angles and designing its macro layout is an excellent approach to improve the performance of composite structures and achieve lighter weight. This article proposes a novel discrete–continuous optimization model for the concurrent optimization of the macro structural topology and local fiber angle for the fiber-reinforced composite structure. Therein, the novel modeling consists of a two-step strategy, where the normal distribution function is adopted to form the discrete material interpolation scheme firstly and then the continuous angle design on fine angle subintervals is conducted. Because it requires the fixed discrete design variables regardless of the number of angle subintervals and simultaneously achieves the continuous fiber angle design, it can save computational cost and provide an infinite design domain. Furthermore, the subdividing angle interval reduces the risks of falling into the local optimum solution for continuous angle design. Besides, the fiber discontinuity may cause stress concentration and manufacturing problems in the intersection of discontinuous fiber paths. Therefore, a linear density technique is used to achieve fiber continuity. Finally, a concurrent topology optimization model is built based on the proposed novel model. And several numerical examples are well-organized and presented to illustrate the validity of the proposed model.

Power-Efficient Wireless Streaming of Multi-Quality Tiled 360 VR Video in MIMO-OFDMA Systems

In this paper, we study the optimal wireless streaming of a multi-quality tiled 360 virtual reality (VR) video from a multi-antenna server to multiple single-antenna users in a multiple-input multiple-output (MIMO)-orthogonal frequency division multiple access (OFDMA) system. In the scenario without user transcoding, we jointly optimize beamforming and subcarrier, transmission power, and rate allocation to minimize the total transmission power. This problem is a challenging mixed discrete-continuous optimization problem. We obtain a globally optimal solution for small multicast groups, an asymptotically optimal solution for a large antenna array, and a suboptimal solution for the general case. In the scenario with user transcoding, we jointly optimize the quality level selection, beamforming, and subcarrier, transmission power, and rate allocation to minimize the weighted sum of the average total transmission power and the transcoding power. This problem is a two-timescale mixed discrete-continuous optimization problem, which is even more challenging than the problem for the scenario without user transcoding. We obtain a globally optimal solution for small multicast groups, an asymptotically optimal solution for a large antenna array, and a low-complexity suboptimal solution for the general case. Finally, numerical results demonstrate the significant gains of proposed solutions over the existing solutions.

Multi-view 2D–3D alignment with hybrid bundle adjustment for visual metrology

High-precision measurement based on multi-view geometry benefits from aligning a template CAD model to the multi-view observations of a target object. It not only improves the multi-camera calibration but also assists the measurement by serving as a “scaffold.” A straightforward approach is to reconstruct the target object and perform a 3D registration with the CAD model. However, the accuracy of such 3D alignment cannot meet the high-precision requirement. We formulate the problem as a bundle adjustment where we jointly optimize the 6DoF poses of both the template model and the multiple cameras. To accommodate the manufacturing error of products, we propose a simple and robust solution based on a discrete–continuous optimization which interleaves between correspondence selection and pose optimization. In the discrete step, a robust RANSAC-based selection process selects well-matched 2D–3D feature points according to the current model/camera poses. In the continuous step, it jointly optimizes the 6D poses of the CAD model and the multiple cameras. This interleaving optimization constitutes a novel hybrid bundle adjustment (HBA). In HBA, the reprojection error of a feature point is measured either with the 2D–3D correspondence between the observation image and the CAD model or with the cross-view correspondence between multiple images, whichever is more reliable according to the discrete selection step. Through extensive evaluation on real-world data, we demonstrate that HBA achieves high-precision multi-camera calibration, outperforming alternative approaches significantly.

A Discrete-Continuous Algorithm for Free Flight Planning

We propose a hybrid discrete-continuous algorithm for flight planning in free flight airspaces. In a first step, our discrete-continuous optimization for enhanced resolution (DisCOptER) method computes a globally optimal approximate flight path on a discretization of the problem using the A* method. This route initializes a Newton method that converges rapidly to the smooth optimum in a second step. The correctness, accuracy, and complexity of the method are governed by the choice of the crossover point that determines the coarseness of the discretization. We analyze the optimal choice of the crossover point and demonstrate the asymtotic superority of DisCOptER over a purely discrete approach.

Autonomous Outdoor Scanning via Online Topological and Geometric Path Optimization

Autonomous 3D acquisition of outdoor environments poses special challenges. Different from indoor scenes, where the room space is delineated by clear boundaries and separations (e.g., walls and furniture), an outdoor environment is spacious and unbounded (thinking of a campus). Therefore, unlike for indoor scenes where the scanning effort is mainly devoted to the discovery of boundary surfaces, scanning an open and unbounded area requires actively delimiting the extent of scanning region and dynamically planning a traverse path within that region. Thus, for outdoor scenes, we formulate the planning of an energy-efficient autonomous scanning through a discrete-continuous optimization of robot scanning paths. The discrete optimization computes a topological map, through solving an online traveling sales problem (Online TSP), which determines the scanning goals and paths on-the-fly. The dynamic goals are determined as a collection of visit sites with high reward of visibility-to-unknown. A visit graph is constructed via connecting the visit sites with edges weighted by traversing cost. This topological map evolves as the robot scans via deleting outdated sites that are either visited or become rewardless and inserting newly discovered ones. The continuous part optimizes the traverse paths geometrically between two neighboring visit sites via maximizing the information gain of scanning along the paths. The discrete and continuous processes alternate until the traverse cost of the current graph exceeds the remaining energy capacity of the robot. Our approach is evaluated with both synthetic and field tests, demonstrating its effectiveness and advantages over alternatives. The project is at http://vcc.szu.edu.cn/research/2020/Husky, and the codes are available at https://github.com/alualu628628/Autonomous-Outdoor-Scanning-via-Online-Topological-and-Geometric-Path-Optimization.

Pareto optimal synthesis of eight-bar mechanism using meta-heuristic multi-objective search approaches: application to bipedal gait generation

The structural complexity of the bipedal locomotion may be reduced by using mechanisms with fewer actuators, which ideally do not degrade the performance and functionality of the robot. However, when the mechanism design considers conflicting design objectives, the search for design solutions with specific trade-offs becomes hard. This work studies different multi-objective search approaches the dominance-based, the decomposition-based and the metric-driven in the dimensional synthesis of the eight-bar mechanism for the bipedal locomotion application in the sagittal plane stated as a mixed discrete-continuous nonlinear multi-objective optimisation problem. This work also proposes a dominance-based multi-objective differential evolution algorithm called Multi-Objective Specialist Population-based Differential Evolution (MOSPDE), which endows specialised subpopulations to exploit different regions of the Pareto front to favour the search for design trade-offs with a suitable generation of the gait and force transmission during the stance phase. The comparative study includes algorithms reported in the specialised literature based on dominance (MOPSO, MODE variants and NSGA-II), decomposition (MOEA/D-DE) and metric-driven (SMS-EMOA). The statistical analysis reveals that using the Pareto dominance search approach based on differential evolution and the inclusion of exhaustive exploitation can promote the reconfigurability of the bipedal locomotion mechanism with better trade-offs that satisfy the conflicting design objectives.

Optimization of transmission capacity of energy water pipeline networks with a tree-shaped configuration and multiple sources

The problem of pipeline network transmission capacity optimization is solved during network design, optimization, and development. It consists in determining the optimal pipeline diameters, installation sites, and parameters of pumps and valves. This study proposes a methodology for optimizing the transmission capacity of tree-shaped water pipeline networks of energy systems that serve various purposes and have multiple sources. A method for constructing a network model that ensures the versatility of its representation regardless of the purpose of the network and the composition of its equipment is proposed. A method for modeling of the branch that enables one to flexibly alter the configuration of the branch in the process of running the optimization algorithm without changing the network topology is presented. The mathematical statement of the problem is formulated as the discrete-continuous optimization problem. This statement is based on the proposed methods of representing the network model and its elements. An algorithm based on dynamic programming is proposed, which implements a new approach to setting up the computational procedure and is versatile enough for calculating networks that serve various purposes. The results obtained during real network optimization are presented.

Simultaneous optimization of placement and parameters of rotational friction dampers for seismic-excited steel moment-resisting frames

The present paper proposes the simultaneous optimization of the placement and parameters of rotational friction dampers (RFDs) for the seismic protection of a nonlinear steel moment-resisting frame (SMRF). In that respect, the placement and parameters of RFDs in the framework of an optimization problem is simultaneously optimized for a ten-story SMRF subjected to an artificial earthquake. In the optimization problem, the placement of RFDs and their frictional moment and the length of their vertical rigid beam are considered as the design variables, and the ratio of the maximum seismic input energy to the maximum cumulative dissipated energy by the RFDs is selected as the objective function. The mixed discrete-continuous optimization problem is solved by a hybrid of binary and real-coded modified particle swarm optimization (BRPSO). Furthermore, the effect of the number of RFDs is investigated on the seismic performance of the SMRF. The optimization results reveal that a significant portion of the input seismic energy is dissipated by the RFDs and at the same time reducing the amount of the cumulative hysteretic energy of the structure. Therefore, the seismic damage in the structure is significantly reduced. Furthermore, the appreciable reduction of the seismic responses of the structure is not observed by increasing the number of the RFDs. Finally, the results of the seismic assessment indicate that the optimized RFDs obtained for the artificial earthquake can guarantee the structure subjected to real earthquakes.

A new thief and police algorithm and its application in simultaneous reconfiguration with optimal allocation of capacitor and distributed generation units

This work aims to present a new optimization approach named Thief and Police Algorithm (TPA) to solve discrete-continuous optimization problems. The TPA performance was assessed by solving a discrete-continuous optimization problem in distribution networks. The optimization problem in this study was simultaneous reconfiguration with optimal allocation of capacitor, photovoltaic (PV) and Wind Turbine (WT) units. The objectives were considered as minimization of power loss, minimization of operational cost and improvement of voltage stability of the network. The proposed technique was evaluated using the IEEE 33 bus distribution network with eight different scenarios. Their results were compared with each other on different scenarios and with other well-known algorithms such as Symbiotic Organisms Search (SOS), Ant Colony Optimization (ACO), Particle Swarm Optimization (PSO) and Genetic Algorithm (GA). The results reveals that the TPA outperforms its counterparts in solving the mentioned problem. Also, the performance of the TPA was assessed using the 30 CEC 2017 test functions and the results were compared to the other optimization methods which confirms high capability of the TPA to solve complex optimization problems.

Achieving Channel Capacity of Visible Light Communication

In this article, we propose an efficient inexact gradient descent method to obtain the channel capacity-achieving input distribution of visible light communication (VLC). Under both a peak and an average optical power constraints, finding the channel capacity of VLC is formulated as a mixed continuous-discrete optimization problem without an analytical expression of the objective function. To solve this challenging problem, we first adopt the numerical integration method to approximate the objective function and its gradient. Here, we prove that the gaps between the original functions and the approximations can be arbitrarily small. Then, based on the approximated functions, we describe the method, and theoretically show that the obtained solution sequence converges to the channel capacity-achieving discrete distribution of VLC. We also provide simulation results to verify the effectiveness and optimality of the proposed method. More importantly, the simulations numerically reveal that on-off keying (OOK) modulation achieves the capacity of VLC channel at low signal-to-noise ratio (SNR), while pulse amplitude modulation (PAM) achieves the capacity at high SNR.

A Tutorial on Clique Problems in Communications and Signal Processing

Since its first use by Euler on the problem of the seven bridges of Konigsberg, graph theory has shown excellent abilities in solving and unveiling the properties of multiple discrete optimization problems. The study of the structure of some integer programs reveals equivalence with graph theory problems making a large body of the literature readily available for solving and characterizing the complexity of these problems. This tutorial presents a framework for utilizing a particular graph theory problem, known as the clique problem, for solving communications and signal processing problems. In particular, this article aims to illustrate the structural properties of integer programs that can be formulated as clique problems through multiple examples in communications and signal processing. To that end, the first part of the tutorial provides various optimal and heuristic solutions for the maximum clique, maximum weight clique, and ${k}$ -clique problems. The tutorial, further, illustrates the use of the clique formulation through numerous contemporary examples in communications and signal processing, mainly in maximum access for nonorthogonal multiple access networks, throughput maximization using index and instantly decodable network coding, collision-free radio-frequency identification networks, and resource allocation in cloud-radio access networks. Finally, the tutorial sheds light on the recent advances of such applications, and provides technical insights on ways of dealing with mixed discrete-continuous optimization problems.

Distributed Passive Actuation Schemes for Seismic Protection of Multibuilding Systems

In this paper, we investigate the design of distributed damping systems (DDSs) for the overall seismic protection of multiple adjacent buildings. The considered DDSs contain interstory dampers implemented inside the buildings and also interbuilding damping links. The design objectives include mitigating the buildings seismic response by reducing the interstory-drift and story-acceleration peak-values and producing small interbuilding approachings to decrease the risk of interbuilding collisions. Designing high-performance DDS configurations requires determining convenient damper positions and computing proper values for the damper parameters. That allocation-tuning optimization problem can pose serious computational difficulties for large-scale multibuilding systems. The design methodology proposed in this work—(i) is based on an effective matrix formulation of the damped multibuilding system; (ii) follows an H ∞ approach to define an objective function with fast-evaluation characteristics; (iii) exploits the computational advantages of the current state-of-the-art genetic algorithm solvers, including the usage of hybrid discrete-continuous optimization and parallel computing; and (iv) allows setting actuation schemes of particular interest such as full-linked configurations or nonactuated buildings. To illustrate the main features of the presented methodology, we consider a system of five adjacent multistory buildings and design three full-linked DDS configurations with a different number of actuated buildings. The obtained results confirm the flexibility and effectiveness of the proposed design approach and demonstrate the high-performance characteristics of the devised DDS configurations.

A framework for Data-Driven Structural Analysis in general elasticity based on nonlinear optimization: The static case

Data-Driven Computational Mechanics is a novel computing paradigm that enables the transition from standard data-starved approaches to modern data-rich approaches. At this early stage of development, one can distinguish two mainstream directions. The first one, which can be classified as a direct approach, relies on a discrete-continuous optimization problem and seeks to assign to each material point a point in the phase space that satisfies compatibility and equilibrium, while being closest to the data set provided. The second one, which can be classified as an inverse approach, seeks to reconstruct a constitutive manifold from data sets by manifold learning techniques, relying on a well-defined functional structure of the underlying constitutive law. In this work, we propose a hybrid approach that combines the strengths of the two existing directions and mitigates some of their weaknesses. This is achieved by the formulation of an approximate nonlinear optimization problem, which can be robustly solved, is computationally efficient, and does not rely on any special functional structure of the reconstructed constitutive manifold. Additional benefits include the natural incorporation of kinematic constraints and the possibility to operate with implicitly defined stress–strain relations. We discuss important mathematical aspects of our approach for a data-driven truss element and investigate its key numerical behavior for a data-driven beam element that makes use of all components of our methodology.