Mixed Discrete-continuous Optimization Problem 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.

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.

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 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.

Optimal Multi-View Video Transmission in OFDMA Systems

In this letter, we study the transmission of a multi-view video (MVV) to multiple users in an Orthogonal Frequency Division Multiple Access (OFDMA) system. To maximally improve transmission efficiency, we exploit both natural multicast opportunities and view synthesis-enabled multicast opportunities. First, we establish a communication model for transmission of a MVV to multiple users in an OFDMA system. Then, we formulate the minimization problem of the average weighted sum energy consumption for view transmission and synthesis with respect to view selection and transmission power and subcarrier allocation. The optimization problem is a challenging mixed discrete-continuous optimization problem with huge numbers of variables and constraints. A low-complexity algorithm is proposed to obtain a suboptimal solution. Finally, numerical results further demonstrate the value of view synthesis-enabled multicast opportunities for MVV transmission in OFDMA systems.

Optimal Multi-View Video Transmission in Multiuser Wireless Networks by Exploiting Natural and View Synthesis-Enabled Multicast Opportunities

Multi-view videos (MVVs) provide immersive viewing experience, at the cost of traffic load increase for wireless networks. In this paper, we would like to optimize MVV transmission in a multiuser wireless network by exploiting both natural multicast opportunities and view synthesis-enabled multicast opportunities. Specifically, we first establish a mathematical model to specify view synthesis at the server and each user, and characterize its impact on multicast opportunities. This model is highly nontrivial and fundamentally enables the optimization of view synthesis-based multicast opportunities. For given video quality requirements of all users, we consider the optimization of view selection, transmission time and power allocation to minimize the average weighted sum energy consumption for view transmission and synthesis. In addition, under the energy consumption constraints at the server and each user respectively, we consider the optimization of view selection, transmission time and power allocation and video quality selection to maximize the total utility. These two optimization problems are challenging mixed discrete-continuous optimization problems. For the first problem, we propose an algorithm to obtain an optimal solution with reduced computational complexity by exploiting optimality properties. For each problem, to reduce computational complexity, we also propose a low-complexity algorithm to obtain a suboptimal solution, using Difference of Convex (DC) programming. Finally, numerical results show the advantage of the proposed solutions over existing ones, and demonstrate the importance of the optimization of view synthesis-enabled multicast opportunities in MVV transmission.

Optimal Resource Allocation for Scalable Mobile Edge Computing

In this letter, we would like to investigate the optimal tradeoff between user experience and consumptions of communications and computation resources in designing mobile edge computing (MEC) systems. First, we propose a novel computation task model, where each computation task is scalable with different service levels, corresponding to different user experiences (i.e., utilities) and consumptions of communications and computation resources. Then, we formulate the system utility maximization problem to optimize the service level selection and transmission resource allocation under the transmission power constraints, transmission energy constraints, and deadline constraint. The problem is a challenging mixed discrete-continuous optimization problem. We develop an algorithm to obtain an optimal solution by exploiting optimality properties. We also propose a low-complexity algorithm to obtain a suboptimal solution by the difference-of-convex (DC) programming. This letter provides an effective model for trading off user experience against the consumptions of communications and computation resources in designing efficient MEC systems.

Efficient hybrid algorithms to solve mixed discrete-continuous optimization problems

PurposeIn real-world cases, it is common to encounter mixed discrete-continuous problems where some or all of the variables may take only discrete values. To solve these non-linear optimization problems, the use of finite element methods is very time-consuming. The purpose of this study is to investigate the efficiency of the proposed hybrid algorithms for the mixed discrete-continuous optimization and compare it with the performance of genetic algorithms (GAs).Design/methodology/approachIn this paper, the enhanced multipoint approximation method (MAM) is used to reduce the original nonlinear optimization problem to a sequence of approximations. Then, the sequential quadratic programing technique is applied to find the continuous solution. Following that, the implementation of discrete capability into the MAM is developed to solve the mixed discrete-continuous optimization problems.FindingsThe efficiency and rate of convergence of the developed hybrid algorithms outperforming GA are examined by six detailed case studies in the ten-bar planar truss problem, and the superiority of the Hooke–Jeeves assisted MAM algorithm over the other two hybrid algorithms and GAs is concluded.Originality/valueThe authors propose three efficient hybrid algorithms, the rounding-off, the coordinate search and the Hooke–Jeeves search-assisted MAMs, to solve nonlinear mixed discrete-continuous optimization problems. Implementations include the development of new procedures for sampling discrete points, the modification of the trust region adaptation strategy and strategies for solving mix optimization problems. To improve the efficiency and effectiveness of metamodel construction, regressors f defined in this paper can have the form in common with the empirical formulation of the problems in many engineering subjects.

Porcellio scaber algorithm (PSA) for solving constrained optimization problems

In this paper, we extend a bio-inspired algorithm called the porcellio scaber algorithm (PSA) to solve constrained optimization problems, including a constrained mixed discrete-continuous nonlinear optimization problem. Our extensive experiment results based on benchmark optimization problems show that the PSA has a better performance than many existing methods or algorithms. The results indicate that the PSA is a promising algorithm for constrained optimization.

Analysis and Optimization of Caching and Multicasting in Large-Scale Cache-Enabled Heterogeneous Wireless Networks

Heterogeneous wireless networks (HetNets) provide a powerful approach to meeting the dramatic mobile traffic growth, but also impose a significant challenge on backhaul. Caching and multicasting at macro and pico base stations (BSs) are two promising methods to support massive content delivery and reduce backhaul load in HetNets. In this paper, we jointly consider caching and multicasting in a large-scale cache-enabled HetNet with backhaul constraints. We propose a hybrid caching design consisting of identical caching in the macro-tier and random caching in the pico-tier, and a corresponding multicasting design. By carefully handling different types of interferers and adopting appropriate approximations, we derive tractable expressions for the successful transmission probability in the general signal-to-noise ratio (SNR) and user density region as well as the high SNR and user density region, utilizing tools from stochastic geometry. Then, we consider the successful transmission probability maximization by optimizing design parameters, which is a very challenging mixed discrete-continuous optimization problem. By exploring structural properties, we obtain a near optimal solution with superior performance and manageable complexity. This solution achieves better performance in the general region than any asymptotically optimal solution, under a mild condition. The analysis and optimization results provide valuable design insights for practical cache-enabled HetNets.

Population-based heuristic algorithms for continuous and mixed discrete-continuous optimization problems

Continuous optimization problems are optimization problems where all variableshave a domain that typically is a subset of the real numbers; mixed discrete-continuousoptimization problems have additionally other types of variables, sothat some variables are continuous and others are on an ordinal or categoricalscale. Continuous and mixed discrete-continuous problems have a wide rangeof applications in disciplines such as computer science, mechanical or electricalengineering, economics and bioinformatics. These problems are also often hard tosolve due to their inherent difficulties such as a large number of variables, manylocal optima or other factors making problems hard. Therefore, in this thesis ourfocus is on the design, engineering and configuration of high-performing heuristicoptimization algorithms.We tackle continuous and mixed discrete-continuous optimization problemswith two classes of population-based heuristic algorithms, ant colony optimization(ACO) algorithms and evolution strategies. In a nutshell, the main contributionsof this thesis are that (i) we advance the design and engineering of ACO algorithms to algorithms that are competitive or superior to recent state-of-the-artalgorithms for continuous and mixed discrete-continuous optimization problems,(ii) we improve upon a specific state-of-the-art evolution strategy, the covariancematrix adaptation evolution strategy (CMA-ES), and (iii) we extend CMA-ES totackle mixed discrete-continuous optimization problems.More in detail, we propose a unified ant colony optimization (ACO) frameworkfor continuous optimization (UACOR). This framework synthesizes algorithmiccomponents of two ACO algorithms that have been proposed in the literatureand an incremental ACO algorithm with local search for continuous optimization,which we have proposed during my doctoral research. The design of UACORallows the usage of automatic algorithm configuration techniques to automaticallyderive new, high-performing ACO algorithms for continuous optimization. We alsopropose iCMAES-ILS, a hybrid algorithm that loosely couples IPOP-CMA-ES, aCMA-ES variant that uses a restart schema coupled with an increasing populationsize, and a new iterated local search (ILS) algorithm for continuous optimization.The hybrid algorithm consists of an initial competition phase, in which IPOP-CMA-ES and the ILS algorithm compete for further deployment during a secondphase. A cooperative aspect of the hybrid algorithm is implemented in the formof some limited information exchange from IPOP-CMA-ES to the ILS algorithmduring the initial phase. Experimental studies on recent benchmark functionssuites show that UACOR and iCMAES-ILS are competitive or superior to otherstate-of-the-art algorithms.To tackle mixed discrete-continuous optimization problems, we extend ACOMV and propose CESMV, an ant colony optimization algorithm and a covariance matrix adaptation evolution strategy, respectively. In ACOMV and CESMV , the decision variables of an optimization problem can be declared as continuous, ordinal, or categorical, which allows the algorithm to treat them adequately. ACOMV andCESMV include three solution generation mechanisms: a continuous optimizationmechanism, a continuous relaxation mechanism for ordinal variables, and a categorical optimization mechanism for categorical variables. Together, these mechanisms allow ACOMV and CESMV to tackle mixed variable optimization problems.We also propose a set of artificial, mixed-variable benchmark functions, which cansimulate discrete variables as ordered or categorical. We use them to automatically tune ACOMV and CESMV's parameters and benchmark their performance.Finally we test ACOMV and CESMV on various real-world continuous and mixed-variable engineering optimization problems. Comparisons with results from theliterature demonstrate the effectiveness and robustness of ACOMV and CESMVon mixed-variable optimization problems.Apart from these main contributions, during my doctoral research I have accomplished a number of additional contributions, which concern (i) a note on thebound constraints handling for the CEC'05 benchmark set, (ii) computational results for an automatically tuned IPOP-CMA-ES on the CEC'05 benchmark set and(iii) a study of artificial bee colonies for continuous optimization. These additionalcontributions are to be found in the appendix to this thesis.

Monte Carlo Simulation Approach with Ordinal Optimization Based STATCOM Installation Strategy for Transmission System Loading Margin Enhancement

The power system loading margin enhancement problem to determine an optimal Static Synchronous Compensator (STATCOM) installation can be formulated as a mixed discrete-continuous nonlinear optimization problem (MDCP). In the paper, long-term efficiency of STATCOM installation to enhance transmission system loading margin (LM) is evaluated using a Monte Carlo Simulation (MCS) procedure with an Ordinal Optimization (OO) based STATCOM installation strategy to solve the MDCP. First under each N-1 contingency with bigger risk index (RI) value, the crude model is used to estimate the rough solution for each candidate, and then the exact method is used to a smaller set to determine the good enough solutions. Obtained from the union of the respective good enough solutions for all considered contingencies, the scheme with the smallest number of STATCOM units enabling the network to provide required average LM is recommended.

OO-Based STATCOM Installation for Power System Loading Margin Improvement

As electricity demands and power transactions continuously increase, it becomes vulnerable to voltage instability for power systems, generally incurred by over-utilized facilities or any contingency. The transmission system loading margin (LM) enhancement problem with Static Synchronous Compensator (STATCOM) installation can be formulated as a mixed discrete-continuous nonlinear optimization problem (MDCP); due to the complexity of the MDCP, the computing burden might be very heavy. In the paper, the proposed ordinal optimization (OO) based STATCOM installation method is applied to the MDCP to solve for good enough solutions rather than the best solution, so as to largely reduce the computation burden. In the method, the crude method is first used to solve the MDCP and, based on OO theory, the exact method is then used to determine the good enough solutions. Finally, the good enough solution, as uses the fewest STATCOM device units for installation and makes the power system able to provide the required LM, is recommended for network reinforcement.

Improved Quantum-Inspired Evolutionary Algorithm for Engineering Design Optimization

An improved quantum-inspired evolutionary algorithm is proposed for solving mixed discrete-continuous nonlinear problems in engineering design. The proposed Latin square quantum-inspired evolutionary algorithm (LSQEA) combines Latin squares and quantum-inspired genetic algorithm (QGA). The novel contribution of the proposed LSQEA is the use of a QGA to explore the optimal feasible region in macrospace and the use of a systematic reasoning mechanism of the Latin square to exploit the better solution in microspace. By combining the advantages of exploration and exploitation, the LSQEA provides higher computational efficiency and robustness compared to QGA and real-coded GA when solving global numerical optimization problems with continuous variables. Additionally, the proposed LSQEA approach effectively solves mixed discrete-continuous nonlinear design optimization problems in which the design variables are integers, discrete values, and continuous values. The computational experiments show that the proposed LSQEA approach obtains better results compared to existing methods reported in the literature.

The Real-Valued Model of Hierarchical Classes

We propose a non-negative real-valued model of hierarchical classes (HICLAS) for two-way two-mode data. Like the other members of the HICLAS family, the non-negative real-valued model (NNRV-HICLAS) implies simultaneous hierarchically organized classifications of all modes involved in the data. A distinctive feature of the novel model is that it yields continuous, non-negative real-valued reconstructed data, which considerably expands the application range of the HICLAS family. The expansion implies a major algorithmic challenge as it involves a move from the typical discrete optimization problems in HICLAS to a mixed discrete-continuous one. To solve this mixed discrete-continuous optimization problem, a two-stage algorithm combining a simulated annealing and an alternating local descent stage is proposed. Subsequently it is evaluated in a simulation study. Finally, the NNRVHICLAS model is applied to an empirical data set on anger.

Enterprise Profit Maximization in the Range of Products and Production Rate Selection Problems

In conditions of monopolistic position of enterprises, they exhibit unrestrained tendency to increase the prices of their products, without taking care for full utilization of their production capacities. Such a situation may occur in the planned and as well as the free-market national economy. The decision center is able to counteract this tendency by introducing a proper system of income taxes, which results in the optimal prices providing the maximal profit. Simultaneously, due to the high level of the prime costs of small production series, there exists some threshold value in the relation between profit and production rate. If the production rate is less than this value, it is unprofitable. For reasons mentioned, the problem of enterprise profit optimization in the case of production capacities constrained, is a non-trivial mathematical programming problem. It is a mixed discrete-continuous optimization problem: the decision variables connected with selecting ranges of products are of discrete (zero-one) type: the decision variables associated with the choice of the optimal production rate are continuous. The paper presents an optimization algorithm which can be used to solve this mixed discrete-continuous decision problem. The worked out method can be also applied to other socio-economic decision problems of similar type.