Global Optimization Techniques Research Articles

Restricted global optimization for QAOA

The Quantum Approximate Optimization Algorithm (QAOA) has emerged as a promising variational quantum algorithm for addressing NP-hard combinatorial optimization problems. However, a significant limitation lies in optimizing its classical parameters, which is in itself an NP-hard problem. To circumvent this obstacle, initialization heuristics, enhanced problem encodings and beneficial problem scalings have been proposed. While such strategies further improve QAOA’s performance, their remaining problem is the sole utilization of local optimizers. We show that local optimization methods are inherently inadequate within the complex cost landscape of QAOA. Instead, global optimization techniques greatly improve QAOA’s performance across diverse problem instances. While global optimization generally requires high numbers of function evaluations, we demonstrate how restricted global optimizers still show better performance without requiring an exceeding amount of function evaluations.

Optimal Placement of Sensors in Traffic Networks Using Global Search Optimization Techniques Oriented towards Traffic Flow Estimation and Pollutant Emission Evaluation

The relationship between estimating traffic flow and evaluating pollutant emissions lies in understanding how vehicular traffic patterns affect air quality. Traffic flow estimation is a complex field that involves a variety of analytical techniques to understand, predict, and manage the flow of vehicles on road networks. Different types of analyses commonly employed in this area are statistical analysis (e.g., descriptive statistics, inferential statistics, time series analysis), mathematical modeling (macroscopic models, microscopic models, mesoscopic models), computational methods (e.g., simulation modeling, machine learning, and AI techniques), geospatial analysis (e.g., geographic information systems (GISs), spatial data analysis), network analysis (e.g., graph theory and network flow models). In sensor network setups, the strategic placement of sensors is crucial, primarily due to the challenges posed by limited energy supplies, restricted storage capabilities, and the demands on processing and communication, all of which significantly impact maintenance costs and hardware limitations. To mitigate the burden on processing and communication, it is essential to deploy a limited number of sensors strategically. In practical applications, achieving an optimal layout of physical sensors (i.e., placing sensors within the network in such a way as to meet a specific optimality criterion, such as identifying the minimum number of sensors required to ensure the ability to design reliable state observers capable of reconstructing the network’s state based on the available data) is essential for the accurate monitoring of large-scale systems, including traffic flow or the distribution networks of water and gas. In the context of traffic systems, addressing the challenge of full link flow observability, that is, the ability to accurately monitor and assess the flow of entities (i.e., vehicles) across all the links or pathways within a network, entails selecting the smallest number of traffic sensors from a larger set to install. The goal is to choose a subset of p sensors, which may include redundancies, from a pool of n>>p potential sensors. This is conducted to maintain the structural observability of the entire traffic network. This concept pertains to deducing the complete internal state (traffic volume on each road link in the network) from external outputs and inputs (measurements from sensors). The traditional concept of system observability serves as a criterion for sensor placement. This article presents the development of a simulated annealing heuristic to address the selection problem. The selected sensors are then applied to construct a Luenberger observer, a mathematical construct used in control theory to accurately estimate the internal state of a dynamic system based on its inputs and outputs. Numerical simulations are carried out to demonstrate the effectiveness of this method, and a performance analysis using a digital twin of a transport network, designed using the Aimsun Next software, are also carried out to assess traffic flow and associated pollutant emissions. In particular, we examine a traffic network comprising 21 roads. We address the sensor selection problem by identifying an optimal set of six sensors, which facilitates the design of a Luenberger observer. This observer enables the reconstruction of traffic flow across the network with minimal estimation error. Furthermore, by integrating this observer with data from the Aimsun Next software, we assess the pollutant emissions related to traffic flow. The results indicate a high accuracy in estimating pollutant levels.

Size Optimization of Truss Structures Based on Genetic Algorithm by Developing Single-point Crossover with Bit-wise Mutation Using FORTRAN Programming

A truss is a collection of axially loaded structural elements connected through pin joints. Optimization of truss structures indicates the determination of best possible conditions that are necessary for achieving the most economical design in terms of lowest possible weights of the truss elements. According to recent studies, Size, Topology and Shape based optimizations are the three possible independent optimization scopes for finding the optimum weight of a truss. The current study focuses on the size optimization of truss structure and aims to identify the most cost-effective sections of the truss using a powerful evolutionary optimization technique named as Genetic Algorithm (GA). Here single point cross over method and bit wise mutation operator are adopted for expanding the search space with the assigned displacement and stress constraints. Fortran based sub routines are developed for each individual steps of GA such as fitness evaluation, selection, cross-over and mutation. Finaly the complete program is run for validation of its outcome. While comparing with the solution of an existing literature regarding a minimization of objective function having two design variables the outcomes tend to show a good correlation. Further, the study is extended for the weight minimization of two widely used truss having 10 members and 17 members respectively with standard Finite Element Method of analysis. Minimum weight of each truss is obtained via convergence plot and compared with the existing literatures. With this simple but efficient global optimization technique optimum weight of truss structure can easily be achieved.

Dynamic Events in the Flexible Job-Shop Scheduling Problem: Rescheduling with a Hybrid Metaheuristic Algorithm

In the real world of manufacturing systems, production planning is crucial for organizing and optimizing various manufacturing process components. The objective of this paper is to present a methodology for both static scheduling and dynamic scheduling. In the proposed method, a hybrid algorithm is utilized to optimize the static flexible job-shop scheduling problem (FJSP) and dynamic flexible job-shop scheduling problem (DFJSP). This algorithm integrates the genetic algorithm (GA) as a global optimization technique with a simulated annealing (SA) algorithm serving as a local search optimization approach to accelerate convergence and prevent getting stuck in local minima. Additionally, variable neighborhood search (VNS) is utilized for efficient neighborhood search within this hybrid algorithm framework. For the FJSP, the proposed hybrid algorithm is simulated on a 40-benchmark dataset to evaluate its performance. Comparisons among the proposed hybrid algorithm and other algorithms are provided to show the effectiveness of the proposed algorithm, ensuring that the proposed hybrid algorithm can efficiently solve the FJSP, with 38 out of 40 instances demonstrating better results. The primary objective of this study is to perform dynamic scheduling on two datasets, including both single-purpose machine and multi-purpose machine datasets, using the proposed hybrid algorithm with a rescheduling strategy. By observing the results of the DFJSP, dynamic events such as a single machine breakdown, a single job arrival, multiple machine breakdowns, and multiple job arrivals demonstrate that the proposed hybrid algorithm with the rescheduling strategy achieves significant improvement and the proposed method obtains the best new solution, resulting in a significant decrease in makespan.

Multilayer Kirkpatrick-Baez focusing mirrors with phase compensation for sub-20 nm focusing at the hard X-ray nanoprobe beamline of SSRF

The hard X-ray nanoprobe beamline is the first beamline to take advantage of the full coherent beam to attain the nanoscale focusing at the Shanghai Synchrotron Radiation Facility (SSRF). Here we introduce the beamline and specially go over the features of the multilayer Kirkpatrick-Baez focusing system and its supporting phase compensator system. The performance and stability of the phase compensator are also put to the test. By using the speckle scanning metrology, the wavefront of a focused beam was characterized and intensity distribution near the focus was reconstructed. The focusing performance was greatly enhanced by two phase compensations based on a global optimization technique, and a two-dimensional focal spot of 26 nm × 17 nm was achieved and maintained with good stability.

Global optimization for the design of shell and tube horizontal thermosyphon reboiler

Shell and tube horizontal thermosyphon reboilers (STHTs) find wide applications in the process industry particularly due to their inherent advantages in handling large duties, high viscosity fluids and low fouled susceptibility. However, current design commercial packages suffer from obvious limitations including extensive user manipulations, trial-and-error necessity, increased human factor influence. To overcome these limitations, we develop in this work a rigorous MINLP model for STHT sizing, by achieving automated selection of detailed exchanger geometries with optimal overall heat transfer coefficient. The corrected minimum cross-flow area in shell-side is incorporated to accommodate full cross-flow patterns in tube bundle, thus handling popular service of X shell without baffle construction. A dependable convective boiling heat transfer model is proposed to obtain the shell-side heat transfer coefficient, validated with Aspen EDR. The critical variable, heat flux, which affects nucleate boiling heat transfer performance, is optimized through a simultaneous equation-solving process using a global solver, reducing excessive unit overdesign. Pressure balance between STHT and distillation column is fully considered by a proposed optimization model incorporating pressure drop and pipe sizing. By employing global optimization techniques and automating the design process, this innovative approach minimizes manual adjustments and provides more accurate and cost-effective designs.

Image Synthesis for Solar Flare Prediction

Solar flare prediction is a topic of interest to many researchers owing to the potential of solar flares to affect various technological systems, both terrestrial and in orbit. In recent years, the forecasting task has become progressively more reliant on data-driven computations and machine-learning algorithms. Although these efforts have improved solar flare predictions, they still falter in doing so for large solar flares, in particular under operational conditions, since large-flare data are very scarce and labeled data are heavily imbalanced. In this work, we seek to address this fundamental issue and present a scheme for generating synthetic magnetograms to reduce the imbalance in the data. Our method consists of (1) synthetic oversampling of line-of-sight magnetograms using Gaussian mixture model representation, followed by (2) a global optimization technique to ensure consistency of both physical features and flare precursors, and (3) the mapping of the generated representations to realistic magnetogram images using deep generative models. We show that these synthetically generated data indeed improve the capacity of solar flare prediction models and that, when tested on such a state-of-the-art model, it significantly enhances its forecasting performance, achieving an F1-score as high as 0.43 ± 0.08 and a true skill statistic of 0.64 ± 0.10 for X-class flares in the 24 hr operational solar flare data split.

H-ADCP-based real-time sediment load monitoring system using support vector regression calibrated by global optimization technique and its applications

Reliable and continuous measurement of sediment load in rivers is essential for understanding and managing sediment transport dynamics. However, traditional sediment load monitoring methods depend on labor-intensive field sampling techniques, leading to limitations in spatiotemporal coverage. This study introduces a novel approach for machine learning-aided real-time suspended sediment concentration monitoring using a horizontal acoustic Doppler current profiler. Support vector regression models for suspended sediment concentration estimation are derived from eight flow monitoring stations with sediment sampling data. A new model selection framework is proposed, integrating a global optimization algorithm in order to calibrate input variables and hyperparameters simultaneously. The three-fold cross-validation score is utilized to enhance the generalization accuracy and reliability of sediment load estimation. This work assessed the impact of considering additional input variables in addition to the backscattering signal. The findings confirm that incorporating flowrate, water level, and their time derivatives during model training improves predictability. Spatial differences between flow monitoring and sediment sampling locations significantly influence the model accuracy. This study also discusses the potential application of the H-ADCP-based monitoring system to obtain suspended and total sediment loads at an arbitrary flow monitoring station. By offering real-time monitoring capabilities and reduced reliance on labor-intensive field sampling, this work contributes to the advancement of sediment transport data acquisition methods.

Computationally efficient parameter estimation for high-dimensional ocean biogeochemical models

Abstract. Biogeochemical (BGC) models are widely used in ocean simulations for a range of applications but typically include parameters that are determined based on a combination of empiricism and convention. Here, we describe and demonstrate an optimization-based parameter estimation method for high-dimensional (in parameter space) BGC ocean models. Our computationally efficient method combines the respective benefits of global and local optimization techniques and enables simultaneous parameter estimation at multiple ocean locations using multiple state variables. We demonstrate the method for a 17-state-variable BGC model with 51 uncertain parameters, where a one-dimensional (in space) physical model is used to represent vertical mixing. We perform a twin-simulation experiment to test the accuracy of the method in recovering known parameters. We then use the method to simultaneously match multi-variable observational data collected at sites in the subtropical North Atlantic and Pacific. We examine the effects of different objective functions, sometimes referred to as cost functions, which quantify the disagreement between model and observational data. We further examine increasing levels of data sparsity and the choice of state variables used during the optimization. We end with a discussion of how the method can be applied to other BGC models, ocean locations, and mixing representations.

Design of sustainable energy harvesters using a 3D car suspension system model and half-width speed bump

This study aims to not only design a vibration energy harvester integrated in a three-dimensional (3D) multi-particle system but also investigate the longitudinal and lateral body sway and the effect of front and back wheel interlocking caused by the eccentric loading of the vehicle. Using multidimensional vehicle suspension system vibration modes for analysis, this study evaluates the impact of lateral and longitudinal eccentricity on the displacement of the four-suspension system wheel axes. In addition, this study discusses the effect of discontinuous half-width sinusoidal speed bump on power generation and maximum power generation capacity under good ride comfort conditions. To maximize the efficient electrical power from the car's suspension system through the use of energy harvesters, we have employed the simulated annealing algorithm, a robust global optimization technique. The objective function chosen for this optimization is the root-mean-square (RMS) electrical power, represented as WTT(RMS). In addition to this, we have considered the comfort experienced by passengers, incorporating the ride comfort efficiency into the objective function during the optimization process. As a result of this optimization, a 3D car model equipped with four energy harvesters has been fine-tuned to achieve a significant increase in induced electrical power, ultimately reaching an impressive 0.2 W.

Trakya Havzasının Güneybatısında Fay Parametrelerinin Gravite Gradyent Verilerinden Kestirimi

Characterization of fault zones plays an important role for earthquake studies, geothermal and mineral resource exploration. This problem can be achieved by optimizing parameters of a defined synthetic model iteratively using gravity, gravity gradients or magnetic data. Curvature gradients indicate deviations of an equipotential surface from a spherical surface, therefore, reflecting deviations of mass distributions from a source point. In this study, differential curvature gradient is used as it identifies linear structures such as dip-slip faults and geological contacts. Curvature gradients are computed from gravity gradient observations which are derived from the high-resolution Earth Gravitational Model 2008. Estimating the fault parameters from gravity gradients is a complex nonlinear geophysical inverse problem that requires finding the minimum of a multi-variable cost function. A global optimization technique called simulated annealing (SA) is used to estimate location, dip angle and some depth parameters of the faults located in southwest side of the Thrace basin in Türkiye. In the optimization technique, a straight solution is inevitably applied to calculate the theoretical anomaly to be compared with the observed anomaly in each search step. In this case, the problem requires constructing a mathematical or geophysical interpretation model. Such a model is designed as a dip-slip fault model. The results show that dip angle estimates indicate high-angle faults which are found to be consistent with the seismic studies.

Two efficient logarithmic formulations to solve nonconvex economic dispatch

To solve economic dispatch (ED) with nonconvex nonlinear generation cost functions, this paper adopts a global optimization technique where piecewise linear approximation (PLA) functions replace the nonlinear functions. However, a problem with this technique is that the PLA model requires incorporating many discrete variables into the optimization model, which may render the model computationally expensive. This paper presents two efficient representation techniques of the PLAs, which scale logarithmically, including binary zig-zag (ZZB) and general integer zig-zag (ZZI). We replace the nonconvex cost functions in the ED with their PLAs and obtain an alternative ED formulation where two logarithmic techniques model the PLAs. Then, one can efficiently solve the alternative ED model with the standard branch-and-cut/bound-based algorithms to the desired optimality gap. To verify the techniques’ efficacy, we consider test systems having 13, 38, 40, 80, and 160 units. The numerical evaluations show that the logarithmic techniques presented can yield high-quality optimal solutions in less than 2 seconds in the considered cases, illustrating the techniques’ efficiency.

Leaf properties estimation enhancement over heterogeneous vegetation by correcting for terrestrial laser scanning beam divergence effect

The need for an accurate description of forest structure is increased as it is a key for monitoring carbon storage, energy, and mass exchange, and different ecosystem functions. The leaf area index (LAI), the leaf area density (ul), and the leaf inclination distribution function (LIDF) are regarded as important and critical keys allowing the characterization of the forest structure. The dependency between these properties and the heterogeneity of the forest makes their estimation challenging and delicate. In this context, despite the expanding body of works reporting forest structure properties retrieval using infinitesimal footprint size terrestrial LiDAR system (TLS) data, there seem to be few works evaluating the influence of the TLS beam divergence effect (i.e. its size increase with distance from the sensor), on the estimation accuracy, even though the influence is not negligible. In this study, we aim to jointly estimate LAI/ul and LIDF using non-infinitesimal footprint size TLS data based on forward/inverse radiative transfer (RT) modeling. Our approach assumes a uniform distribution of azimuth angles enabling us to attain valuable insights into inclination angles and leaf area density while simplifying our computational efforts. The forward modeling allows an understanding of the interaction between vegetation structure properties and TLS light beams, while the inversion allows retrieving the properties given the TLS data. The inversion technique is based on the maximum likelihood estimator (MLE). Our problem turns out to be a non-linear and high-dimensional cost function to be optimized. To tackle this complexity, the global optimization technique, the shuffled complex evolution (SCE-UA), is adapted to find the globally optimal solution. Our developed approach is validated with simulated and actual forest TLS point clouds acquired from Estonian Pine, Birch, and Spruce stands. Our findings using simulated data prove that our estimates are in agreement with the actual ul and LIDF values with (MAEul∈0.00120.0043) and (RMSEul∈0.01370.0224). In the actual case, RMSELAI for Pine, Birch and Spruce stands are 0.1066, 0.014, and 0.070, respectively. In all cases, our approach outperforms state-of-the-art techniques.

Reduced order modeling of delta operator systems by optimal frequency fitting approach

The delta operator modeling provides a unified framework for both continuous-time and discrete-time modeling in system theory. At high sampling rate, the shift operator fails to provide meaningful information whereas, the delta operator parameterized system provides the same results as of continuous time systems. In this paper reduced order modeling of delta operator parameterized systems is considered. A complex domain (δ) optimal frequency matching (OFM) technique is proposed and frequency points are optimized using Particle Swarm Optimization (PSO) algorithm. This OFM is then utilized to find the reduced order model of the higher order system. PSO algorithm is a robust, global optimization technique, used to find these OFMs and thereby used to find the coefficients of the reduced order model by minimizing a cost function developed based on the responses of the higher order model and that of the reduced order model when both are excited by pseudo random binary sequences (PRBS). The performance characteristics are evaluated in software simulation using MATLAB considering example of higher order system in delta domain and time & frequency responses of the corresponding reduced model.

Solving differential equations with global optimization techniques

Solving differential equations with global optimization techniques

Exploring the energy landscape of graphynes for noble gas adsorption using swarm intelligence

Gas adsorption on one-atom-thick membranes is a growing technology for separation applications owing to its excellent energy efficiency. Herein, we investigate the adsorption of the noble gases, Ne, Ar and Kr on graphynes (GYs), a novel class of one-atom-thick carbon membranes using a swarm intelligence technique, namely particle swarm optimization (PSO). Modeling the adsorption of noble gas clusters on two-dimensional substrates requires a thorough examination of the energy landscape. The high dimensionality of the problem makes it tricky to employ ab initio methods for such studies, necessitating the use of a metaheuristic global optimization technique such as PSO. We explored the adsorption of 1–30 atoms of Ne, Ar and Kr on α-, β-, γ- and rhombic-GYs to predict the most suitable form of GY for the adsorption of each of the gases. Employing the dispersion-corrected density functional theory (DFT-D) data for the adsorption of single gas atoms as the reference data, we parametrized two empirical pairwise potentials, namely, Lennard-Jones (LJ) and improved Lennard-Jones (ILJ) potentials. We then analyzed the growth pattern as well as the energetics of adsorption using the parametrized potentials, in combination with the PSO technique, which enabled us to predict the best possible membrane for the adsorption of the noble gases: α-GY for Ne and γ-GY for Ar and Kr. The accuracy of our modeling approach is further validated against DFT-D computations thereby establishing that PSO, when combined with the ILJ potential, can serve as a computationally feasible approach for modeling gas adsorption on GYs.

A Comparative Analysis of Optimization Algorithms for Finite Element Model Updating on Numerical and Experimental Benchmarks

Finite Element Model Updating (FEMU) is a common approach to model-based Non-Destructive Evaluation (NDE) and Structural Health Monitoring (SHM) of civil structures and infrastructures. Its application can be further utilized to produce effective digital twins of a permanently monitored structure. The FEMU concept, simple yet effective, involves calibrating and/or updating a numerical model based on the recorded dynamic response of the target system. This enables to indirectly estimate its material parameters, thus providing insight into its mass and stiffness distribution. In turn, this can be used to localize structural changes that may be induced by damage occurrence. However, several algorithms exist in the scientific literature for FEMU purposes. This study benchmarks three well-established global optimization techniques—namely, Generalized Pattern Search, Simulated Annealing, and a Genetic Algorithm application—against a proposed Bayesian sampling optimization algorithm. Although Bayesian optimization is a powerful yet efficient global optimization technique, especially suitable for expensive functions, it is seldom applied to model updating problems. The comparison is performed on numerical and experimental datasets based on one metallic truss structure built in the facilities of Cranfield University. The Bayesian sampling procedure showed high computational accuracy and efficiency, with a runtime of approximately half that of the alternative optimization strategies.

Bottom-up dust nucleation theory in oxygen-rich evolved stars

Context. Spinel (MgAl2O4) and krotite (CaAl2O4) are alternative candidates to alumina (Al2O3) as primary dust condensates in the atmospheres of oxygen-rich evolved stars. Moreover, spinel was proposed as a potential carrier of the circumstellar 13 μm feature. However, the formation of nucleating spinel clusters is challenging; in particular, the inclusion of Mg constitutes a kinetic bottleneck. Aims. We aim to understand the initial steps of cosmic dust formation (i.e. nucleation) in oxygen-rich environments using a quantum-chemical bottom-up approach. Methods. Starting with an elemental gas-phase composition, we constructed a detailed chemical-kinetic network that describes the formation and destruction of magnesium-, calcium-, and aluminium-bearing molecules as well as the smallest dust-forming (MgAl2O4)1 and (CaAl2O4)1 monomer clusters. Different formation scenarios with exothermic pathways were explored, including the alumina (Al2O3) cluster chemistry studied in Paper I of this series. The resulting extensive network was applied to two model stars, a semi-regular variable and a Mira-type star, and to different circumstellar gas trajectories, including a non-pulsating outflow and a pulsating model. We employed global optimisation techniques to find the most favourable (MgAl2O4)n, (CaAl2O4)n, and mixed (MgxCa(1−x)Al2O4)n isomers, with n = 1–7 and x∈[0..1], and we used high level quantum-chemical methods to determine their potential energies. The growth of larger clusters with n = 2–7 is described by the temperature-dependent Gibbs free energies. Results. In the considered stellar outflow models, spinel clusters do not form in significant amounts. However, we find that in the Mira-type non-pulsating model CaAl2O3(OH)2, a hydroxylated form of the calcium aluminate krotite monomer forms at abundances as large as 2 × 10−8 at 3 stellar radii, corresponding to a dust-to-gas mass ratio of 1.5 × 10−6. Moreover, we present global minimum (GM) candidates for (MgAl2O4)n and (CaAl2O4)n, where n = 1–7. For cluster sizes n = 3–7, we find new, hitherto unreported GM candidates. All spinel GM candidates found are energetically more favourable than their corresponding magnesium-rich silicate clusters with an olivine stoichiometry, namely (Mg2SiO4)n. Moreover, calcium aluminate clusters, (CaAl2O4)n, are more favourable than their Mg-rich counterparts; the latter show a gradual enhancement in stability when Mg atoms are substituted step by step with Ca. Conclusions. Alumina clusters with a dust-to-gas mass ratio of the order of 10−4 remain the favoured seed particle candidate in our physico-chemical models. However, CaAl2O4 could contribute to stellar dust formation and the mass-loss process. In contrast, the formation of MgAl2O4 is negligible due to the low reactivity of the Mg atom.

Research on antenna design based on low-resolution digital-to-analog converters

In the ever-evolving field of communication, the optimization of antenna design and structure has garnered significant attention from professionals. Effective antenna design plays a crucial role in ensuring reliable and efficient communication systems. However, the challenge lies in overcoming the limitations imposed by current low-resolution digital-to-analog converters (DACs) on the attainment of highly efficient antenna structures. This study aims to address this issue by exploring the potential of low-resolution DACs for improving the structural design of antennas. To achieve this, we propose gradient-based and low-complexity heuristic solutions that leverage global optimization techniques. By optimizing the antenna structure using these innovative approaches, we aim to enhance the overall performance and efficiency of communication systems. The significance of this research lies in its potential to revolutionize antenna design by overcoming the constraints of low-resolution DACs. By incorporating Sigma-Delta modulation in the filter design, we anticipate significant improvements in performance when the number of transmitting antennas is comparable to the number of users. Furthermore, our investigation reveals that the application of Compressed Sensing techniques yields results that closely align with the optimal solution in scenarios where the number of transmitting antennas greatly exceeds the number of users.

Clustering Sequence Data with Mixture Markov Chains with Covariates Using Multiple Simplex Constrained Optimization Routine (MSiCOR)

Mixture Markov Model (MMM) is a widely used tool to cluster sequences of events coming from a finite state-space. However, the MMM likelihood being multi-modal, the challenge remains in its maximization. Although Expectation-Maximization (EM) algorithm remains one of the most popular ways to estimate the MMM parameters, however, convergence of EM algorithm is not always guaranteed. Given the computational challenges in maximizing the mixture likelihood on the constrained parameter space, we develop a pattern search-based global optimization technique which can optimize any objective function on a collection of simplexes, which is eventually used to maximize MMM likelihood. This is shown to outperform other related global optimization techniques. In simulation experiments, the proposed method is shown to outperform the expectation-maximization (EM) algorithm in the context of MMM estimation performance. The proposed method is applied to cluster Multiple sclerosis (MS) patients based on their treatment sequences of disease-modifying therapies (DMTs). We also propose a novel method to cluster people with MS based on DMT prescriptions and associated clinical features (covariates) using MMM with covariates. Based on the analysis, we divided MS patients into three clusters. Further cluster-specific summaries of relevant covariates indicate patient differences among the clusters. Supplementary materials for this article are available online.