MATLAB Optimization Research Articles

Global optimization of MSF seawater desalination processes

This article addresses the global optimal design of multi-stage flash desalination processes. The mathematical formulation accounts for non-linear programming (NLP) based process models that are supplemented with the non-deterministic optimization algorithm. MSF-once through, -simple mixture (MSF-M) and -brine recycle (MSF-BR) process configurations have been evaluated for their optimality. While freshwater production cost has been set as the objective function for minimization, mass, energy and enthalpy balances with relevant supplementary equations constitute the equality constraints. Differential evolution algorithm (DE/rand/bin) was adopted to evaluate the global optimal solutions. Further, obtained solutions have been compared with those obtained with MATLAB optimization toolbox solvers such as SQP and MS-SQP. The global optimal solution corresponds to a variable value set of [2794.4m3/h, 1.0499, 7.62m, 3.359kW/m2∙K, 3.297kW/m2∙K, 3.042kW/m2∙K and 22] for decision variables [WM, RH, LT, UB, UR, Uj, NR] in the MSF-BR process to yield an optimal freshwater production cost of 1.0785$/m3. Compared to the literature, the obtained global solution from DE is 2.31% better. Further, inequality constraint resolution has been excellent for DE but not other methods such as MS-SQP, SQP and DE-SQP.

PID Controller Design Based on Global Optimization Technique with Additional Constraints

Abstract This paper deals with design of PID controller with the use of methods of global optimization implemented in Matlab environment and Optimization Toolbox. It is based on minimization of a chosen integral criterion with respect to additional requirements on control quality such as overshoot, phase margin and limits for manipulated value. The objective function also respects user-defined weigh coefficients for its particular terms for a different penalization of individual requirements that often clash each other such as for example overshoot and phase margin. The described solution is designated for continuous linear time-invariant static systems up to 4th order and thus efficient for the most of real control processes in practice.

Feasibility Investigation of a Hybrid On-Grid Wind Photovoltaic Retrofitting System

This paper investigates the feasibility of wind–photovoltaic (PV) penetration into an existing utility grid system for the city of Ibrahimyya in Jordan. Ibrahimyya is selected because it enjoys both high annual wind speed of 7.27 m/s and high annual solar radiation of $\text{6.05}\;{\text{kWh}}/{\text{m}}^2/{\text{day}}$ . Two sizing methods are presented using MATLAB and hybrid optimization multiple energy resources software. Thousands of iterations have been carried out in order to get the global autonomous sizing solution that is used for economic analysis. Results show that three CS6X-310 PV panels and eight GE1.5sle-77 wind turbines are the optimal choice. A step-by-step analysis of the proposed system is presented. The net present cost (NPC) is 65 069 349$. The cost of energy (COE) is 0.0817$/kWh. A sensitivity analysis on interest rate, inflation rate, wind power law exponent, annual average daily energy demand, and fuel price is implemented to assess the robustness of the system. The results prove the feasibility to apply the proposed hybrid wind–PV system for this city. The same procedure can be applied anywhere.

Numerical Modeling of Dependent Credit Rating Transitions with Asynchronously Moving Industries

Two models of dependent credit rating migrations governed by industry-specific Markovian matrices, are considered. Caused by macroeconomic factors, positive and negative unobserved tendencies, encoded as values “1” or “0” of the corresponding variables, modify the transition probabilities and render the evolutions dependent. They are neither synchronized across industry sectors, nor over credit classes: an upswing in some of them can coexist with a decline of the rest. The models are tested on Standard and Poor’s data. MATLAB optimization software and maximum likelihood estimators are used. Obtained distributions of the hidden variables demonstrate that the considered industries migrate asynchronously trough credit classes. Since downgrading probabilities are less affected by the unobserved tendencies, estimated by Monte-Carlo simulations distributions of defaults, exhibit lighter, than for the known coupling models, tails for schemes with asynchronously moving industries. Moreover, the lightest tails were obtained in the case of industry-specific transition matrices.

Dosimetric investigation of proton therapy on CT-based patient data using Monte Carlo simulation

The aim of radiotherapy is to deliver high radiation dose to the tumor with low radiation dose to healthy tissues. Protons have Bragg peaks that give high radiation dose to the tumor but low exit dose or dose tail. Therefore, proton therapy is promising for treating deep- seated tumors and tumors locating close to organs at risk. Moreover, the physical characteristic of protons is suitable for treating cancer in pediatric patients. This work developed a computational platform for calculating proton dose distribution using the Monte Carlo (MC) technique and patient's anatomical data. The studied case is a pediatric patient with a primary brain tumor. PHITS will be used for MC simulation. Therefore, patient-specific CT-DICOM files were converted to the PHITS input. A MATLAB optimization program was developed to create a beam delivery control file for this study. The optimization program requires the proton beam data. All these data were calculated in this work using analytical formulas and the calculation accuracy was tested, before the beam delivery control file is used for MC simulation. This study will be useful for researchers aiming to investigate proton dose distribution in patients but do not have access to proton therapy machines.

Nonlinear residual minimization by iteratively reweighted least squares

In this paper we address the numerical solution of minimal norm residuals of nonlinear equations in finite dimensions. We take particularly inspiration from the problem of finding a sparse vector solution of phase retrieval problems by using greedy algorithms based on iterative residual minimizations in the $$\ell _p$$lp-norm, for $$1 \le p \le 2$$1≤p≤2. Due to the mild smoothness of the problem, especially for $$p \rightarrow 1$$pź1, we develop and analyze a generalized version of iteratively reweighted least squares (IRLS). This simple and efficient algorithm performs the solution of optimization problems involving non-quadratic possibly non-convex and non-smooth cost functions, which can be transformed into a sequence of common least squares problems. The latter can be tackled eventually by more efficient numerical optimization methods. While its analysis has been by now developed in many different contexts (e.g., for sparse vector, low-rank matrix optimization, and for the solution of PDE involving p-Laplacians) when the model equation is linear, no results are up to now provided in case of nonlinear ones. We address here precisely the convergence and the rate of error decay of IRLS for such nonlinear problems. The analysis of the convergence of the algorithm is based on its reformulation as an alternating minimization of an energy functional. In fact its main variables are the competitors to solutions of the intermediate reweighted least squares problems and their weights. Under a specific condition of coercivity often verified in practice and assumptions of local convexity, we are able to show convergence of IRLS to minimizers of the nonlinear residual problem. For the case where we are lacking the local convexity, we propose an appropriate convexification by quadratic perturbations. Eventually we are able to show convergence of this modified procedure to at least a very good approximation of stationary points of the original problem. In order to illustrate the theoretical results we conclude the paper with several numerical experiments. We first compare IRLS with standard Matlab optimization functions for a simple and easily presentable example. Furthermore we numerically validate our theoretical results in the more complicated framework of phase retrieval problems, which are our main motivation. Finally we examine the recovery capability of the algorithm in the context of data corrupted by impulsive noise where the sparsification of the residual is desired.

Forming District Metered Areas in a Water Distribution Network Using Genetic Algorithms

Dividing a water distribution network into District Metered Areas (DMAs) is the first vital step towards water loss management. It is a multivariate problem. Techniques based on Genetic Algorithms is a proven way to provide a very good solution to optimization problems. Basic requirement is that each problem must be well described by an objective function. The formation of the objective function is tested through Matlab's optimization tools. The results of the genetic algorithm are compared to the results of an algorithm (in C++ Language) developed in an earlier stage to provide optimal system pressure reduction by closing indicated pipes. The process for the formation of the objective function and a case study on a real network are presented.

Analysis and Optimization of Section-Thickness of the Manipulator for GIS Detection

In order to find out the influence of different section-thickness of the manipulator for GIS detection on the mass, strength and stiffness, the performance of the arm rod with different section thickness is analysed by MATLAB under the condition of a certain section height. Firstly, the influence of the parameters of each section-thickness on the mass, strength and stiffness is analysed by single variable analysis. Then the mathematical model is established, and the section-thickness of the arm is optimized using the method of nonlinear programming. The results are similar to the results of single variable analysis. Finally, the section thickness of the virtual prototype is optimized by the finite element analysis method. The correctness of MATLAB optimization results is verified. The optimal solution of the section-thickness of the GIS-detection-arm is obtained.

Disturbance Rejection wi h a Highly Oscillating Second-order-like Process, Part VI: PPI Controller

A PPI controller is investigated for disturbance rejection associated with a highly oscillating second-order-like process. The controller is tuned using the MATLAB optimization toolbox for different error-based objective functions. The best objective function suitable for this type of controllers for the highly oscillating second order process is assigned. The unit step disturbance input time response of the control system has zero steady-state error and low response levels. The effect of the proportional gain of the PPI controller on the system dynamics is investigated. The PPI controller when used for disturbance rejection associated with the highly oscillating second-order process can compete well with PD-PI and PI-PD controllers used for the same purpose.

Study on blasting erosion arc machining of Ti–6Al–4V alloy

Blasting erosion arc machining (BEAM) was applied to improve the machining efficiency of Ti–6Al–4V alloy. 5-factor, 2-level fractional factorial experiment was firstly employed to find out the main factors for MRR (material removal rate). According to the results of fractional factorial experiment, 3-factor, 2-level full factorial experiment was conducted to find out the relationship between machining performance and the main factors (peak current, pulse duration, and pulse interval) under the negative electrode machining. Results revealed that when peak current was 500 A and pulse duration was 8 ms, MRR could achieve 16,800 mm3/min, which means the specific energy efficiency was 33.6 mm3/(A · min). Then, MRR was optimized base on MATLAB optimization toolbox and could reach 20,100 mm3/min when adopting the optimized parameters (peak current 600 A, pulse duration 8.8 ms, and pulse interval 3.0 ms). Although TWR (tool wear ratio) can be effected by the machining parameters, it appeared to be stable (around 3 ± 1 %). Besides, the polarity effect was also studied, and flow field simulations were employed to illustrate the influence of feeding direction on the surface roughness. In addition, the machined surface was analyzed by utilizing SEM, EDS, XRD, etc. Results of the surface analysis disclosed that the oxygen content in the negative electrode machined surface was obviously higher than that in the positive electrode machined surface. Since the positive electrode machined surface was much smoother, it can be used to improve the surface quality while the negative electrode machining is suitable for the bulk material removal with high energy. Finally, a Ti–6Al–4V turbine blade sample was machined to demonstrate the feasibility of high efficiency BEAM in difficult-to-cut material processing.

Modeling dependent credit rating transitions: a comparison of coupling schemes and empirical evidence

Three coupling schemes for generating dependent credit rating transitions are compared and empirically tested. Their distributions, the corresponding variances and default correlations are characterized. Using Standard and Poor’s data for OECD countries, parameters of the models are estimated by the maximum likelihood method and MATLAB optimization software. Two pools of debtors are considered: with 5 and with 12 industry sectors. They are classified into two non-default credit classes. First portfolio mimics the Dow Jones iTraxx EUR market index. The default correlations evaluated for 12 industry sectors are confronted with their counterparts known for the US economy.

Method of determining effects of heat-induced irregular refractive index on an optical system.

The effects of an irregular refractive index on optical performance are examined. A method was developed to express a lens's irregular refractive index distribution. An optical system and its mountings were modeled by a thermomechanical finite element (FE) program in the predicted operating temperature range, -45°C-50°C. FE outputs were elaborated using a MATLAB optimization routine; a nonlinear least squares algorithm was adopted to determine which gradient equation best fit each lens's refractive index distribution. The obtained gradient data were imported into Zemax for sequential ray-tracing analysis. The root mean square spot diameter, modulation transfer function, and diffraction ensquared energy were computed for an optical system under an irregular refractive index and under thermoelastic deformation. These properties are greatly reduced by the irregular refractive index effect, which is one-third to five-sevenths the size of the thermoelastic deformation effect. Thus, thermal analyses of optical systems should consider not only thermoelastic deformation but also refractive index irregularities caused by inhomogeneous temperature.

Ill-Conditioning in Matlab Computation of Optimal Control with Time- Delays

A direct transcription method transforms an optimal control problem (OCP) into a nonlinear programming problem (NLP). The resulting NLP can be solved by any NLP solver, such as the Matlab's optimization toolbox, the fsqp, etc. On solving optimization problems using the Matlab's optimization toolbox does not obtain an accurate Hessian matrix at the optimal solution due to the fact that the Hessian matrix is not being evaluated directly from the optimal solution. In this paper we compute the condition numbers associated with the optimal control computation, where the classical forth-order Runge-Kutta method is used for the discretization of the state equations. The computations of optimal solutions are done for different numbers of switching points and quadrature points per a switching interval. Test examples show that the condition numbers of the active constraints, projected Hessian and the whole Lagrangian system are more likely to increase with the number of the switching intervals per a delay interval than by the number of the quadrature intervals per a switching interval. Also, the three medium scale optimization algorithm of the Matlab’s optimization toolbox give almost similar condition numbers when used to solve the optimal control problem.

Theoretical analysis and design optimization of wideband input/output branch waveguide couplers for a sheet beam traveling wave tube

The branch waveguide coupler (BWC) with an intrinsically excellent microwave property in an extremely wide frequency range was proposed to build the input/output (I/O) structure for the sheet beam traveling wave tube. The odd and even mode theory is utilized to analyze the BWC. Based on the theory, a code is written to calculate its microwave performance parameters. The numerical calculation has a good agreement with the computation simulation. To improve the efficiency of developing the BWC, a novel design optimization approach based on the MATLAB optimization tool is proposed. A BWC with nine branch waveguides is optimized using multi-objective genetic algorithm. In addition to the excellent microwave property, a compact configuration is achieved. It is predicted that the I/O structure possesses a very good performance with coupling above −0.3 dB, port reflectivity under −19 dB, and isolation less than −15 dB in a broad frequency range of 37.4–51.0 GHz, corresponding to a bandwidth of 13.6 GHz (30.7%).

Kinetic modeling of catalytic cracking of C4 alkanes over La/HZSM-5 catalysts in light olefin production

Kinetic modeling of catalytic cracking (CC) of C4 alkanes (mixture of i-butane and n-butane), over Lanthanum modified HZSM-5 catalyst has been studied at 582–630°C in a plug flow reactor. The elementary reaction steps, i.e., protolytic cracking (PC) of CC and CH bonds in alkane molecules, hydrogen transfer between butane and light olefins, olefin cracking, β-scission, adsorption and desorption were taken into account. The combination of theory of reaction routes and LHHW has been employed in the complicated CC network to drive explicit overall reaction rate of reactantand product molecules. Then, the parameter estimation and model discrimination were performed based on the minimization of mean residual sum of squares between experimental and theoretical rates. The intrinsic kinetic parameters were estimated by non-linear least-square regression using MATLAB optimization toolbox.

Design of materials using topology optimization and energy-based homogenization approach in Matlab

This paper presents a Matlab code for the optimal topology design of materials with extreme properties. For code compactness, an energy-based homogenization approach is adopted rather than the asymptotic approach. The effective constitutive parameters are obtained in terms of element mutual energies. A corresponding solution scheme with periodic boundary conditions is implemented. With a single constraint on material volume fraction, this code allows to maximize or minimize objective functions constituted by homogenized stiffness tensors such as bulk modulus, shear modulus and Poisson's ratio. The complete Matlab code built on top of the 88-line code (Andreassen et al. Struct Multidiscip Optim 43(1):1---16, 2011) is given in the Appendix.

Analysis of failure loads and optimal design of composite lattice cylinder under axial compression

Application of large-scale composite lattice cylinders in aeronautic and astronautic engineering is an effective way to reduce structure mass and achieve high mass efficiency. In this paper, structural stiffness and critical axial force of the lattice cylinder are analyzed theoretically by equivalent continuum method. For the lattice cylinder under axial compression, there are four failure modes, which are global buckling, out-of-plane strut buckling, in-plane strut buckling and strength failure. Curves of load capacity and failure maps related to the variation of non-dimensional variables are plotted respectively. According to the failure maps, failure of the large-scale lattice cylinder is generally dominated by global buckling due to relatively small strut thickness of the lattice cylinder. Based on finite element simulations, influence of four key geometrical parameters on stiffness and critical buckling force of the large-scale lattice cylinder under axial compression is discussed. Strut thickness and number of oblique strut rows have relatively great effects on the stiffness and critical buckling force, while number of horizontal strut rows should maintain a relatively small value in order to achieve high mass efficiency. Aimed to minimize mass of a specific Kagome lattice cylinder in practical engineering, a multi-parameter optimization model is built in MATLAB Optimization Toolbox™ and successfully applied to optimize the four geometrical parameters of the large-scale lattice cylinder under axial compression.

Optimization in MATLAB for Cardiac Excitation Modeling towards FPGA Standalone Simulation Tools

In past few decades, most of the modern electrophysiological concepts and methods were developed by the computational technique extensively to compute the cardiac action potential in nerve cells. Thus, tissue models consisting of a large number of single cell models cause a problem in the amount of computation required to obtain meaningful results from simulations. One of the solutions to this problem is by implementing the simulation through hardware modeling using a Field Programmable Gate Array (FPGA). Here, a research on developing a real-time simulation tool responsible for reentrant excitations in a ring of cardiac tissue based on the FitzHugh-Nagumo (FHN) model has been carried out by using a Xilinx Virtex-6 XC6VLX240T ML605 development board FPGA. In order to invest some of the time savings for creating the FPGA prototype, rapid prototyping method introduced by MathWorks which are MATLAB Simulink and its HDL Coder toolbox have been used to automate the algorithm design process by converting Simulink blocks into Hardware Description Language (HDL) code for the FPGA using a fixed-point data type in discrete-time framework. In this paper, the method and the optimization of the HDL design through the MATLAB Simulink have been discussed and the FPGA hardware performance in terms of speed, area and power consumption has also been analyzed.

DISTURBANCE REJECTION WITH A HIGHLY OSCILLATING SECOND-ORDER-LIKE PROCESS, PART VI: PPI CONTROLLER

A PPI controller is investigated for disturbance rejection associated with a highly oscillating second-order-like process. The controller is tuned using the MATLAB optimization toolbox for different error-based objective functions. The best objective function suitable for this type of controllers for the highly oscillating second order process is assigned. The unit step disturbance input time response of the control system has zero steady-state error and low response levels. The effect of the proportional gain of the PPI controller on the system dynamics is investigated. The PPI controller when used for disturbance rejection associated with the highly oscillating second-order process can compete well with PD-PI and PI-PD controllers used for the same purpose. 

A multi-layer line search method to improve the initialization of optimization algorithms

We introduce a novel metaheuristic methodology to improve the initialization of a given deterministic or stochastic optimization algorithm. Our objective is to improve the performance of the considered algorithm, called core optimization algorithm, by reducing its number of cost function evaluations, by increasing its success rate and by boosting the precision of its results. In our approach, the core optimization is considered as a sub-optimization problem for a multi-layer line search method. The approach is presented and implemented for various particular core optimization algorithms: Steepest Descent, Heavy-Ball, Genetic Algorithm, Differential Evolution and Controlled Random Search. We validate our methodology by considering a set of low and high dimensional benchmark problems (i.e., problems of dimension between 2 and 1000). The results are compared to those obtained with the core optimization algorithms alone and with two additional global optimization methods (Direct Tabu Search and Continuous Greedy Randomized Adaptive Search). These latter also aim at improving the initial condition for the core algorithms. The numerical results seem to indicate that our approach improves the performances of the core optimization algorithms and allows to generate algorithms more efficient than the other optimization methods studied here. A Matlab optimization package called “Global Optimization Platform” (GOP), implementing the algorithms presented here, has been developed and can be downloaded at: http://www.mat.ucm.es/momat/software.htm