Multidimensional Optimization Problem Research Articles

Generating $\alpha $-dense curves in non-convex sets to solve a class of non-smooth constrained global optimization

This paper deals with the dimensionality reduction approach to study multi-dimensional constrained global optimization problems where the objective function is non-differentiable over a general compact set $D$ of $\mathbb{R}^{n}$ and H\{o}lderian. The fundamental principle is to provide explicitly a parametric representation $x_{i}=\ell _{i}(t),1\leq i\leq n$ of $\alpha $-dense curve $\ell_{\alpha }$ in the compact $D$, for $t$ in an interval $\mathbb{I}$ of $\mathbb{R}$, which allows to convert the initial problem to a one dimensional H\{o}lder unconstrained one. Thus, we can solve the problem by using an efficient algorithm available in the case of functions depending on a single variable. A relation between the parameter $\alpha $ of the curve $\ell _{\alpha }$ and the accuracy of attaining the optimal solution is given. Some concrete $\alpha $ dense curves in a non-convex feasible region $D$ are constructed. The numerical results show that the proposed approach is efficient.

Low-Complexity DOA Estimation Based on Constraint Solution Space

The Weighted Subspace Fitting (WSF) algorithm is one of the universal algorithms in Direction-Of-Arrival (DOA) estimation, which is of high accuracy. However, it involves the multi-dimensional nonlinear optimization problem, and the computational complexity is usually high. In this paper, we propose a low-complexity DOA estimation algorithm based on constraint solution space. Firstly, we use ESPRIT algorithm to limit the solution space around the best solution and reduce the computational range. Then, we find the best solution in a smaller solution space constraint by Cramr-Rao Bound (CRB), and seek repeatedly until reaching the global optimal solution of WSF algorithm by using the space of the best solution. By limiting the searching process in smaller solution space, this strategy controls the direction of convergence and reduces computational complexity. The experimental results show that this algorithm needs less iterations when the same DOA accuracy is required, and the computational complexity is apparently reduced.

Electric Propulsion Systems Design Supported by Multi-Objective Optimization Strategies

Electric drive systems consisting of battery, inverter, electric motor and gearbox are applied in hybridor purely electric vehicles. The layout process of such propulsion systems is performed on system level under consideration of various component properties and their interfering characteristics. In addition, different boundary conditions are taken under account, e. g. performance, efficiency, packaging, costs. In this way, the development process of the power train involves a broad range of influencing parameters and periphery conditions and thus represents a multi-dimensional optimization problem. Stateof-the-art development processes of mechatronic systems are usually executed according to the V-model, which represents a fundamental basis for handling the complex interactions of the different disciplines involved. In addition, stage-gate processes and spiral models are applied to deal with the high level of complexity during conception, design and testing. Involving a large number of technical and economic factors, these sequential, recursive processes may lead to suboptimal solutions since the system design processes do not sufficiently consider the complex relations between the different, partially conflicting domains. In this context, the present publication introduces an integrated multi-objective optimization strategy for the effective conception of electric propulsion systems, which involves a holistic consideration of all components and requirements in a multi-objective manner. The system design synthesis is based on component-specific Pareto-optimal designs to handle performance, efficiency, package and costs for given system requirements. The results are displayed as Pareto-fronts of electric power train system designs variants, from which decision makers are able to choose the best suitable solution. In this way, the presented system design approach for the development of electrically driven axles enables a multi-objective optimization considering efficiency, performance, costs and package. It is capable to reduce development time and to improve overall system quality at the same time.

Parameter estimation for fractional-order chaotic systems by improved bird swarm optimization algorithm

The essence of parameter estimation for fractional-order chaotic systems is a multi-dimensional parameter optimization problem, which is of great significance for implementing fractional-order chaos control and synchronization. Aiming at the parameter estimation problem of fractional-order chaotic systems, an improved algorithm based on bird swarm algorithm is proposed. The proposed algorithm further studies the social behavior of the original bird swarm algorithm and optimizes the foraging behavior in the original bird swarm algorithm. This method is applied to parameter estimation of fractional-order chaotic systems. Fractional-order unified chaotic system and fractional-order Lorenz system are selected as two examples for parameter estimation systems. Numerical simulation shows that the algorithm has better convergence accuracy, convergence speed and universality than bird swarm algorithm, artificial bee colony algorithm, particle swarm optimization and genetic algorithm.

Multilevel Thresholding based Image Segmentation using Whale Optimization Algorithm

Whale Optimization Algorithm (WOA) was proposed by Seyedali Mirjalili and Andrew Lewis in 2016. WOA is nature-inspired, meta-heuristic (randomization and deterministic) algorithm, which is being used to solve various single objective, multi objective and multi-dimensional optimization problems. To determine threshold value for image segmentation Otsu, kapur, thresholding etc. methods are used. In this paper multilevel threshold values are computed using WOA and these multilevel threshold values are used for image segmentation. Fitness is computed using Otsu thresholding. Minimum fitness score is considered as best optimal value. WOA has capability to explore, exploit the search s pace and avoid local optima. In multilevel thresholding, complex images are segmented into L+1 levels for multiple threshold values L =2, 3 etc. This paper addresses about performance of Whale Optimization Algorithm (WOA) and Particle Swarm Optimization (PSO) for various benchmark objective functions such as unimodel, multimodel, fix dimension multimodel based on their convergence curves for different number of iterations400,500 600 and compute multilevel threshold values for various level image segmentation using Whale Optimization Algorithm.

Stochastic global optimization using tangent minorants for Lipschitz functions

This paper deals with global stochastic optimization where the decision variable belongs to a compact subset X of R. The objective function is the mathematical expectation of a partial bivariate Lipschitz function fx,Θ depending on a decision variable x and a random variableΘ, whose probability distribution depends on x. In the first part of the present paper, we propose a branch and bound algorithm based on tangent minorants that provides a global minimum. In the second part, we consider the case where the function f is discontinuous. We show the manner we correct that discontinuity without modifying the global minimum of Efx,Θ. We also illustrate how to extend this framework to multidimensional stochastic optimization problems by using the Alienor method. Then we validate the proposed method by applying it to some test functions and compare it to known algorithms.

Saddle point criteria for multi-dimensional control optimisation problem involving first-order PDE constraints

In this paper, the multi-dimensional control optimisation problem involving first-order PDE constraints is considered. The absolute value penalty function method is used to solve the considered problem. Further, the saddle point of the Lagrange functional defined for the aforesaid control optimisation problem is discussed. Thereafter, the exactness property of the absolute value penalty function method is studied with the help of saddle-point theory. Then, the relationship between a minimiser of the penalised multi-dimensional control optimisation problem and a saddle point of the aforesaid control problem is established under the convexity assumption. Moreover, we have verified the theoretical results with applications.

Parameters estimation using mABC algorithm applied to distributed tracking control of unknown nonlinear fractional-order multi-agent systems

This paper addresses the parameters estimation and distributed tracking control of unknown nonlinear fractional-order multi-agent systems (FOMASs) with external disturbances from the viewpoint of optimization. Firstly, a modified artificial bee colony (mABC) algorithm is put forward to enhance the exploration and exploitation abilities of the original ABC algorithm. Secondly, a parameters estimation approach based on the above mABC algorithm is proposed to estimate the unknown fractional orders and system parameters, where the parameters estimation is converted into a multi-dimensional functional optimization problem. Thirdly, the obtained parameters estimation values are further utilized to construct a distributed control algorithm for solving the tracking control problem. Finally, the experimental results illustrate that the proposed approaches can accurately estimate and robustly control such unknown nonlinear FOMASs.

On a modified optimal control problem with first-order PDE constraints and the associated saddle-point optimality criterion

In this paper, based on a new multidimensional optimal control problem, in short (OCP), we introduce a modified multidimensional optimal control problem (which is simpler or easier to solve than the original one) involving first-order partial differential equations (PDEs) and inequality-type constraints. Optimality conditions associated with this new optimal control problem are formulated and proved, as well. Furthermore, under some generalized convexity assumptions, we establish an equivalence between an optimal solution of (OCP) and a saddle-point associated with the Lagrange functional corresponding to the modified multidimensional optimal control problem. Also, in order to illustrate the main results and their effectiveness, some applications are presented.

Multidimensional global optimization using numerical estimates of objective function derivatives

This paper proposes a method for solving the computation-costly multidimensional global optimization problems. The method is efficient in the one-dimensional case and its combination with the nested reduction scheme competitive with optimization methods reducing multidimensional problems by using space-filling (Peano) curves. The developed method is based on an approach, in which not only the minimized function values but also the values of derivatives of these functions are used to increase the efficiency of global optimization. The required values of the derivatives are estimated numerically by handling the available search information. The results of the executed experiments confirm the developed approach is promising.

Enhanced global flower pollination algorithm for parameter identification of chaotic and hyper-chaotic system

The problem of system parameter identification is a fundamental problem in the field of nonlinear science, which can be described as a multidimensional optimization problem. In this paper, an enhanced global flower pollination algorithm (GFPA) is proposed for parameter identification of chaotic and hyper-chaotic systems. The motion trajectory of the flower pollination algorithm is analyzed for the first time, and the equation of the algorithm exploration phase is improved by the chaotic mapping method to ensure the convergence of the algorithm in the exploration phase. In addition, in order to improve the convergence speed of the algorithm, the update method of the exploitation phase is reset by using the best information to guide the searching. Through analysis, the proposed new algorithm can guarantee the convergence of the algorithm without increasing the time complexity. Finally, we identify and validate the system of the Lorenz, Rossler, Chen and the system of the Rossler hyper-chaotic, Chen hyper-chaotic. The experimental results show that GFPA has better identification effect.

ДЕРЖАВНЕ РЕГУЛЮВАННЯ РОЗВИТКУ РЕСУРСНОГО ПОТЕНЦІАЛУ ПІДПРИЄМСТВ ЗАЛІЗНИЧНОГО ТРАНСПОРТУ

The article presents an optimization model for efficient allocation of budget funds between structural units of JSC Ukrzaliznytsya. The model represents the problem of multidimensional nonlinear optimization with constraints, in which the dependence of the efficiency of the operation of the n-th regional branch on the volume of received investments is described by the logistic function. The definition of the efficiency of regional branches of rail transport, which is identified with the level of development of their resource potential, is given, and its mathematical description is given also. The model includes such exogenous parameters as: the number of regional branches, the assessment of efficiency of their functioning, the amount of state funds allocated to increase the efficiency of JSC “Ukrzaliznytsya”, the level of depreciation of fixed assets of regional branches and the assessment of the significance of each of the regional branches. The model is based on the assumption that the level of efficiency of regional branches at the beginning of the planning period is known, the planning period is one year. Next, the article gives a solution to the above problem by the Lagrange multiplier method, while this solution is represented as a system of differential equations. This system of equations is solved using Newton's method. The advantage of the author's model is that it allows to maintain the achieved level of development of the resource potential of the railway industry and to direct funds, first of all, to those structural units of JSC Ukrzaliznytsya, which most need to increase the level of development of resource potential, and, consequently, the efficiency of functioning . The author's model requires further research and testing on the basis of the data of the economic accounting of existing railway transport enterprises.

Maximum Likelihood Joint Angle and Delay Estimation from Multipath and Multicarrier Transmissions with Application to Indoor Localization over IEEE 802.11ac Radio

In this paper, we tackle the problem of joint angle and delays estimation (JADE) of multiple reflections of a known signal impinging on multiple receiving antennae. Based on the importance sampling (IS) concept, we propose a new non-iterative maximum likelihood (ML) estimator that enjoys guaranteed global optimality and enhanced high-resolution capabilities for both single- and multi-carrier models. The new ML approach succeeds in transforming the original multi-dimensional optimization problem into multiple two-dimensional ones thereby resulting in huge computational savings. Moreover, it does not suffer from the off-grid problems that are inherent to most existing JADE techniques. By exploiting the sparsity feature of a carefully designed pseudo-pdf that is intrinsic to the new estimator, we also propose a novel approach that enables the accurate detection of the unknown number of paths over a wide range of practical signal-to-noise ratios (SNRs). Computer simulations show the distinct advantage of the new ML estimator over state-of-the art JADE techniques both in the single- and multi-carrier scenarios. Most remarkably, they suggest that the proposed IS-based ML JADE is statistically efficient as it almost reaches the Camer-Rao lower bound (CRLB) even in the adverse conditions of low SNR levels. Using real-world channel measurements collected from four access points (APs) with IEEE 802.11ac standard’s setup parameters in an indoor environment, we also show that the proposed ML estimator achieves a localization performance below 15 cm accuracy.

On the Rank-One Approximation of Positive Matrices Using Tropical Optimization Methods

An approach to the problem of rank-one approximation of positive matrices in the Chebyshev metric in logarithmic scale is developed in this work, based on the application of tropical optimization methods. The theory and methods of tropical optimization constitute one of the areas of tropical mathematics that deals with semirings and semifields with idempotent addition and their applications. Tropical optimization methods allow finding a complete solution to many problems of practical importance explicitly in a closed form. In this paper, the approximation problem under consideration is reduced to a multidimensional tropical optimization problem, which has a known solution in the general case. A new solution to the problem in the case when the matrix has no zero columns or rows is proposed and represented in a simpler form. On the basis of this result, a new complete solution of the problem of rank-one approximation of positive matrices is developed. To illustrate the results obtained, an example of the solution of the approximation problem for an arbitrary two-dimensional positive matrix is given in an explicit form.

Design and Implementation of Adaptive Multi-Tap Analog Interference Canceller

Due to the severity of self-interferences, analog interference canceller is an essential part of a full duplex radio system. As the self-interference can potentially be even much stronger than the received signal, it must be meritoriously suppressed before analog-to-digital conversion. This paper is aimed to provide an optimal radio frequency domain multi-tap interference cancellation design. To this end, the following two key problems are addressed: the hardware design problem and the adaptive weight adjusting problem. For optimal hardware design, a trade-off between widening effective delay range and maximizing the minimum cancellation capability is formulated. The principles and guidelines for optimal solution and hardware parameter design are derived. We also formulate adaptive weight adjusting as a multi-dimensional optimization problem, which is proved to be non-convex while adjusting the attenuators and phase shifters independently. By using a coordinate transformation technique, it is then converted into a convex optimization problem, and a convergent iterative algorithm based on the gradient descent method is applied to obtain the global optimum. The hardware implementation and experimental results substantiated the effectiveness of the proposed principles and algorithms.

Improved Particle Swarm Optimization Using Wolf Pack Search

Particle Swarm Optimization (PSO) has excellent global exploration ability, but its local exploitation ability is not ideal. Wolf Pack Search (WPS), which is abstracted from the intelligent predatory behavior of the wolf pack, is an excellent local exploitation strategy and can be used to replace or improve the local exploitation capabilities of other heuristic algorithms. In order to improve the local exploitation ability of Particle Swarm Optimization without affecting the global exploration ability, a hybrid improved algorithm, named as WPS-PSO, based on wolf pack search is proposed. Though the simulation of fixed-dimension and multi-dimensional benchmark functions, and compared with the simulation results of the basic particle swarm algorithm, θ-PSO and Quantum Particle Swarm Optimization (QPSO), the results of 50 times simulations show that wolf pack search can improve the local exploit ability of PSO, and can better solve the multi-dimensional optimization problem.

Dimension-by-dimension enhanced cuckoo search algorithm for global optimization

Cuckoo search (CS) algorithm is an efficient meta-heuristic algorithm that has been successfully applied in many fields. However, the algorithm uses the whole updating and evaluating strategy on solutions. For solving multi-dimensional optimization problems, solutions with partial dimension evolution may be discarded due to mutual interference among dimensions. Therefore, this strategy may deteriorate the quality solution and convergence rate of algorithm. To overcome this defect and enhance the algorithm performance, a dimension-by-dimension enhanced CS algorithm is proposed. In the global explorative random walk, the improved algorithm uses the dimension-by-dimension updating and evaluating strategy on solutions. This strategy combines the updated values of each dimension with the values of other dimensions into a new solution. In addition, a greedy strategy is adopted to accept new solution and the search center is set as the current optimal solution. The proposed algorithm was tested on fourteen well-known benchmark functions. The numerical results show that the improved algorithm can effectively enhance the quality solution and convergence rate for the global optimization problems.

An algorithm inspired by bee colonies coupled to an adaptive penalty method for truss structural optimization problems

The constrained optimization problems are very common in the engineering field. For instance, in civil, aeronautical, mechanical engineering and so on, this type of problem is largely used to find the best designs of structures leading to a structural optimization problem to be solved. Commonly, these problems consist in to find structures with the minimum weight, subject to a set of constraints such as allowable stress, displacements, natural frequencies of vibration and stability criteria. Besides the traditional optimization methods, consolidated through the decades, the evolutionary algorithms, in general inspired by natural phenomenona, have been playing an important role showing robustness to solve this kind of problem. In 2005, the artificial bee colony algorithm (ABC), inspired by the foraging of bee colonies, was proposed to solve multimodal and multidimensional optimization problems. This paper proposes, analyzes and discusses the coupling of ABC to variants of an adaptive penalty method, handling the constraints, to solve traditional problems of truss structural optimization. The results obtained are compared with the literature showing that the proposed strategy can be efficient and competitive.

Global best-guided oppositional algorithm for solving multidimensional optimization problems

This paper presents an alternative optimization algorithm to the literature optimizers by introducing global best-guided oppositional-based learning method. The procedure at hand uses the active and recent manipulation schemes of oppositional learning procedure by applying some modifications to them. The first part of the algorithm deals with searching the optimum solution around the current best solution by means of the ensemble learning-based strategy through which unfeasible and semi-optimum solutions have been straightforwardly eliminated. The second part of the algorithm benefits the useful merits of the quasi-oppositional learning strategy to not only improve the solution diversity but also enhance the convergence speed of the whole algorithm. A set of 22 optimization benchmark functions have been solved and corresponding results have been compared with the outcomes of the well-known literature optimization algorithms. Then, a bunch of parameter estimation problem consisting of hard-to-solve real world applications has been analyzed by the proposed method. Following that, eight widely applied constrained benchmark problems along with well-designed 12 constrained test cases proposed in CEC 2006 session have been solved and evaluated in terms of statistical analysis. Finally, a heat exchanger design problem taken from literature study has been solved through the proposed algorithm and respective solutions have been benchmarked against the prevalent optimization algorithms. Comparison results show that optimization procedure dealt with in this study is capable of achieving the utmost performance in solving multidimensional optimization algorithms.

A Novel Multicomponent PSO Algorithm Applied in FDE–AJTF Decomposition

The echo of maneuvering targets can be expressed as a multicomponent polynomial phase signal (mc-PPS), which should be processed by time frequency analysis methods, while, as a modified maximum likelihood (ML) method, the frequency domain extraction-based adaptive joint time frequency (FDE–AJTF) decomposition method is an effective tool. However, the key procedure in the FDE–AJTF method is searching for the optimal parameters in the solution space, which is essentially a multidimensional optimization problem with different extremal solutions. To solve the problem, a novel multicomponent particle swarm optimization (mc-PSO) algorithm is presented and applied in the FDE–AJTF decomposition with the new characteristic that can extract several components simultaneously based on the feature of the standard PSO, in which the population is divided into three groups and the neighborhood of the best particle in the optimal group is set as the forbidden area for the suboptimal group, and then two different independent components can be obtained and extracted in one extraction. To analyze its performance, three simulation tests are carried out and compared with a standard PSO, genetic algorithm, and differential evolution algorithm. According to the tests, it is verified that the mc-PSO has the best performance in that the convergence, accuracy, and stability are improved, while its searching times and computation are reduced.