Non-differentiable Objective Functions Research Articles

Economic Dispatch Using Improved Differential Evolution Approach: A Case Study of the Algerian Electrical Network

Differential evolution (DE) is a simple but powerful evolutionary optimization algorithm with continually outperforming many of the already existing stochastic and direct search global optimization techniques. DE algorithm is a new optimization method that can handle non-differentiable, nonlinear, and multimodal objective functions. This paper presents an efficient modified differential evolution algorithm for solving economic dispatch problem. A new mutation strategy of the conventional DE is suggested to improve the performance and avoid premature convergence. Numerical results on the IEEE 30 bus test system and the practical Algerian 59 bus system show that the proposed approach is faster and more robust compared with those reported recently in the literature. The comparison results prove the capability of the proposed method in real-time implementation for the economic dispatch problem.

Interactive Polyhedral Outer Approximation (IPOA) strategy for general multiobjective optimization problems

We propose an interactive polyhedral outer approximation (IPOA) method to solve a broad class of multiobjective optimization problems (MOP) with, possibly, nonlinear and nondifferentiable objective and constraint functions, and with continuous or discrete decision variables. During the interactive optimization phase, the method progressively constructs a polyhedral approximation of the decision-maker’s (DM’s) unknown preference structure and a polyhedral outer-approximation of the feasible set of MOP. The piecewise linear approximation of the DM’s preferences also provides a mechanism for testing the consistency of the DM’s assessments and removing inconsistencies; it also allows post-optimality analysis. All the feasible trial solutions are non-dominated (efficient, or Pareto-optimal) so preference assessments are made in the context of non-dominated alternatives only. Upper and lower bounds on the yet unknown optimal value are produced at every iteration, allowing terminating the search prematurely at a good-enough solution and providing information about the closeness of this solution to the optimal solution. The IPOA method includes a preliminary phase in which a limited probe of the efficient set is conducted in order to find a good initial trial solution for the interactive phase. The computational requirements of the algorithm are relatively simple. The results of an extensive computational study are reported.

A Kind of Nonlinear Programming Problem Based on Mixed Fuzzy Relation Equations Constraints

Abstract In this work, a kind of nonlinear programming problem with non-differential objective function and under the constraints expressed by a system of mixed fuzzy relation equations is investigated. First, some properties of this kind of optimization problem are obtained. Then, a polynomial-time algorithm for this kind of optimization problem is proposed based on these properties. Furthermore, we show that this algorithm is optimal for the considered optimization problem in this paper. Finally, numerical examples are provided to illustrate our algorithms.

An Improved Trigonometric Differential Evolution

Differential evolution is an efficient and powerful population-based stochastic technique capable of handling non-differentiable, non-linear and multi-modal objective functions. In order to improve its performance, this paper introduces a best-trigonometric mutation strategy and applies a crossover rate update strategy to the proposed algorithm. The performance of the proposed algorithm is investigated on a set of benchmark functions. The numerical experimental results show that the convergence rate of proposed algorithm is higher and the robustness of proposed algorithm is better than DE and TDE algorithm.

Determination of the Optimum Capacity and Type of Wind Turbine Generators in a Power System Considering Reliability and Cost

The economic and reliability benefits of adding a wind energy conversion system (WECS) to a power system are highly dependent on the wind turbine generator (WTG) installation design, i.e., the type and number of WTG. This paper presents an approach to determine the optimum installation design at each wind site considering the WTG parameters, the total cost of WECS and the power system reliability performance. The model is formulated as a combinatorial constrained optimization problem with a nonlinear, nondifferentiable objective function, which consists of the capital, maintenance and operating costs, and the costumer interruption costs. A number of constraints, including the total installed capacity at a specific wind site and the capital cost, are considered in the model. The model is solved using a genetic algorithm with adaptive crossover and mutation probabilities based on the standard deviation of chromosome fitness in a generation population. The proposed model and algorithm are tested on two power systems. The results show that the proposed approach can be used to obtain the maximum economic and reliability benefits associated with a WECS installation and is a powerful search technique for determining the optimum number and type of WTG considering reliability performance and costs.

Minimising value-at-risk in a portfolio optimisation problem using a multi-objective genetic algorithm

In this paper, we develop a general framework for market risk optimisation that focuses on VaR. The reason for this choice is the complexity and problems associated with risk return optimisation (non-convex and non-differential objective function). Our purpose is to obtain VaR efficient frontiers using a multi-objective genetic algorithm (GA) and to show the potential utility of the algorithm to obtain efficient portfolios when the risk measure does not allow calculating an optimal solution. Furthermore, we measure differences between VaR efficient frontiers and variance efficient frontiers in VaR-return space and we evaluate out-sample capacity of portfolios on both bullish and bearish markets. The results indicate the reliability of VaR-efficient portfolios on both bullish and bearish markets and a significant improvement over Markowitz efficient portfolios in the VaR-return space. The improvement decreases as the portfolios level of risk increases. In this particular case, efficient portfolios do not depend on the risk measure minimised.

Optimal system harmonic filters design

In this paper, we formulate the design harmonic filter problem taking by considering investment cost and operational constraints. System quality can generally be achieved by installing harmonic filters in industry systems, thereby reducing the harmonics. Users prioritize the performance and cost of fi lters as primary concerns. This new formulation uses a combinatorial optimization problem with a non-diff erentiable objective function. A solution methodology based on an optimization technique-simulated annealing is also proposed to determine the size of filters with minimum cost. As a result, the proposed solution methodology can off er the optimal size and minimum cost of design filters. In conclusion, a solution algorithm is developed to derive the global optimal solution and test on a 69kV industrial power system with highly promising results.

Genetic-Algorithm-Based Design of Passive Filters for Offshore Applications

With the proliferation of power electronics converters and increased use of magnetics, power lines have become highly polluted. Both passive and active filters have been used near harmonic-producing loads or at the point of common coupling to mitigate/block current harmonics. Shunt filters still dominate the harmonic compensation at medium/high voltage level, whereas active filters have been proclaimed for low/medium voltage ratings. With diverse applications involving reactive power together with harmonic compensation, passive filters are found suitable. For offshore platforms, having multiple variable frequency drives for oil exploration and the power fed through submerged power cable reduces the reactive power requirement in the power system. Under such reduced reactive power requirement, the design of passive filter conforming to the IEEE 1531 standard is presented in this paper. This paper utilizes genetic algorithm technique to design the tuned harmonic filters to compensate current harmonics together with limited reactive power compensation. The main idea behind the design is minimization of net source rms current with insertion of a passive filter. The formulation thus becomes a combinational optimization system with nondifferentiable objective function. Other environmental and operational constraints are also incorporated in the design technique to ensure their effective performance. The performance and operation of the so-designed passive filter has been studied through simulation under MATLAB environment.

Two derivative-free optimization algorithms for mesh quality improvement

High-quality meshes are essential in the solution of partial differential equations (PDEs), which arise in numerous science and engineering applications, as the mesh quality affects the solution accuracy, the solver execution time, and the problem conditioning. Mesh quality improvement is necessary when the mesh is of less than desirable quality (either from mesh generation or deformation). Nondifferentiable objective functions arise when the goal of the mesh optimization is to improve the worst quality element in the mesh. We propose two derivative-free methods for mesh optimization, namely the pattern search (PS) and multidirectional search (MDS) mesh quality improvement methods, to be used with nondifferentiable objective functions representing the overall mesh quality. Experimental results show that these two methods are successful in improving the worst quality mesh elements. The PS method yielded higher quality 2D meshes than did the MDS method; however, its execution time was longer. In the 3D case, most of the meshes converged to meshes of approximately the same quality because the initial meshes were fairly close to optimal. In 3D, the PS method required longer to execute than did the MDS method.

Design of a Control Lyapunov Function for Stabilizing Specified States

This paper focuses on the design of control Lyapunov function for control affine systems to guarantee the stability for the states of interest in a specified region. Without restrictive assumptions found in previous approaches, a min-max optimization problem is formulated to solve for a quadratic Lyapunov function. A derivative-free coordinate search method is employed to optimize a non-differentiable objective function. Approximation of the objective function as a piecewise linear function gradually reduces search space, leading to an effective Lyapunov function. A CSTR example with actuator saturation is illustrated to demonstrate the efficacy of the proposed approach.

In Vivo Impedance Imaging With Total Variation Regularization

We show that electrical impedance tomography (EIT) image reconstruction algorithms with regularization based on the total variation (TV) functional are suitable for in vivo imaging of physiological data. This reconstruction approach helps to preserve discontinuities in reconstructed profiles, such as step changes in electrical properties at interorgan boundaries, which are typically smoothed by traditional reconstruction algorithms. The use of the TV functional for regularization leads to the minimization of a nondifferentiable objective function in the inverse formulation. This cannot be efficiently solved with traditional optimization techniques such as the Newton method. We explore two implementations methods for regularization with the TV functional: the lagged diffusivity method and the primal dual-interior point method (PD-IPM). First we clarify the implementation details of these algorithms for EIT reconstruction. Next, we analyze the performance of these algorithms on noisy simulated data. Finally, we show reconstructed EIT images of in vivo data for ventilation and gastric emptying studies. In comparison to traditional quadratic regularization, TV regularization shows improved ability to reconstruct sharp contrasts.

A hybrid PSO technique for procuring VAR ancillary service in the deregulated electricity markets

This paper develops a new market-based technique for acquiring VAR ancillary service in the electricity market. The main objective of the developed market is to enable transmission operator “TO” to procure VAR service in a long term contract from the critical VAR providers that satisfy minimum VAR service payment while maintaining system security. Reactive power control problem for voltage stability is introduced into the VAR market problem in an explicit manner for normal and emergency states. An integration of particle swarm optimization “PSO” is presented with successive linear programming “SLP” for dealing with the VAR ancillary service problem. The problem is formulated as a large-scale nonlinear constrained optimization problem with a non-differentiable objective function representing VAR payment and operational costs. This type of problem is hard to be treated straightforwardly by the classical optimization methods. Therefore, we propose here a two-layer hybrid PSO/SLP approach, which is suited for carrying out the difficulties associated with non-differentiable and discontinuous objective functions. The proposed method has been examined on the standard IEEE 57 bus-system and compared with GA/SLP method to demonstrate its capability.

Probabilistic model identification of uncertainties in computational models for dynamical systems and experimental validation

We present a methodology to perform the identification and validation of complex uncertain dynamical systems using experimental data, for which uncertainties are taken into account by using the nonparametric probabilistic approach. Such a probabilistic model of uncertainties allows both model uncertainties and parameter uncertainties to be addressed by using only a small number of unknown identification parameters. Consequently, the optimization problem which has to be solved in order to identify the unknown identification parameters from experiments is feasible. Two formulations are proposed. The first one is the mean-square method for which a usual differentiable objective function and an unusual non-differentiable objective function are proposed. The second one is the maximum likelihood method coupling with a statistical reduction which leads us to a considerable improvement of the method. Three applications with experimental validations are presented in the area of structural vibrations and vibroacoustics.

Some remarks on the Minty vector variational principle

In scalar optimization it is well known that a solution of a Minty variational inequality of differential type is a solution of the related optimization problem. This relation is known as “Minty variational principle.” In the vector case, the links between Minty variational inequalities and vector optimization problems were investigated in [F. Giannessi, On Minty variational principle, in: New Trends in Mathematical Programming, Kluwer Academic, Dordrecht, 1997, pp. 93–99] and subsequently in [X.M. Yang, X.Q. Yang, K.L. Teo, Some remarks on the Minty vector variational inequality, J. Optim. Theory Appl. 121 (2004) 193–201]. In these papers, in the particular case of a differentiable objective function f taking values in R m and a Pareto ordering cone, it has been shown that the vector Minty variational principle holds for pseudoconvex functions. In this paper we extend such results to the case of an arbitrary ordering cone and a nondifferentiable objective function, distinguishing two different kinds of solutions of a vector optimization problem, namely ideal (or absolute) efficient points and weakly efficient points. Further, we point out that in the vector case, the Minty variational principle cannot be extended to quasiconvex functions.

Optimal Design for LC Power Harmonic Filters

This study describes a novel method for designing LC-type (single-tuned) filters to reduce harmonic distortion. The proposed method minimizes the total cost of the filters, such that the harmonic distortion is within an acceptable range. The filter design problem is formulated as a combinatorial optimization problem with a nondifferentiable objective function and multiple constraints. The objective function of the filter design problem is the filter cost function, which includes both purchasing and installation costs. The harmonic current limit for the given utility, the system impedance variations, and resonance conditions are also considered in the problem formulation. Furthermore, a solution methodology based on an optimization technique-simulated annealing-is proposed to determine the size of filters that can be made at minimum cost. Test cases with promised results are reported.

A novel ACS-based optimum switch relocation method

In this paper, a cooperative agent algorithm, the ant colony system (ACS), for optimum switch relocation is proposed. Switch relocation is a useful tool for distribution automation, since it can reduce the interruption costs without additional capital investments. The formulation of switch relocation is a combinatorial optimization problem with nonlinear and nondifferential objective functions. In this paper, the authors choose the ACS to solve the problem since the ACS has the characteristics of positive feedback, distributed computation, and the use of a constructive greedy heuristic. One of the main goals of this paper is to investigate the applicability of the ACS-based algorithm in the power system optimization problems. Test results show that the proposed ACS-based algorithm can offer a near-optimum solution for switch relocation. The comparisons of the proposed method with a genetic-algorithm (GA)-based method are also shown in the test results to demonstrate the values of the proposed method.

An evolutionary model for simultaneously optimizing three-dimensional highway alignments

Optimizing highway alignments is a very complex engineering problem. The factors that should be considered in the design process are complex and interrelated. Although several mathematical models have been developed to solve the alignment optimization problem, most of them emphasize either horizontal or vertical alignments, and only yield a suboptimal solution to the problem. Models for simultaneously optimizing three-dimensional alignments are rare in the literature and their capabilities are quite limited. In this paper, an evolutionary model (a search algorithm that imitates the natural evolution process) for solving three-dimensional alignment optimization problems is developed. It overcomes some drawbacks in existing models. The cost components and design constraints embedded in it can be comprehensive. The proposed algorithm can optimize complex, comprehensive, and non-differentiable objective function. The model can also exploit detailed geographical information for highway analysis. The resulting alignments are smooth everywhere and can have backward bends (i.e., “backtracking”) to better fit terrain and land-use patterns. A numerical example is presented to illustrate the proposed model and the performance of the solution algorithm.

OPTIMIZATION OF COMPOSITE LAMINATES WITH CUTOUTS USING GENETIC ALGORITHM, VARIABLE METRIC AND COMPLEX SEARCH METHODS

Optimization of composite laminates with cutouts is a complex problem, involving non-differentiable objective function and constraints. Choice of the optimization method is generally based on the nature and complexity of the objective function, constraints and how easily and/or accurately the first derivatives can be found. Many researchers have attempted and applied different classical optimization techniques for non-convex optimization problems. This paper clearly brings out the advantages of a non-traditional optimization method-Genetic algorithm (GA) over conventional techniques, the limitations of conventional techniques and GA's ability to approach the global optimum in an n-dimensional search space, for composite laminates.

A graph search algorithm for optimal placement of fixed and switched capacitors on radial distribution systems

Proper placement of fixed and switched shunt capacitors on primary distribution feeders reduces power and energy losses, increases the available capacity of the feeders and improves the feeder voltage profile. This paper presents a practical solution technique to the capacitor placement problem that is easy to implement. The proposed approach uses a graph search algorithm. The method has been tested along with other methods on existing feeders using actual data. Test results indicate that the proposed approach is well suited to the problem and determines a near-optimal solution. The method can handle standard capacitor sizes and associated costs, and the resulting nondifferentiable objective function, with ease. It can also determine the type of capacitors to be installed, and the switching times of the switched ones. The algorithm is currently being used for placement of shunt capacitors on distribution systems of a large electric utility.

Lipschitz-Type Stability in Nonsmooth Convex Programs

This paper deals with upper Lipschitzian continuity of the optimal solution to parametrized convex programs with linear equality and inequality constraints and with a convex nondifferentiable objective function. Under quadratic growth conditions for the objective function, some accurate bound for the rate of the upper Lipschitzian continuity is provided.