Firefly Algorithm for solving non-convex economic dispatch problems with valve loading effect

The growing costs of fuel and operation of power generating units warrant improvement of optimization methodologies for economic dispatch (ED) problems. The practical ED problems have non-convex objective functions with equality and inequality constraints that make it much harder to find the global optimum using any mathematical algorithms. Modern optimization algorithms are often meta-heuristic, and they are very promising in solving nonlinear programming problems. This paper presents a novel approach to determining the feasible optimal solution of the ED problems using the recently developed Firefly Algorithm (FA). Many nonlinear characteristics of power generators, and their operational constraints, such as generation limitations, prohibited operating zones, ramp rate limits, transmission loss, and nonlinear cost functions, were all contemplated for practical operation. To demonstrate the efficiency and applicability of the proposed method, we study four ED test systems having non-convex solution spaces and compared with some of the most recently published ED solution methods. The results of this study show that the proposed FA is able to find more economical loads than those determined by other methods. This algorithm is considered to be a promising alternative algorithm for solving the ED problems in practical power systems.     

Stochastic radial basis function algorithms for large-scale optimization involving expensive black-box objective and constraint functions

This paper presents a new algorithm for derivative-free optimization of expensive black-box objective functions subject to expensive black-box inequality constraints. The proposed algorithm, called ConstrLMSRBF, uses radial basis function (RBF) surrogate models and is an extension of the Local Metric Stochastic RBF (LMSRBF) algorithm by Regis and Shoemaker (2007a) [1] that can handle black-box inequality constraints. Previous algorithms for the optimization of expensive functions using surrogate models have mostly dealt with bound constrained problems where only the objective function is expensive, and so, the surrogate models are used to approximate the objective function only. In contrast, ConstrLMSRBF builds RBF surrogate models for the objective function and also for all the constraint functions in each iteration, and uses these RBF models to guide the selection of the next point where the objective and constraint functions will be evaluated. Computational results indicate that ConstrLMSRBF is better than alternative methods on 9 out of 14 test problems and on the MOPTA08 problem from the automotive industry (Jones, 2008 [2]). The MOPTA08 problem has 124 decision variables and 68 inequality constraints and is considered a large-scale problem in the area of expensive black-box optimization. The alternative methods include a Mesh Adaptive Direct Search (MADS) algorithm (Abramson and Audet, 2006 [3]; Audet and Dennis, 2006 [4]) that uses a kriging-based surrogate model, the Multistart LMSRBF algorithm by Regis and Shoemaker (2007a) [1] modified to handle black-box constraints via a penalty approach, a genetic algorithm, a pattern search algorithm, a sequential quadratic programming algorithm, and COBYLA (Powell, 1994 [5]), which is a derivative-free trust-region algorithm. Based on the results of this study, the results in Jones (2008) [2] and other approaches presented at the ISMP 2009 conference, ConstrLMSRBF appears to be among the best, if not the best, known algorithm for the MOPTA08 problem in the sense of providing the most improvement from an initial feasible solution within a very limited number of objective and constraint function evaluations.     

Distributed Multi-Parametric Quadratic Programming

One of the fundamental problems in the area of large-scale optimization is to study locality features of spatially distributed optimization problems in which the variables are coupled in the cost function as well as constraints. Such problems can motivate the development of fast and well-conditioned distributed algorithms. In this paper, we study spatial locality features of large-scale multi-parametric quadratic programming (MPQP) problems with linear inequality constraints. Our main application focus is receding horizon control of spatially distributed linear systems with input and state constraints. We propose a new approach for analysis of large-scale MPQP problems by blending tools from duality theory with operator theory. The class of spatially decaying matrices is introduced to capture couplings between optimization variables in the cost function and the constraints. We show that the optimal solution of a convex MPQP is piecewise affine- represented as convolution sums. More importantly, we prove that the kernel of each convolution sum decays in the spatial domain at a rate proportional to the inverse of the corresponding coupling function of the optimization problem.     

An artificial bee colony algorithm with feasibility enforcement and infeasibility toleration procedures for cardinality constrained portfolio optimization

One of the most studied variant of portfolio optimization problems is with cardinality constraints that transform classical mean–variance model from a convex quadratic programming problem into a mixed integer quadratic programming problem which brings the problem to the class of NP-Complete problems. Therefore, the computational complexity is significantly increased since cardinality constraints have a direct influence on the portfolio size. In order to overcome arising computational difficulties, for solving this problem, researchers have focused on investigating efficient solution algorithms such as metaheuristic algorithms since exact techniques may be inadequate to find an optimal solution in a reasonable time and are computationally ineffective when applied to large-scale problems. In this paper, our purpose is to present an efficient solution approach based on an artificial bee colony algorithm with feasibility enforcement and infeasibility toleration procedures for solving cardinality constrained portfolio optimization problem. Computational results confirm the effectiveness of the solution methodology.     

控制系统参数优化的一种方法

本文给出一种在时间域里实现带稳定性约束的线性定常系统的优化方法,它不同于最优控制理论所采用的解析方法,而是建立在用计算机解非线性规划问题的基础上的。优化的目标函数采用二次型积分泛函,利用模态矩阵导出了它的一个规范化的算法,然后用增广的拉格朗日乘子法来求解。这一方法适合于那些不能满足最优控制论要求的许多实际系统。对于可稳定的系统,无论原始设计参数是否满足稳定性条件,经优化后总能获得稳定的最优解。     

A new meta-heuristic algorithm for continuous engineering optimization: harmony search theory and practice

Most engineering optimization algorithms are based on numerical linear and nonlinear programming methods that require substantial gradient information and usually seek to improve the solution in the neighborhood of a starting point. These algorithms, however, reveal a limited approach to complicated real-world optimization problems. If there is more than one local optimum in the problem, the result may depend on the selection of an initial point, and the obtained optimal solution may not necessarily be the global optimum. This paper describes a new harmony search (HS) meta-heuristic algorithm-based approach for engineering optimization problems with continuous design variables. This recently developed HS algorithm is conceptualized using the musical process of searching for a perfect state of harmony. It uses a stochastic random search instead of a gradient search so that derivative information is unnecessary. Various engineering optimization problems, including mathematical function minimization and structural engineering optimization problems, are presented to demonstrate the effectiveness and robustness of the HS algorithm. The results indicate that the proposed approach is a powerful search and optimization technique that may yield better solutions to engineering problems than those obtained using current algorithms.     

Evolutionary programming techniques for constrained optimizationproblems

Two evolutionary programming (EP) methods are proposed for handling nonlinear constrained optimization problems. The first, a hybrid EP, is useful when addressing heavily constrained optimization problems both in terms of computational efficiency and solution accuracy. But this method offers an exact solution only if both the mathematical form of the objective function to be minimized/maximized and its gradient are known. The second method, a two-phase EP (TPEP) removes these restrictions. The first phase uses the standard EP, while an EP formulation of the augmented Lagrangian method is employed in the second phase. Through the use of Lagrange multipliers and by gradually placing emphasis on violated constraints in the objective function whenever the best solution does not fulfill the constraints, the trial solutions are driven to the optimal point where all constraints are satisfied. Simulations indicate that the TPEP achieves an exact global solution without gradient information, with less computation time than the other optimization methods studied here, for general constrained optimization problems     

Hybrid Nested Partitions and Mathematical Programming Approach and Its Applications

Large-scale discrete optimization problems are difficult to solve, especially when different kinds of real constraints are considered. Conventionally, standard mathematical programming is a general approach for discrete optimization, but may suffer from the unacceptable long solution time in applications. On the other hand, some heuristics/metaheuristics methods are more powerful in finding approximate solutions efficiently, but mostly are problem and constraint dependent. In this paper, we develop a new hybrid nested partitions and mathematical programming approach, which creates compliance between mathematical programming and the heuristics/metaheuristics methods. Potentially applicable to many different types of problems, the hybrid approach can provide approximate solutions efficiently, and in the meantime can easily handle different kinds of constraints. The applications of the hybrid approach to the local pickup and delivery problem (LPDP) and the discrete facility location problem (DFLP) are presented in this paper.     

Weighted residual methods in optimal control

Weighted residual methods (WRM) afford a viable approach to the numerical solution of differential equations. Application of WRM results in the transformation of differential equations into systems of algebraic equations in the modal coefficients. This suggests that WRM can be used as a tool for reducing optimal control problems to mathematical programming problems. Thereby, the optimal control problem is replaced by the minimization of a cost function of static coefficients subject to algebraic constraints. The motivation for this approach lies in the profusion of sophisticated computational algorithms and digital computer codes for the solution of mathematical programming problems. In this note the solution of optimal control problems as mathematical programming problems via WRM is illustrated. The example presented indicates that reasonable accuracy is obtained for modest computational effort. While the simplest types of modes-polynomials and piecewise constants-are employed in this note, the ideas delineated can be applied in conjunction with cubic splines for the generation of computational algorithms of enhanced efficiency.     

A new approach to differential dynamic programming for discrete time systems

This paper proposes a new differential dynamic programming algorithm for solving discrete time optimal control problems with equality and inequality constraints on both control and state variables and proves its convergence. The present algorithm is different from differential dynamic programming algorithms developed in [10]-[15], which can hardly solve optimal control problems with inequality constraints on state variables and whose convergence has not been proved. Composed of iterative methods for solving systems of nonlinear equations, it is based upon Kuhn-Tucker conditions for recurrence relations of dynamic programming. Numerical examples show file efficiency of the present algorithm.     

Penalty guided genetic search for reliability design optimization

Reliability optimization has been studied in the literature for decades, usually using a mathematical programming approach. Because of these solution methodologies, restrictions on the type of allowable design have been made, however heuristic optimization approaches are free of such binding restrictions. One difficulty in applying heuristic approaches to reliability design is the highly constrained nature of the problems, both in terms of number of constraints and the difficulty of satisfying constraints. This paper presents a penalty guided genetic algorithm which efficiently and effectively searches over promising feasible and infeasible regions to identify a final, feasible optimal, or near optimal, solution. The penalty function is adaptive and responds to the search history. Results obtained on 33 test problems from the literature dominate previous solution techniques.     

Some aspects of truss topology optimization

The present paper studies some aspects of formulations of truss topology optimization problems. The ground structure approach-based formulations of three types of truss topology optimization problems, namely the problems of minimum weight design for a given compliance, of minimum weight design with stress constraints and of minimum weight design with stress constraints and local buckling constraints are examined. The common difficulties with the formulations of the three problems are discussed. Since the continuity of the constraint or/and objective function is an important factor for the determination of the mathematical structure of optimization problems, the issue of the continuity of stress, displacement and compliance functions in terms of the cross-sectional areas at zero area is studied. It is shown that the bar stress function has discontinuity at zero crosssectional area, and the structural displacement and compliance are continuous functions of the cross-sectional area. Based on the discontinuity of the stress function we point out the features of the feasible domain and global optimum for optimization problems with stress and/or local buckling constraints, and conclude that they are mathematical programming with discontinuous constraint functions and that they are essentially discrete optimization problems. The difference between topology optimization with global constraints such as structural compliance and that with local constraints on stress or/and local buckling is notable and has important consequences for the solution approach.     

Large scale structural optimization: Computational methods and optimization algorithms

Summary The objective of this paper is to investigate the efficiency of various optimization methods based on mathematical programming and evolutionary algorithms for solving structural optimization problems under static and seismic loading conditions. Particular emphasis is given on modified versions of the basic evolutionary algorithms aiming at improving the performance of the optimization procedure. Modified versions of both genetic algorithms and evolution strategies combined with mathematical programming methods to form hybrid methodologies are also tested and compared and proved particularly promising. Furthermore, the structural analysis phase is replaced by a neural network prediction for the computation of the necessary data required by the evolutionary algorithms. Advanced domain decomposition techniques particularly tailored for parallel solution of large-scale sensitivity analysis problems are also implemented. The efficiency of a rigorous approach for treating seismic loading is investigated and compared with a simplified dynamic analysis adopted by seismic codes in the framework of finding the optimum design of structures with minimum weight. In this context a number of accelerograms are produced from the elastic design response spectrum of the region. These accelerograms constitute the multiple loading conditions under which the structures are optimally designed. The numerical tests presented demonstrate the computational advantages of the discussed methods, which become more pronounced in large-scale optimization problems.     

Multi-objective optimization and selection for the PI control of ALSTOM gasifier problem

Based on the baseline PI control structure, the control parameters of non-linear ALSTOM gasifier benchmark problem are optimized. Firstly, taking all the input and output limits under three load conditions as constraints, the relative IAE indices at six scenarios are calculated and optimized by using multi-objective optimization algorithm NSGA-II. A set of non-dominated solutions are obtained which facilitate the further improvement on the performance under coal quality change. Then among those non-dominated solutions, the solution with best coal quality flexibility comes to the fore through a selection procedure. The simulation results show that the optimization and selection procedure presented in this paper improves the baseline PI control performance with better dynamic responses and coal quality flexibility.     

On a Stabilization Problem of Nonlinear Programming Neural Networks

Intrinsically, Lagrange multipliers in nonlinear programming algorithms play a regulating role in the process of searching optimal solution of constrained optimization problems. Hence, they can be regarded as the counterpart of control input variables in control systems. From this perspective, it is demonstrated that constructing nonlinear programming neural networks may be formulated into solving servomechanism problems with unknown equilibrium point which coincides with optimal solution. In this paper, under second-order sufficient assumption of nonlinear programming problems, a dynamic output feedback control law analogous to that of nonlinear servomechanism problems is proposed to stabilize the corresponding nonlinear programming neural networks. Moreover, the asymptotical stability is shown by Lyapunov First Approximation Principle.     

Pathbook: Cross-layer optimization for full-duplex wireless networks

Recently, Choi et al. designed the first practical full-duplex wireless system, which challenges the basic assumption in wireless communications that a radio cannot transmit and receive on the same frequency at the same time. In this paper, we study cross-layer optimization for full-duplex wireless networks, comprehensively considering various resource and social constraints. We focus on (1) the problem of allocating resources to maximize the total profit of multiple users subject to node constraints and (2) the problem of allocating resources to minimize the network power consumption subject to user rate demands and node constraints. We formulate these problems as convex programming systems. By combining Lagrangian decomposition and subgradient methods, we design distributed iterative algorithms to solve these problems, which compute the optimized user information flow (i.e. user behavior) for the network layer and the optimized node broadcast rate (i.e. node behavior) for the MAC layer. Our algorithms allow each user and each node to adjust its own behavior individually in each iteration. We analyze the convergence rate, the amount of feasibility violation, and the gap between the optimal solution and our solution in each iteration. We also use the dual space information to analyze node load constraint violation.     

A NEW EXPERIMENTAL STUDY OF GENETIC ALGORITHM AND SIMULATED ANNEALING WITH BOUNDED VARIABLES

Genetic algorithms (GA) can work in very large and complex spaces, which gives them the ability to solve many complex real-world problems. The bounded variables linear programming is formulated as genetic algorithms and simulated annealing (SA). This article demonstrates that genetic algorithms and simulated annealing are much easier to implement for solving network problems compared with constructing mathematical programming formulations, because it is a very simple matter to implement a new cost function and solution constraints when using a GA and SA. Finally, the presented results show that the genetic algorithm and simulated annealing provide a good scheduling methodology to bounded variables programming.     

Approach to decision making in fuzzy environment

A general approach to solving a wide class of optimization problems with fuzzy coefficients in objective functions and constraints is described. It is based on a modification of traditional mathematical programming methods and consists in formulating and solving one and the same problem within the framework of interrelated models with constructing equivalent analogs with fuzzy coefficients in objective function alone. This approach allows one to maximally cut off dominated alternatives from below as well as from above. The subsequent contraction of the decision uncertainty region is associated with reduction of the problem to multicriteria decision making in a fuzzy environment. The approach is applied within the context of fuzzy discrete optimization models, that is based on a modification of discrete optimization algorithms. The results of the paper are of a universal character and are already being used to solve problems of the design and control of power systems and subsystems.     

用优选法解系统可靠性最优化问题的一个简捷算法

冗余系统可靠性最优化问题在本质上属于非线性整数规划.现有的严格解法和近似解法 要求浩繁的计算,因而较简单的直接寻查法引起了重视.本文提出,用优选法解可靠性最优化 问题是值得探讨的;在一定范围内灵活掌握工程性约束条件,有可能简化可靠性最优化问题的 求解,具有实际意义. 本文采用优选法的分批试验法,对文献[8]的算法作了进一步的简化,从而提出一个更简 捷的算法.文中举例说明这种算法的应用,并和文献结果进行了比较.