State-control parameterization method based on using hybrid functions of block-pulse and Legendre polynomials for optimal control of linear time delay systems

In this paper, a method based on using hybrid functions of block-pulse and Legendre polynomials for finding the optimal solution of systems with delay in state and control variables is presented. The state-control parameterization method is used to convert the original optimal control problem with time delays into an optimization problem. This method does not require operational matrices of delay, product and integration of hybrid functions for obtaining this goal. The validity of this method is examined by illustrative examples.     

Optimal Control of Delay Systems by Using a Hybrid Functions Approximation

In this paper, a new numerical method for solving the optimal control of linear time-varying delay systems with quadratic performance index is presented. The method is based upon hybrid functions approximation. The properties of hybrid functions, consisting of block-pulse functions and Bernoulli polynomials, are presented. The operational matrices of integration, product, delay and the integration of the cross product of two hybrid functions of block-pulse and Bernoulli polynomials vectors are given. These matrices are then utilized to reduce the solution of the optimal control of delay systems to the solution of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the technique.     

Global optimization of bounded factorable functions with discontinuities

A deterministic global optimization method is developed for a class of discontinuous functions. McCormick’s method to obtain relaxations of nonconvex functions is extended to discontinuous factorable functions by representing a discontinuity with a step function. The properties of the relaxations are analyzed in detail; in particular, convergence of the relaxations to the function is established given some assumptions on the bounds derived from interval arithmetic. The obtained convex relaxations are used in a branch-and-bound scheme to formulate lower bounding problems. Furthermore, convergence of the branch-and-bound algorithm for discontinuous functions is analyzed and assumptions are derived to guarantee convergence. A key advantage of the proposed method over reformulating the discontinuous problem as a MINLP or MPEC is avoiding the increase in problem size that slows global optimization. Several numerical examples for the global optimization of functions with discontinuities are presented, including ones taken from process design and equipment sizing as well as discrete-time hybrid systems.     

Hybrid Approximate Proximal Method with Auxiliary Variational Inequality for Vector Optimization

This paper studies a general vector optimization problem of finding weakly efficient points for mappings from Hilbert spaces to arbitrary Banach spaces, where the latter are partially ordered by some closed, convex, and pointed cones with nonempty interiors. To find solutions of this vector optimization problem, we introduce an auxiliary variational inequality problem for a monotone and Lipschitz continuous mapping. The approximate proximal method in vector optimization is extended to develop a hybrid approximate proximal method for the general vector optimization problem under consideration by combining an extragradient method to find a solution of the variational inequality problem and an approximate proximal point method for finding a root of a maximal monotone operator. In this hybrid approximate proximal method, the subproblems consist of finding approximate solutions to the variational inequality problem for monotone and Lipschitz continuous mapping, and then finding weakly efficient points for a suitable regularization of the original mapping. We present both absolute and relative versions of our hybrid algorithm in which the subproblems are solved only approximately. The weak convergence of the generated sequence to a weak efficient point is established under quite mild assumptions. In addition, we develop some extensions of our hybrid algorithms for vector optimization by using Bregman-type functions.     

Fractional-Order Legendre Functions and Their Application to Solve Fractional Optimal Control of Systems Described by Integro-differential Equations

In this paper, we introduce a set of functions called fractional-order Legendre functions (FLFs) to obtain the numerical solution of optimal control problems subject to the linear and nonlinear fractional integro-differential equations. We consider the properties of these functions to construct the operational matrix of the fractional integration. Also, we achieved a general formulation for operational matrix of multiplication of these functions to solve the nonlinear problems for the first time. Then by using these matrices the mentioned fractional optimal control problem is reduced to a system of algebraic equations. In fact the functions of the problem are approximated by fractional-order Legendre functions with unknown coefficients in the constraint equations, performance index and conditions. Thus, a fractional optimal control problem converts to an optimization problem, which can then be solved numerically. The convergence of the method is discussed and finally, some numerical examples are presented to show the efficiency and accuracy of the method.     

Recognizing underlying sparsity in optimization

Exploiting sparsity is essential to improve the efficiency of solving large optimization problems. We present a method for recognizing the underlying sparsity structure of a nonlinear partially separable problem, and show how the sparsity of the Hessian matrices of the problem’s functions can be improved by performing a nonsingular linear transformation in the space corresponding to the vector of variables. A combinatorial optimization problem is then formulated to increase the number of zeros of the Hessian matrices in the resulting transformed space, and a heuristic greedy algorithm is applied to this formulation. The resulting method can thus be viewed as a preprocessor for converting a problem with hidden sparsity into one in which sparsity is explicit. When it is combined with the sparse semidefinite programming relaxation by Waki et al. for polynomial optimization problems, the proposed method is shown to extend the performance and applicability of this relaxation technique. Preliminary numerical results are presented to illustrate this claim. S. Kim’s research was supported by Kosef R01-2005-000-10271-0. M. Kojima’s research was supported by Grant-in-Aid for Scientific Research on Priority Areas 16016234.     

基于符号数值混合计算的混成系统Lyapunov函数构造

基于平方和松弛和有理向量恢复,提出了一种符号数值混合计算方法来构造多项式Lyapunov函数以判定非线性混成系统的稳定性,首先,为Lyapunov函数预定一个给定次数的多项式模板,则Lyapunov函数构造问题可转化为相应的带参数的多项式优化问题,然后运用平方和松弛方法求得一个近似的数值多项式Lyapunov函数,再应用高斯-牛顿精化和有理向量恢复将数值多项式转化为验证的有理多项式Lyapunov函数.     

Application of an optimization problem in Max-Plus algebra to scheduling problems

The problem tackled in this paper deals with products of a finite number of triangular matrices in Max-Plus algebra, and more precisely with an optimization problem related to the product order. We propose a polynomial time optimization algorithm for 2×2 matrices products. We show that the problem under consideration generalizes numerous scheduling problems, like single machine problems or two-machine flow shop problems. Then, we show that for 3×3 matrices, the problem is NP-hard and we propose a branch-and-bound algorithm, lower bounds and upper bounds to solve it. We show that an important number of results in the literature can be obtained by solving the presented problem, which is a generalization of single machine problems, two- and three-machine flow shop scheduling problems. The branch-and-bound algorithm is tested in the general case and for a particular case and some computational experiments are presented and discussed.     

Multi-objective design optimization of an engine accessory drive system with a robustness analysis

Designing a good engine accessory drive system becomes a hard work with its increasingly complicated configuration and high demands on its dynamic characteristics. In this work, a hybrid mutation particle swarm optimization (HMPSO) algorithm is presented to optimize the key structure parameters of an engine accessory drive system for its vibration control. The superiority of the HMPSO algorithm against several other concerned metaheuristic algorithms in terms of solution quality and stability are verified by non-parametric statistical tests on ten benchmark functions. The design problem of the engine accessory drive system is a multi-objective optimization problem; the weighted sum method and main target method are applied to convert it to a single-objective one. Optimization on an example engine accessory drive system using the HMPSO algorithm demonstrates obvious improvement in system vibration after optimization. A robustness analysis is conducted to identify the robustness of dynamic responses of the engine accessory drive system with respect to small variations of the design variables relative to the optimal design in the design space, and suggestions on design of an engine accessory drive system are given according to it.     

Conditional Gradient Method for Double-Convex Fractional Programming Matrix Problems

We consider the problem of optimizing the ratio of two convex functions over a closed and convex set in the space of matrices. This problem appears in several applications and can be classified as a double-convex fractional programming problem. In general, the objective function is nonconvex but, nevertheless, the problem has some special features. Taking advantage of these features, a conditional gradient method is proposed and analyzed, which is suitable for matrix problems. The proposed scheme is applied to two different specific problems, including the well-known trace ratio optimization problem which arises in many engineering and data processing applications. Preliminary numerical experiments are presented to illustrate the properties of the proposed scheme.     

Epsilon-Ritz Method for Solving a Class of Fractional Constrained Optimization Problems

In this paper, epsilon and Ritz methods are applied for solving a general class of fractional constrained optimization problems. The goal is to minimize a functional subject to a number of constraints. The functional and constraints can have multiple dependent variables, multiorder fractional derivatives, and a group of initial and boundary conditions. The fractional derivative in the problem is in the Caputo sense. The constrained optimization problems include isoperimetric fractional variational problems (IFVPs) and fractional optimal control problems (FOCPs). In the presented approach, first by implementing epsilon method, we transform the given constrained optimization problem into an unconstrained problem, then by applying Ritz method and polynomial basis functions, we reduce the optimization problem to the problem of optimizing a real value function. The choice of polynomial basis functions provides the method with such a flexibility that initial and boundary conditions can be easily imposed. The convergence of the method is analytically studied and some illustrative examples including IFVPs and FOCPs are presented to demonstrate validity and applicability of the new technique.     

Chaotic invasive weed optimization algorithm with application to parameter estimation of chaotic systems

This paper introduces a novel hybrid optimization algorithm by taking advantage of the stochastic properties of chaotic search and the invasive weed optimization (IWO) method. In order to deal with the weaknesses associated with the conventional method, the proposed chaotic invasive weed optimization (CIWO) algorithm is presented which incorporates the capabilities of chaotic search methods. The functionality of the proposed optimization algorithm is investigated through several benchmark multi-dimensional functions. Furthermore, an identification technique for chaotic systems based on the CIWO algorithm is outlined and validated by several examples. The results established upon the proposed scheme are also supplemented which demonstrate superior performance with respect to other conventional methods.     

Solving an Inverse Problem for an Elliptic Equation by d.c. Programming

An inverse problem of determination of a coefficient in an elliptic equation is considered. This problem is ill-posed in the sense of Hadamard and Tikhonov's regularization method is used for solving it in a stable way. This method requires globally solving nonconvex optimization problems, the solution methods for which have been very little studied in the inverse problems community. It is proved that the objective function of the corresponding optimization problem for our inverse problem can be represented as the difference of two convex functions (d.c. functions), and the difference of convex functions algorithm (DCA) in combination with a branch-and-bound technique can be used to globally solve it. Numerical examples are presented which show the efficiency of the method.     

Global optimization for a class of fractional programming problems

This paper presents a canonical dual approach to minimizing the sum of a quadratic function and the ratio of two quadratic functions, which is a type of non-convex optimization problem subject to an elliptic constraint. We first relax the fractional structure by introducing a family of parametric subproblems. Under proper conditions on the “problem-defining” matrices associated with the three quadratic functions, we show that the canonical dual of each subproblem becomes a one-dimensional concave maximization problem that exhibits no duality gap. Since the infimum of the optima of the parameterized subproblems leads to a solution to the original problem, we then derive some optimality conditions and existence conditions for finding a global minimizer of the original problem. Some numerical results using the quasi-Newton and line search methods are presented to illustrate our approach.     

A new hybrid method for solving global optimization problem

In this paper we present a new hybrid method, called the SASP method. The purpose of this method is the hybridization of the simulated annealing (SA) with the descent method, where we estimate the gradient using simultaneous perturbation. Firstly, the new hybrid method finds a local minimum using the descent method, then SA is executed in order to escape from the currently discovered local minimum to a better one, from which the descent method restarts a new local search, and so on until convergence.The new hybrid method can be widely applied to a class of global optimization problems for continuous functions with constraints. Experiments on 30 benchmark functions, including high dimensional functions, show that the new method is able to find near optimal solutions efficiently. In addition, its performance as a viable optimization method is demonstrated by comparing it with other existing algorithms. Numerical results improve the robustness and efficiency of the method presented.     

On optimization of continuous-time Markov networks in distributed computing

The paper presents a new stochastic model for studying the optimization of functioning rules in distributed computing. In this model a network is represented by a finite number of continuous-time homogeneous Markov processes which are connected by relations between entries of their intensity matrices. Good functioning rules are those optimizing a guide function defined according to the context. Two specific optimization problems are studied: a problem of resource allocation with conflicts between processes, and a problem of access to shared resources. The latter is a linearly constrained nonconvex problem with an objective function which is a sum of ratios of linear functions of special form.     

Solution of Volterra's population model via block‐pulse functions and Lagrange‐interpolating polynomials

A numerical method for solving Volterra's population model for population growth of a species in a closed system is proposed. Volterra's model is a nonlinear integro‐differential equation where the integral term represents the effects of toxin. The approach is based on hybrid function approximations. The properties of hybrid functions that consist of block‐pulse and Lagrange‐interpolating polynomials are presented. The associated operational matrices of integration and product are then utilized to reduce the solution of Volterra's model to the solution of a system of algebraic equations. The method is easy to implement and computationally very attractive. Applications are demonstrated through an illustrative example. Copyright © 2008 John Wiley & Sons, Ltd.     

An Improved Hybrid Adjoint Method in External Aerodynamics Using Variational Technique for the Boundary Integral Based Optimal Objective Function Gradient

An improved hybrid adjoint method to the viscous, compressible Reynold-Averaged Navier-Stokes Equation (RANS) is developed for the computation of objective function gradient and demonstrated for external aerodynamic design optimization. In this paper, the main idea is to extend the previous coupling of the discrete and continuous adjoint method by the grid-node coordinates variation technique for the computation of the variation in the gradients of flow variables. This approach in combination with the Jacobian matrices of flow fluxes refrained the objective function from field integrals and coordinate transformation matrix. Thus, it opens up the possibility of employing the hybrid adjoint method to evaluate the subsequent objective function gradient analogous to many shape parameters, comprises of only boundary integrals. This avoids the grid regeneration in the geometry for every surface perturbation in a structured and unstructured grid. Hence, this viable technique reduces the overall CPU cost. Moreover, the new hybrid adjoint method has been successfully applied to the computation of accurate sensitivity derivatives. Finally, for the investigation of the presented numerical method, simulations are carried out on NACA0012 airfoil in a transonic regime and its accuracy and effectiveness related to the new gradient equation have been verified with the Finite Difference Method (FDM). The analysis reveals that the presented methodology for the optimization provides the designer with an indispensable CPU-cost effective tool to reshape the complex geometry airfoil surfaces, useful relative to the state-of-the-art, in a less computing time.     

Hybrid Population-Based Algorithms for the Bi-Objective Quadratic Assignment Problem

We present variants of an ant colony optimization (MO-ACO) algorithm and of an evolutionary algorithm (SPEA2) for tackling multi-objective combinatorial optimization problems, hybridized with an iterative improvement algorithm and the robust tabu search algorithm. The performance of the resulting hybrid stochastic local search (SLS) algorithms is experimentally investigated for the bi-objective quadratic assignment problem (bQAP) and compared against repeated applications of the underlying local search algorithms for several scalarizations. The experiments consider structured and unstructured bQAP instances with various degrees of correlation between the flow matrices. We do a systematic experimental analysis of the algorithms using outperformance relations and the attainment functions methodology to asses differences in the performance of the algorithms. The experimental results show the usefulness of the hybrid algorithms if the available computation time is not too limited and identify SPEA2 hybridized with very short tabu search runs as the most promising variant. This research was mainly done while Luís Paquete and Thomas Stützle were with the Intellectics Group at the Computer Science Department of Darmstadt University of Technology, Germany     

A smoothing Newton's method for the construction of a damped vibrating system from noisy test eigendata

In this paper we consider an inverse problem for a damped vibration system from the noisy measured eigendata, where the mass, damping, and stiffness matrices are all symmetric positive‐definite matrices with the mass matrix being diagonal and the damping and stiffness matrices being tridiagonal. To take into consideration the noise in the data, the problem is formulated as a convex optimization problem involving quadratic constraints on the unknown mass, damping, and stiffness parameters. Then we propose a smoothing Newton‐type algorithm for the optimization problem, which improves a pre‐existing estimate of a solution to the inverse problem. We show that the proposed method converges both globally and quadratically. Numerical examples are also given to demonstrate the efficiency of our method. Copyright © 2008 John Wiley & Sons, Ltd.