A semidefinite programming approach for solving fractional optimal control problems

This paper presents a numerical scheme for solving fractional optimal control. The fractional derivative in this problem is in the Riemann–Liouville sense. The proposed method, based upon the method of moments, converts the fractional optimal control problem to a semidefinite optimization problem; namely, the nonlinear optimal control problem is converted to a convex optimization problem. The Grunwald–Letnikov formula is also used as an approximation for fractional derivative. The solution of fractional optimal control problem is found by solving the semidefinite optimization problem. Finally, numerical examples are presented to show the performance of the method.     

Bimodal optimal design of vibrating plates using theory and methods of nondifferentiable optimization

This paper is concerned with an optimal design problem of vibrating plates. The optimization problem consists in maximizing the smallest eigenvalue of the elliptic eigenvalue problem describing the free plate vibration. The thickness of the plate is the variable subject to optimization. The volume of the plate is constant and the thickness of the plate is bounded.In this paper, we consider the case where the smallest eigenvalue is multiple. This implies that the optimization problem is nondifferentiable. A necessary optimality condition is formulated. The finite-element method is employed as an approximation method. A nonsmooth optimization method is used to solve this optimization problem. Numerical examples are provided.This work was supported by the Polish Academy of Sciences and the Education Ministry of Japan. Lemarechal's implementation of his method was used for numerical computations.on leave from Systems Research Institute, Warsaw, Poland.     

Successive Optimization Method via Parametric Monotone Composition Formulation*

In this paper a successive optimization method for solving inequality constrained optimization problems is introduced via a parametric monotone composition reformulation. The global optimal value of the original constrained optimization problem is shown to be the least root of the optimal value function of an auxiliary parametric optimization problem, thus can be found via a bisection method. The parametric optimization subproblem is formulated in such a way that it is a one-parameter problem and its value function is a monotone composition function with respect to the original objective function and the constraints. Various forms can be taken in the parametric optimization problem in accordance with a special structure of the original optimization problem, and in some cases, the parametric optimization problems are convex composite ones. Finally, the parametric monotone composite reformulation is applied to study local optimality.     

基于动力系统的线性不等式组的解法

本文提出了一种新的求解线性不等式组可行解的方法-基于动力系统的方法．假设线性不等式组的可行域为非空,在可行域的相对内域上建立一个非线性关系表达式,进而得到一个结构简单的动力系统模型．同时,定义了穿越方向。文章最后的数值实验结果表明此算法是有效的．     

求解线性不等式组的方法

本提出了一个新的求解线性不等式组可行解的方法－－无约束极值方法。通过在线性不等式组的非空可行域的相对内域上建立一个非线性极值问题，根据对偶关系，得到了一个对偶空间的无约束极值及原始，对偶变量之间的简单线性映射关系，这样将原来线性不等式组问题的求解转化为一个无约束极值问题。中主要讨论了求解无约束极值问题的共轭梯度算法。同时，在寻找不等式组可行解的过程中，定义了穿越方向，这样大大减少计算量。中最后数值实验结果表明此算法是有效的。     

A primal-dual interior point method for optimal zero-forcing beamformer design under per-antenna power constraints

In this paper, we consider an optimal zero-forcing beamformer design problem in multi-user multiple-input multiple-output broadcast channel. The minimum user rate is maximized subject to zero-forcing constraints and power constraint on each base station antenna array element. The natural formulation leads to a nonconvex optimization problem. This problem is shown to be equivalent to a convex optimization problem with linear objective function, linear equality and inequality constraints and quadratic inequality constraints. Here, the indirect elimination method is applied to reduce the convex optimization problem into an equivalent convex optimization problem of lower dimension with only inequality constraints. The primal-dual interior point method is utilized to develop an effective algorithm (in terms of computational efficiency) via solving the modified KKT equations with Newton method. Numerical simulations are carried out. Compared to algorithms based on a trust region interior point method and sequential quadratic programming method, it is observed that the method proposed is much superior in terms of computational efficiency.     

置换流水车间调度问题的中心引力优化算法求解

目前求解置换流水车间调度问题的智能优化算法都是随机型优化方法,存在的一个问题是解的稳定性较差。针对该问题,本文给出一种确定型智能优化算法——中心引力优化算法的求解方法。为处理基本中心引力优化算法对初始解选择要求高的问题,利用低偏差序列生成初始解,提高初始解质量;利用加速度和位置迭代方程更新解的状态;利用两位置交换排序法进行局部搜索,提高算法的优化性能。采用置换流水车间调度问题标准测试算例进行数值实验,并和基本中心引力优化算法、NEH启发式算法、微粒群优化算法和萤火虫算法进行比较。结果表明该算法不仅具有更好的解的稳定性,而且具有更高的计算精度,为置换流水车间调度问题的求解提供了一种可行有效的方法。     

基于遗传算法的车间作业计划仿真研究

将仿真技术和遗传算法相结合,根据生产车间的资源情况、优化目标等建立了生产调度仿真模型,然后对仿真输出结果进行统计,针对统计结果应用遗传算法对调度决策进行优化.仿真优化结果说明了该集成优化方法是有效性的.     

Computational Discretization Algorithms for Functional Inequality Constrained Optimization

In this paper, a functional inequality constrained optimization problem is studied using a discretization method and an adaptive scheme. The problem is discretized by partitioning the interval of the independent parameter. Two methods are investigated as to how to treat the discretized optimization problem. The discretization problem is firstly converted into an optimization problem with a single nonsmooth equality constraint. Since the obtained equality constraint is nonsmooth and does not satisfy the usual constraint qualification condition, relaxation and smoothing techniques are used to approximate the equality constraint via a smooth inequality constraint. This leads to a sequence of approximate smooth optimization problems with one constraint. An adaptive scheme is incorporated into the method to facilitate the computation of the sum in the inequality constraint. The second method is to apply an adaptive scheme directly to the discretization problem. Thus a sequence of optimization problems with a small number of inequality constraints are obtained. Convergence analysis for both methods is established. Numerical examples show that each of the two proposed methods has its own advantages and disadvantages over the other.     

A filled function method for optimal discrete-valued control problems

In this paper, we present a new approach to solve a class of optimal discrete-valued control problems. This type of problem is first transformed into an equivalent two-level optimization problem involving a combination of a discrete optimization problem and a standard optimal control problem. The standard optimal control problem can be solved by existing optimal control software packages such as MISER 3.2. For the discrete optimization problem, a discrete filled function method is developed to solve it. A numerical example is solved to illustrate the efficiency of our method.     

Scenario decomposition of risk-averse multistage stochastic programming problems

For a risk-averse multistage stochastic optimization problem with a finite scenario tree, we introduce a new scenario decomposition method and we prove its convergence. The main idea of the method is to construct a family of risk-neutral approximations of the problem. The method is applied to a risk-averse inventory and assembly problem. In addition, we develop a partially regularized bundle method for nonsmooth optimization.     

Identification of an unknown source depending on both time and space variables by a variational method

The present paper is devoted to solve the problem of identifying an unknown heat source depending simultaneously on both space and time variables. This problem is transformed into an optimization problem and the uniqueness of minimum element is proved rigorously. Then a variational formulation for solving the optimization problem is given. A conjugate gradient method and a finite difference method are used to solve the variational problem. Some numerical examples are also provided to show the efficiency of the proposed method.     

Value-Estimation Function Method for Constrained Global Optimization

A novel value-estimation function method for global optimization problems with inequality constraints is proposed in this paper. The value-estimation function formulation is an auxiliary unconstrained optimization problem with a univariate parameter that represents an estimated optimal value of the objective function of the original optimization problem. A solution is optimal to the original problem if and only if it is also optimal to the auxiliary unconstrained optimization with the parameter set at the optimal objective value of the original problem, which turns out to be the unique root of a basic value-estimation function. A logarithmic-exponential value-estimation function formulation is further developed to acquire computational tractability and efficiency. The optimal objective value of the original problem as well as the optimal solution are sought iteratively by applying either a generalized Newton method or a bisection method to the logarithmic-exponential value-estimation function formulation. The convergence properties of the solution algorithms guarantee the identification of an approximate optimal solution of the original problem, up to any predetermined degree of accuracy, within a finite number of iterations.     

Simulation of controlled uncertain nonlinear systems

In this paper a new method for the simulation of uncertain nonlinear discrete time systems is given. The problem of simulation is reformulated as an optimization problem. Using methods of global optimization, namely, interval analysis, the global solution to the optimization problem stated is computed. Derivatives of the objective function of the optimization problem are computed using methods of automatic differentiation. The presented approach is implemented on a SPARC Station using the language PASCAL-XSC. For a typical example simulation results are presented and discussed.     

A rolling window optimization method for large-scale WCDMA base stations planning problem

The large-scale base station planning problem for wideband code division multiple access (WCDMA) wireless networks is studied in this paper. A new rolling window optimization method is presented, where the global optimization problem is decomposed into small optimization sub-problems, which are defined on a series of successive rolling windows. Effective rolling strategies are designed in the rolling optimization method based on the prediction of the interference among the base stations in the WCDMA wireless network. We show that the proposed method has the property that the global objective is non-increasing in the successive optimization procedure. Simulations are carried out to analyze the performance of the proposed optimization method, which show the importance of the rolling strategy.     

不确定信息下分式半无限优化问题的近似最优性刻画

该文研究了一类带不确定参数的多目标分式半无限优化问题。首先借助鲁棒优化方法,引入该不确定多目标分式优化问题的鲁棒对应优化模型,并借助Dinkelbach方法,将该鲁棒对应优化模型转化为一般的多目标优化问题。随后借助一种标量化方法,建立了该优化问题的标量化问题,并刻画了它们的解之间的关系。最后借助一类鲁棒型次微分约束规格,建立了该不确定多目标分式优化问题拟近似有效解的鲁棒最优性条件。     

A fuzzy goal programming approach to multi-objective optimization problem with priorities

In goal programming problem, the general equilibrium and optimization are often two conflicting factors. This paper proposes a generalized varying-domain optimization method for fuzzy goal programming (FGP) incorporating multiple priorities. According to the three possible styles of the objective function, the varying-domain optimization method and its generalization are proposed. This method can generate the results consistent with the decision-maker (DM)’s expectation, that the goal with higher priority may have higher level of satisfaction. Using this new method, it is a simple process to balance between the equilibrium and optimization, and the result is the consequence of a synthetic decision between them. In contrast to the previous method, the proposed method can make that the higher priority achieving the higher satisfactory degree. To get the global solution of the nonlinear nonconvex programming problem resulting from the original problem and the varying-domain optimization method, the co-evolutionary genetic algorithms (GAs), called GENOCOPIII, is used instead of the SQP method. In this way the DM can get the optimum of the optimization problem. We demonstrate the power of this proposed method by illustrative examples.     

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.     

Riemannian Newton-type methods for joint diagonalization on the Stiefel manifold with application to independent component analysis

The joint approximate diagonalization of non-commuting symmetric matrices is an important process in independent component analysis. This problem can be formulated as an optimization problem on the Stiefel manifold that can be solved using Riemannian optimization techniques. Among the available optimization techniques, this study utilizes the Riemannian Newton’s method for the joint diagonalization problem on the Stiefel manifold, which has quadratic convergence. In particular, the resultant Newton’s equation can be effectively solved by means of the Kronecker product and the vec and veck operators, which reduce the dimension of the equation to that of the Stiefel manifold. Numerical experiments are performed to show that the proposed method improves the accuracy of the approximate solution to this problem. The proposed method is also applied to independent component analysis for the image separation problem. The proposed Newton method further leads to a novel and fast Riemannian trust-region Newton method for the joint diagonalization problem.     

Minimizing rational functions by exact Jacobian SDP relaxation applicable to finite singularities

This paper considers the optimization problem of minimizing a rational function. We reformulate this problem as a polynomial optimization problem by the technique of homogenization. These two problems are shown to be equivalent under some generic conditions. The exact Jacobian SDP relaxation method proposed by Nie is used to solve the resulting polynomial optimization problem. We also prove that the assumption of nonsingularity in Nie’s method can be weakened to the finiteness of singularities. Some numerical examples are given in the end.