Some efficient algorithms for a class of abstract optimization problems arising in optimal control

Three abstract optimization problems are presented along with doubly iterative algorithms for their numerical solution. These algorithms are generalizations of particular algorithms described by Barr and Gilbert [19], [21] and Fujisawa and Yasuda [22]. The supporting theory is fully developed along with proofs of convergence. Practical aspects of computations are considered and procedures which insure rapid convergence are discussed. Two applications to discrete-time optimal control problems are described.     

On level sets and an approximation problem for the numerical solution of a free boundary problem

In optimization theory an essential assumption for the existence of solutions as well as the convergence of algorithms is the compactness of the so-called level sets. Here some general tools for the proof of this property in the case of minimum norm problems are provided. Finally for an approximation problem suggested in [3] for the numerical solution of a class of free boundary problems, compactness of the corresponding sets is shown. Therewith for this approximation problem, existence of a solution and in addition global convergence of the family of algorithms presented in [4] is guaranteed.     

Variational stability of infinite dimensional optimal control problems

We examine the variational stability of infinite dimensional optimal control problems governed by non-linear evolution equations. Our tools are the Kuratowski-Mosco convergence of sets and the corresponding τ-convergence of functions. We prove the τ-convergence of cost functionals, the convergence of the values of the problems and we examine the variational stability of the solution and reachable sets. These results are then applied to a sequence of non-linear parabolic distributed parameter optimal control problems.     

混合遗传算法求解高超音速飞行器弹道优化

高超音速巡航导弹最优上升轨道设计问题是终端时刻未定、终端约束苛刻的最优控制问题,经典算法求解这类问题时对初值选取敏感、局部收敛等问题表现得比较突出.针对上述问题,将具有良好全局收敛性的遗传算法应用到导弹最优上升段设计问题求解中,为了提高遗传算法的收敛速度和克服早熟问题,结合单纯形和Powell算法的优点,设计了两种混合遗传算法.通过所设计的两种混合遗传算法的求解结果和分别用单纯形以及Powell算法的求解结果进行比较,得出所设计的混合遗传算法是更有效的求解高超声速巡航导弹轨迹优化的方法.     

Multi-objective optimal power flow using quasi-oppositional teaching learning based optimization

This paper describes teaching learning based optimization (TLBO) algorithm to solve multi-objective optimal power flow (MOOPF) problems while satisfying various operational constraints. To improve the convergence speed and quality of solution, quasi-oppositional based learning (QOBL) is incorporated in original TLBO algorithm. The proposed quasi-oppositional teaching learning based optimization (QOTLBO) approach is implemented on IEEE 30-bus system, Indian utility 62-bus system and IEEE 118-bus system to solve four different single objectives, namely fuel cost minimization, system power loss minimization and voltage stability index minimization and emission minimization; three bi-objectives optimization namely minimization of fuel cost and transmission loss; minimization of fuel cost and L-index and minimization of fuel cost and emission and one tri-objective optimization namely fuel cost, minimization of transmission losses and improvement of voltage stability simultaneously. In this article, the results obtained using the QOTLBO algorithm, is comparable with those of TLBO and other algorithms reported in the literature. The numerical results demonstrate the capabilities of the proposed approach to generate true and well-distributed Pareto optimal non-dominated solutions of the multi-objective OPF problem. The simulation results also show that the proposed approach produces better quality of the individual as well as compromising solutions than other algorithms.     

Convergence analysis of a computational method for optimal control of non-linear differential-algebraic systems

The convergence analysis of a computational method for optimal control problems of non-linear differential-algebraic systems is considered. The class of admissible controls is taken to be the class of piecewise smooth functions. A control parametrizution technique is used to approximate the optimal control problem into a sequence of optimal parameter selection problems. The solution of each of these approximate problems gives rise to a suboptimal solution to the original optimal control problem in an obvious way. The gradients of the cost functional with respect to parameters are derived. Furthermore, the error bounds between the suboptimal costs and the true optimal cost are derived.     

Convexity and the maximum principle for discrete systems

Halkin has given a derivation of the discrete maximum principle using a convexity requirement. An example given in this paper shows that incorrect results may be obtained when Halkin's convexity requirement is not met. There are, however, systems that do not satisfy the convexity requirement, but for which there is still a maximum principle. The discrete maximum principle is rederived with a requirement, directional convexity, that is weaker than convexity and which considerably extends its applicability. Though convexity has appeared to be basic in the development of optimal control theory, it is only the weaker property of directional convexity which is required for much of the development.     

Contraction analysis of non-linear distributed systems

Contraction theory is a comparatively recent dynamic analysis and non-linear control system design tool based on an exact differential analysis of convergence. This paper extends contraction theory to local and global stability analysis of important classes of non-linear distributed dynamics, such as convection-diffusion-reaction processes, Lagrangian and Hamilton–Jacobi dynamics, and optimal controllers and observers. By contrast with stability proofs based on energy dissipation, stability and convergence can be determined for energy-based systems excited by time-varying inputs.

The Hamilton–Jacobi–Bellman controller and a similar optimal non-linear observer design are studied based on explicitly computable conditions on the convexity of the cost function. These stability conditions extend the well-known conditions on controllability and observability Grammians for linear time-varying systems, without requiring the unknown transition matrix of the underlying differential dynamics.     

Application of the small parameter method to the singularly perturbed linear-quadratic optimal control problem

This paper considers a control problem for a linear singularly perturbed system with minimum energy. The terminal state of the system and the transition time are given. We construct asymptotic approximations of optimal programmed control and optimal feedback control in the problem. A main advantage of the proposed algorithms is decomposing the initial optimal control problem into two unperturbed problems of smaller dimension.     

Comment on “Stability of linear stationary systems with time delay”

Quadratic programming (QP) has previously been applied to the computation of optimal controls for linear systems with quadratic cost criteria. This paper extends the application of QP to non-linear problems through quasi-linearization and the solution of a sequence of linear-quadratic sub-problems whose solutions converge to the solution of the original non-linear problem. The method is called quasi-linearization-quadratic programming or Q-QP.

The principal advantage of the Q-QP method lies in the ease with which optimal controls can be computed when saturation constraints are imposed on the control signals and terminal constraints are imposed on the state vector. Use of a bounded-variable QP algorithm permits solution of constrained problems with a negligible increase in computing time over the corresponding unconstrained problems. Numerical examples show how the method can be applied to certain problems with non-analytic objective functions and illustrate the facility of the method on problems with constraints. The Q-QP method is shown to be competitive with other methods in computation time for unconstrained problems and to be essentially unaffected in speed for problems having saturation and terminal constraints     

求解最小费用最大流问题的信念传播算法

最小费用最大流问题是一种组合优化问题,在经济、工业等领域具有重要研究意义和应用价值.针对部分最小费用最大流问题求解算法效率较低的情况,依据最小费用最大流问题的线性规划方程,将问题模型映射为对应因子图模型,改进描述函数,给出迭代方程,设计了求解最小费用最大流问题的信念传播算法.利用迭代方程优先对最大可行流特征值进行收敛计算,得到最大流,设置最大流阈值,在此基础上进行最小费用计算,从而求得问题最优解.最后选取若干带权有向图模型进行数值实验,验证了算法的可行性及有效性,且算法在求解效率上优于部分算法.     

Optimal periodic control problem: a modified trigonometric approximation and quasi-stationarity conditions

The constrained optimal periodic control problem for a system described by differential equations and endowed with inertial controllers is considered, A sequence of discretized problems using trigonometric polynomials is proposed to approximate the problem. Instantaneous constraints for the state and control are handled by a new and more precise approach that imposes only a small number of non-linear but easily computable constraints. The convergence conditions for a sequence of optimal solutions of discretized problems are derived. The inclusion in the approximating scheme of various quasi-stationarity conditions for the control and state variables is analysed. Extension of a new approximating approach for inertialess and smooth problems is also discussed.     

One-shot methods in function space for PDE-constrained optimal control problems

We state and analyse one-shot methods in function space for the optimal control of nonlinear partial differential equations (PDEs) that can be formulated in terms of a fixed-point operator. A general convergence theorem is proved by generalizing the previously obtained results in finite dimensions. As application examples we consider two nonlinear elliptic model problems: the stationary solid fuel ignition model and the stationary viscous Burgers equation. For these problems we present a more detailed convergence analysis of the method. The resulting algorithms are computationally implemented in combination with an adaptive mesh refinement strategy, which leads to an improvement in the performance of the one-shot approach.     

Gilbert算法研究及其改进

Gilbert算法是求解最接近点对问题的一种算法,广泛应用于碰撞检测、数据分类、运动规划等领域。但是,Gilbert算法的最大缺点是在很多情况下,当它接近最优解时,收敛速度非常慢。在Gilbert算法的基础上提出一个新的迭代策略,可以减少算法的迭代次数,加快收敛速度。实验结果证明,改进后的算法求解速度和收敛速度快。     

基于预设时间收敛的分布式优化算法

考虑一类分布式优化问题,其目标是通过局部信息交互,使得局部成本函数之和构成的全局成本函数最小.针对该类问题,通过引入时基发生器(TBG),提出两种基于预设时间收敛的分布式比例积分(PI)优化算法.与现有的基于有限/固定时间收敛的分布式优化算法相比,所提出算法的收敛时间不依赖于系统的初值和参数,且可以任意预先设计.此外,在全局成本函数关于最优值点有限强凸,局部成本函数为可微的凸函数,且具有局部Lipschitz梯度的条件下,通过Lyapunov理论证明了所提算法都能实现预设时间收敛.最后,通过数值仿真验证了所提出算法的有效性.     

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.     

Linear optimal stochastic control using instantaneous output feedback‡

The problem of determining the linear feedback control of the instantaneous system output which minimizes a quadratic performance measure for a linear system with state and control-dependent noise is solved in this paper. Both the finite and infinite terminal time versions of this problem are treated. For the latter case, a sufficient condition for the existence of an optimal control is obtained. For the finite terminal time problem, it is shown that a two-point boundary value problem must be solved to realize the optimal control. For the infinite terminal time case, two non-linear matrix equations must be solved to realize the optimal control. Some discussion on the numerical methods used by the author to solve these equations is included in the paper.     

Adaptive Grey Wolf Optimization for Weightage-based Combined Economic Emission Dispatch in Hybrid Renewable Energy Systems

Nowadays, the electric power networks comprise diverse renewable energy resources, with the rapid development of technologies. In this scenario, the optimal Economic Dispatch is required by the power system due to the increment of power generation cost and ever growing demand of electrical energy. Thus, the reduction of power generation cost in terms of fuel cost and emission cost has become one of the main challenges in the power system. Accordingly, this article proposes the Grey Wolf Optimization-Extended Searching (GWO-ES) algorithm to provide the excellent solution for the problems regarding Combined Economic and Emission Dispatch (CEED). It validates the robustness of the proposed algorithm in seven Hybrid Renewable Energy Systems (HRES) test bus systems, which combines the wind turbine along with the thermal power plant. Furthermore, it compares the performance of the proposed GWO-ES algorithm with conventional algorithms such as Genetic Algorithm (GA), Particle Swarm Optimization (PSO), Differential Evolution (DE), and GWO. Next, the article emulates a valuable convergence analysis and justification for the quality of CEED through the GWO-ES algorithm. Finally, the result was compared to four other conventional algorithms to assure the efficiency of the proposed algorithm in terms of fuel cost and emission cost reduction.     

Constrained Optimal Hybrid Control of a Flow Shop System

We consider an optimal control problem for the hybrid model of a deterministic flow shop system, in which the jobs are processed in the order they arrive at the system. The problem is decomposed into a higher-level discrete-event system control problem of determining the optimal service times, and a set of lower-level classical control problems of determining the optimal control inputs for given service times. We focus on the higher-level problem which is nonconvex and nondifferentiable. The arrival times are known and the decision variables are the service times that are controllable within constraints. We present an equivalent convex optimization problem with linear constraints. Under some cost assumptions, we show that no waiting is observed on the optimal sample path. This property allows us to simplify the convex optimization problem by eliminating variables and constraints. We also prove, under an additional strict convexity assumption, the uniqueness of the optimal solution and propose two algorithms to decompose the simplified convex optimization problem into a set of smaller convex optimization problems. The effects of the simplification and the decomposition on the solution times are shown on an example problem.     

基于遗传算法的Peer-to-Peer路由算法R-GA

路由算法是制约Peer-to-Peer 系统整体性能的关键因素之一.目前大多数路由算法无法保证全局收敛,而链路延迟、费用、网络带宽等现实制约因素往往在选路时被忽略.针对上述问题,提出了基于遗传算法的R-GA路由算法.通过适度函数和遗传因子,R-GA可以快速地实现全局收敛.同时将链路的延迟、费用、带宽等参数插入到适度函数中, 避免了盲目路由.仿真试验的结果表明,R-GA路由算法在大规模Peer-to-Peer系统中是高效和可扩展的.