An SQP feasible descent algorithm for nonlinear inequality constrained optimization without strict complementarity

In this paper, a kind of nonlinear optimization problems with nonlinear inequality constraints are discussed, and a new SQP feasible descent algorithm for solving the problems is presented. At each iteration of the new algorithm, a convex quadratic program (QP) which always has feasible solution is solved and a master direction is obtained, then, an improved (feasible descent) direction is yielded by updating the master direction with an explicit formula, and in order to avoid the Maratos effect, a height-order correction direction is computed by another explicit formula of the master direction and the improved direction. The new algorithm is proved to be globally convergent and superlinearly convergent under mild conditions without the strict complementarity. Furthermore, the quadratic convergence rate of the algorithm is obtained when the twice derivatives of the objective function and constrained functions are adopted. Finally, some numerical tests are reported.     

Reply to “Further Comment on “Optimal Fuzzy Controller Design: Local Concept Approach””

We proposed two approaches to solve a quadratic optimal problem of Takagi-Sugeno (T-S) fuzzy-model-based nonlinear systems. These two approaches are totally different in both concept and derivation even they deal with the same issue and the adopted notations look similar. Readers are suggested to clear local-concept approach away from their brains and reexamine the global-concept approach. Then, you will find out that in global-concept approach, via the proposed synthetical matrices, a quadratic optimal fuzzy problem is transformed into a general nonlinear quadratic optimal problem; a numerical approach (dynamic decomposition algorithm) is further introduced to speed up numerical solution and to keep the global optimality at the same time. The stability analysis, the derivation of controller, and the minimum energy are derived fully on the basis of nonlinear systems. Their formulation or representation are all in form of entire fuzzy system instead of fuzzy subsystems. Therefore, the mentioned issue is not the case for global approach. Those are only related to local-concept approach. In this article, we clarify the pointed issue and focus on reinforcing our theorem.     

Complex weight control of array pattern nulling

An efficient method based on the sequential quadratic programming (SQP) algorithm for the linear antenna arrays pattern synthesis with prescribed nulls in the interference direction and minimum side lobe levels by the complex weights of each array element is presented. In general, the pattern synthesis technique that generates a desired pattern is a greatly nonlinear optimization problem. SQP method is a versatile method to solve the general nonlinear constrained optimization problems and is much simpler to implement. It transforms the nonlinear minimization problem to a sequence of quadratic subproblem that is easier to solve, based on a quadratic approximation of the Lagrangian function. Several numerical results of Chebyshev pattern with the imposed single, multiple, and broad nulls sectors are provided and compared with published results to illustrate the performance of the proposed method. © 2007 Wiley Periodicals, Inc. Int J RF and Microwave CAE, 2007.     

Nonlinear analysis of truss by energy minimization

New analysis methods for trusses with material and geometric nonlinearity are presented using the energy principles and modified sequential quadratic programming (SQP) algorithms.The analysis problem of a truss with material nonlinearity is formulated as the total complementary energy minimization problem and the member forces are determined by using modified SQP algorithms.The truss subject to nonlinearities both in material properties and finite displacements is solved by the combination of complementary and potential energy minimization algorithms.The problem formulations on the basis of energy principles are quite simple and the proposed analysis methods can be applicable for any type of nonlinear material problem. The efficiency and reliability of the proposed method are clarified by giving numerical results for 3–33-bar statically indeterminate trusses with three types of nonlinear materials.     

Range and Null Space Decomposition applied to analysis of slender concrete columns

The strength analysis of a simply supported slender concrete column subject to biaxial bending is formulated as a nonlinear programming problem. Geometrical imperfections as well as two types of concrete constitutive equations, for local and global verifications, are taken into account. The algorithm of choice is the Sequential Quadratic Programming method (SQP). Large numbers of state variables and equilibrium equality constraints appear in the formulation, which is a characteristic of the optimization of nonlinear structures in general. This considerably hinders the efficiency and robustness of the SQP algorithm. Therefore the Range and Null Space Decomposition (RND) is employed in order to decrease the size of the quadratic programming subproblem that must be solved in each iteration, as well as the size of the approximating Hessian that must be updated. An example is presented to illustrate the efficiency of the proposed approach which took almost one-third of the CPU time required by the standard SQP algorithm to converge to a solution.     

A new non-monotone SQP algorithm for the minimax problem

In this paper, a new sequential quadratic programming (SQP) algorithm is proposed to solve the minimax problem which uses the idea of nonmonotonicity. The problem is transformed into an equivalent inequality constrained nonlinear optimization problem. In order to prevent the scaling problem, we do some modifications to the minimization problem. By the non-monotone SQP method, the new algorithm is globally convergent without using a penalty function. Furthermore, it is shown that the proposed method does not suffer from the Maratos effect, so the locally superlinear convergence is achieved. Numerical results suggest that our algorithm for solving the minmax problem is efficient and robust.     

Interior point techniques for optimal control of variational inequalities

This paper is devoted to a new application of an interior point algorithm to solve optimal control problems of variational inequalities. We propose a Lagrangian technique to obtain a necessary optimality system. After the discretization of the optimality system we prove its equivalence to Karush-Kuhn-Tucker conditions of a nonlinear regular minimization problem. This problem can be efficiently solved by using a modification of Herskovits' interior point algorithm for nonlinear optimization. We describe the numerical scheme for solving this problem and give some numerical examples of test problems in 1-D and 2-D.     

弹性需求下易变质物品定价、营销与生产计划的联合优化

针对易变质的商品,分析基于弹性需求的定价、营销及生产计划的联合优化问题,并建立用于描述该问题的非线性规划模型。考虑到模型是高度非线性的,提出基于几何规划的求解方法。首先将高度非线性的问题简化为只含有一个变量的问题;然后利用黄金分割法获得原问题高质量的近优解;最后通过算例验证了所提出求解方法的可操作性和正确性,并分析了主要参数的灵敏度。     

基于SQP算法的SAR成像导引头三维弹道优化

弹载SAR平台轨迹的设计是研究弹载SAR成像算法的前提。为了在满足SAR成像条件的同时降低导弹打击时间,需要对SAR成像导引头的弹道进行优化。该问题属于非线性最优控制问题,本文采用序列二次规划(SQP)优化算法进行求解。首先以波束驻留时间最小为指标函数,导弹俯仰、偏航加速度为优化变量,建立了SAR成像导引头三维弹道优化模型,模型的约束包括SAR成像约束、过载约束和导弹飞行高度约束。然后,将原最优控制问题进行参数化,转换成非线性规划问题,利用SQP算法进行求解。参数化时,离散节点越多,得到的非线性规划问题规模越大,求解速度就越慢。仿真结果表明,SQP算法能够有效解决SAR成像导引头三维弹道优化问题,得到的解满足模型约束。     

Policy iteration based feedback control

It is well known that stochastic control systems can be viewed as Markov decision processes (MDPs) with continuous state spaces. In this paper, we propose to apply the policy iteration approach in MDPs to the optimal control problem of stochastic systems. We first provide an optimality equation based on performance potentials and develop a policy iteration procedure. Then we apply policy iteration to the jump linear quadratic problem and obtain the coupled Riccati equations for their optimal solutions. The approach is applicable to linear as well as nonlinear systems and can be implemented on-line on real world systems without identifying all the system structure and parameters.     

Robustness,generality and efficiency of optimization algorithms for practical applications

The design of most engineering systems is a complex and time-consuming process. In addition, the need to optimize such systems where multidisciplinary analysis and design procedures are required can cost additional human and computational resources if proper software and numerical algorithms are not used. Several computational aspects of optimization algorithms and the associated software must be considered while making comparative studies and selecting a suitable algorithm for practical applications. Several parameters, such asaccuracy, generality, robustness, efficiency and ease of use, must be considered while deciding the superiority of an optimization approach. Approximate algorithms without sound mathematical basis can be sometimes more efficient for a specific problem, but fail to satisfy other requirements. They are, therefore, not suitable for general applications. An objective of the paper is to emphasize the critical importance of the above-mentioned parameters in large scalestructural optimization and other applications. Theoretical foundations of two promising approaches, thesequential quadratic programming (SQP) andoptimality criteria (OC), are presented and analysed. Recent numerical experiments and experiences with the SQP algorithm satisfying these requirements are described by solving a variety of structural design problems. An important conclusion of the paper is that the SQP method with a potential constraint strategy is a better choice as compared to the currently prevalent mathematical programming (MP) and OC approaches.     

An efficient sequential linear quadratic algorithm for solving nonlinear optimal control problems

We develop a numerically efficient algorithm for computing controls for nonlinear systems that minimize a quadratic performance measure. We formulate the optimal control problem in discrete-time, but many continuous-time problems can be also solved after discretization. Our approach is similar to sequential quadratic programming for finite-dimensional optimization problems in that we solve the nonlinear optimal control problem using sequence of linear quadratic subproblems. Each subproblem is solved efficiently using the Riccati difference equation. We show that each iteration produces a descent direction for the performance measure, and that the sequence of controls converges to a solution that satisfies the well-known necessary conditions for the optimal control.     

Contact shape optimization: a bilevel programming approach

We consider the problem of shape optimization of nonlinear elastic solids in contact. The equilibrium of the solid is defined by a constrained minimization problem, where the body energy functional is the objective and the constraints impose the nonpenetration condition. Then the optimization problem can be formulated in terms of a bilevel mathematical program. We describe new optimality conditions for bilevel programming and construct an algorithm to solve these conditions based on Herskovits’ feasible direction interior point method. With this approach we simultaneously carry out shape optimization and nonlinear contact analysis. That is, the present method is a “one shot” technique. We describe some numerical examples solved in a very efficient way. Received July 27, 1999     

经济调度问题的混合蚁群算法及序列二次规划法解

为了获得整体近似最优解,提出采用蚁群算法,搜索发电机可运行状态的最优组合,并对蚁群算法的数学模型进行分析,以参数的形式给出具有普遍意义的收敛性定理.在此求解过程中,以每只人工蚂蚁来表示符合限制条件的某个可运转状态的发电机组合并以序列二次规划法来求解传统的经济调度问题.以三部机组的数值模拟,验证该方法正确有效.     

Reliability-based design optimization using SORM and SQP

In this work a second order approach for reliability-based design optimization (RBDO) with mixtures of uncorrelated non-Gaussian variables is derived by applying second order reliability methods (SORM) and sequential quadratic programming (SQP). The derivation is performed by introducing intermediate variables defined by the incremental iso-probabilistic transformation at the most probable point (MPP). By using these variables in the Taylor expansions of the constraints, a corresponding general first order reliability method (FORM) based quadratic programming (QP) problem is formulated and solved in the standard normal space. The MPP is found in the physical space in the metric of Hasofer-Lind by using a Newton algorithm, where the efficiency of the Newton method is obtained by introducing an inexact Jacobian and a line-search of Armijo type. The FORM-based SQP approach is then corrected by applying four SORM approaches: Breitung, Hohenbichler, Tvedt and a recent suggested formula. The proposed SORM-based SQP approach for RBDO is accurate, efficient and robust. This is demonstrated by solving several established benchmarks, with values on the target of reliability that are considerable higher than what is commonly used, for mixtures of five different distributions (normal, lognormal, Gumbel, gamma and Weibull). Established benchmarks are also generalized in order to study problems with large number of variables and several constraints. For instance, it is shown that the proposed approach efficiently solves a problem with 300 variables and 240 constraints within less than 20 CPU minutes on a laptop. Finally, a most well-know deterministic benchmark of a welded beam is treated as a RBDO problem using the proposed SORM-based SQP approach.     

Partial‐state stabilization and optimal feedback control

In this paper, we develop a unified framework to address the problem of optimal nonlinear analysis and feedback control for partial stability and partial‐state stabilization. Partial asymptotic stability of the closed‐loop nonlinear system is guaranteed by means of a Lyapunov function that is positive definite and decrescent with respect to part of the system state, which can clearly be seen to be the solution to the steady‐state form of the Hamilton–Jacobi–Bellman equation and hence guaranteeing both partial stability and optimality. The overall framework provides the foundation for extending optimal linear‐quadratic controller synthesis to nonlinear nonquadratic optimal partial‐state stabilization. Connections to optimal linear and nonlinear regulation for linear and nonlinear time‐varying systems with quadratic and nonlinear nonquadratic cost functionals are also provided. Finally, we also develop optimal feedback controllers for affine nonlinear systems using an inverse optimality framework tailored to the partial‐state stabilization problem and use this result to address polynomial and multilinear forms in the performance criterion. Copyright © 2015 John Wiley & Sons, Ltd.     

A data‐driven bilinear predictive controller design based on subspace method

In this paper, a new data‐driven model predictive control (MPC), based on bilinear subspace identification, is considered. The system's nonlinear behavior is described with a bilinear subspace predictor structure in an MPC framework. Thus, the MPC formulation results in a fixed structure objective function with constraints regardless of the underlying nonlinearity. For unconstrained systems, the identified subspace predictor matrices can be directly used as controller parameters. Therefore, we design optimization algorithms that exploit this feature. The open‐loop optimization problem of MPC that is nonlinear in nature is solved with series quadratic programming (SQP) without any approximations. The computational efficiency already demonstrated with the current formulation presents further opportunities to enable online control of nonlinear systems. These improvements and close integration of modeling and control also eliminate the intermediate design step, which provides a means for data‐driven controller design in generalized predictive controller (GPC) framework. Finally, the proposed control approach is illustrated with a verification study of a nonlinear continuously stirred tank reactor (CSTR) system. Copyright © 2011 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

A simple iterative method for sub-optimal control of linear time delay systems with quadratic cost†

The usual approach for obtaining the optimal control For a linear time-delay system with a quadratic cost consists of first deriving a set of necessary conditions for optimality and then using conventional iterative numerical methods to find a control satisfying those conditions. The burden of computation in this approach is enormous. The iterative scheme proposed in this paper does not proceed along these lines. Instead, the delay term is treated liko an extra perturbing input. A linear non-delay system is optimized at each stage, and the resulting sequence of control functions converges rapidly to the sub-optimal control for the original problem, as illustrated by two numerical examples.     

Rough‐fuzzy quadratic minimum spanning tree problem

A quadratic minimum spanning tree problem determines a minimum spanning tree of a network whose edges are associated with linear and quadratic weights. Linear weights represent the edge costs whereas the quadratic weights are the interaction costs between a pair of edges of the graph. In this study, a bi‐objective rough‐fuzzy quadratic minimum spanning tree problem has been proposed for a connected graph, where the linear and the quadratic weights are represented as rough‐fuzzy variables. The proposed model is formulated by using rough‐fuzzy chance‐constrained programming technique. Subsequently, three related theorems are also proposed for the crisp transformation of the proposed model. The crisp equivalent models are solved with a classical multi‐objective solution technique, the epsilon‐constraint method and two multi‐objective evolutionary algorithms: (a) nondominated sorting genetic algorithm II (NSGA‐II) and (b) multi‐objective cross‐generational elitist selection, heterogeneous recombination, and cataclysmic mutation (MOCHC) algorithm. A numerical example is provided to illustrate the proposed model when solved with different methodologies. A sensitivity analysis of the example is also performed at different confidence levels. The performance of NSGA‐II and MOCHC are analysed on five randomly generated instances of the proposed model. Finally, a numerical illustration of an application of the proposed model is also presented in this study.     

Smooth time-optimal tool trajectory generation for CNC manufacturing systems

An optimization approach is proposed in this paper for generating smooth and time-optimal path constrained tool trajectory for Cartesian computer numerical control (CNC) manufacturing systems. The desired smooth time-optimal trajectory generation (STOTG) problem is formulated as a general optimal control problem. And axis jerk (derivative of acceleration with respect to time) constraints are introduced into this problem to remove discontinuities of the acceleration profiles. The desired smoothness of the trajectory can be accomplished by adjusting the values of jerk constraints. A control vector parameterization (CVP) method is applied to convert the optimal control problem into a nonlinear programming (NLP) problem which can be solved conveniently and effectively. The third derivative of the path parameter with respect to time (pseudo-jerk) and jerk act as optimization variables. The pseudo-jerk is approximated as piecewise constant, thus for at least second-order continuous parametric path, the resulted optimized trajectory with respect to time is also at least second-order continuous. Sequential quadratic programming (SQP) method is used to solve the NLP problem, through which numerical solution is obtained. Non-smooth (i.e. without considering jerk constraints) time-optimal trajectory generation (non-STOTG) problem is also considered in this paper for the purpose of comparison. Solutions of time-optimal trajectory generation (TOTG) problems for two test paths are performed to verify the effectiveness of the proposed approach.