A Dual‐Type Method Based Algorithm for Nonlinear Large Network Optimization Problems

In previous research, we have proposed a Dual Projected Pseudo Quasi Newton (DPPQN) method which differs from the conventional Lagrange relaxation method by treating the inequality constraints as the domain of the primal variables in the dual function and using Projection Theory to handle the inequality constraints. We have combined this dual‐type method with a Projected Jacobi (PJ) method to solve nonlinear large network optimization problems with decomposable inequality constraints, and have achieved several attractive features. To retain the attractive features and to remedy the flaw of the previous method, in the current paper, we propose an active set strategy based DPPQN method to solve the projection problem formed by coupling functional inequality constraints. This method associated with the DPPQN method and the PJ method can be used to solve general nonlinear large network optimization problems. We present this algorithm, demonstrate its computational efficiency through numerical simulations and compare it with the previous method.     

Dantzig—Wolfe Decomposition of Variational Inequalities

The creation and ongoing management of a large economic model can be greatly simplified if the model is managed in separate smaller pieces defined, e.g. by region or commodity. For this purpose, we define an extension of Dantzig–Wolfe decomposition for the variational inequality (VI) problem, a modeling framework that is widely used for models of competitive or oligopolistic markets. The subproblem, a collection of independent smaller models, is a relaxed VI missing some “difficult” constraints. The subproblem is modified at each iteration by information passed from the last solution of the master problem in a manner analogous to Dantzig–Wolfe decomposition for optimization models. The master problem is a VI which forms convex combinations of proposals from the subproblem, and enforces the difficult constraints. A valid stopping condition is derived in which a scalar quantity, called the “convergence gap,” is monitored. The convergence gap is a generalization of the primal-dual gap that is commonly monitored in implementations of Dantzig–Wolfe decomposition for optimization models. Convergence is proved under conditions general enough to be applicable to many models. An illustration is provided for a two-region competitive model of Canadian energy markets.     

Well-posedness and attainability of indefinite stochastic linear quadratic control in infinite time horizon

This paper is concerned with a stochastic linear-quadratic (LQ) problem in an infinite time horizon with multiplicative noises both in the state and the control. A distinctive feature of the problem under consideration is that the cost weighting matrices for the state and the control are allowed to be indefinite. A new type of algebraic Riccati equation – called a generalized algebraic Riccati equation (GARE) – is introduced which involves a matrix pseudo-inverse and two additional algebraic equality/inequality constraints. It is then shown that the well-posedness of the indefinite LQ problem is equivalent to a linear matrix inequality (LMI) condition, whereas the attainability of the LQ problem is equivalent to the existence of a “stabilizing solution” to the GARE. Moreover, all possible optimal controls are identified via the solution to the GARE. Finally, it is proved that the solution to the GARE can be obtained via solving a convex optimization problem called semidefinite programming.     

Model predictive tracking control for a linear system under time‐varying input constraints

In this paper, we propose a tracking control law for a linear dynamical system under time‐varying input constraints. The proposed control law consists of a dual‐mode model predictive control (MPC) law and a target recalculation mechanism. As the terminal controller of the dual‐mode MPC, we propose a saturation‐level‐dependent gain‐scheduled feedback control law that ensures closed‐loop stability against arbitrary change of the position limit of the actuators. We also present conditions that guarantee feasibility and stability of the control algorithm under time‐varying input constraints. The control algorithm is reduced to an online optimization problem under linear matrix inequality constraints. Copyright © 2012 John Wiley & Sons, Ltd.     

基于鞍点法的自适应分布式资源分配算法

研究一类带有不等式约束为凸函数的多智能体系统分布式资源分配问题.在资源分配问题中,各智能体拥有仅自身可知的局部成本函数和局部凸不等式约束.分布式资源分配旨在如何利用智能体间的信息交互设计一种分布式优化算法,完成定量资源分配的同时还保证最小化全局成本函数.针对该问题,基于卡罗需-库恩-塔克条件和比例积分控制思想,首先提出一种自适应分布式优化算法,其中凸不等式约束的对偶变量可实现自适应获取;然后,为了降低系统的通信资源消耗,设计一种动态事件触发控制策略以实现离散时间通信的分布式资源分配算法;最后,通过数值仿真验证所设计算法的有效性.     

The quasi-newtonian method of solution of convex variational inequalities with descent decomposition

The problem of numerical solution of convex variational inequalities with nonlinear constraints is considered. An equivalent optimization problem is constructed. To solve it, a Newtonian-type method with descent decomposition with respect to direct and dual variables of a variational inequality is developed. The nonlocal convergence of the algorithm and the superlinear rate of convergence in a neighborhood of the solution are proved. Translated from Kibemetika i Sistemnyi Analiz, No. 4, pp. 90–105, July–August, 1999.     

On the noninferior set for the systems with vector-valued objective function

The necessary and sufficient conditions of the locally noninferior set for the static optimization problem with a vector-valued objective function have been derived. Equality and inequality constraints in the variable space have been discussed to give a more complete formulation. All of these have been done by using the properties of the dual linear inequality systems.     

General redundancy optimization method for cooperating manipulatorsusing quadratic inequality constraints

In redundancy optimization problems related to cooperating manipulators such as optimal force distribution, constraints on the physical limits of the manipulators should be considered. We propose quadratic inequality constraints (QICs), which lead to ellipsoidal feasible regions, to solve the optimization problem more efficiently. We investigate the effect of the use of QICs from the points of view of problem size and change of the feasible region. To efficiently deal with the QICs, we also propose the dual quadratically constrained quadratic programming (QCQP) method. In this method, the size of the optimization problem is reduced so that the computational burden is lightened. The proposed method and another well-known quadratic programming method are applied to the two PUMA robots system and compared with each other. The results show that the use of QICs with the dual QCQP method allows for faster computation than the existing method     

Learning Tetris using the noisy cross-entropy method

The cross-entropy method is an efficient and general optimization algorithm. However, its applicability in reinforcement learning (RL) seems to be limited because it often converges to suboptimal policies. We apply noise for preventing early convergence of the cross-entropy method, using Tetris, a computer game, for demonstration. The resulting policy outperforms previous RL algorithms by almost two orders of magnitude.     

考虑融合算法与交叉熵的毕达哥拉斯决策模型

针对专家评价信息为毕达哥拉斯模糊数并且属性值信息存在相互关联的多属性决策问题,定义了广义的毕达哥拉斯运算法则,并结合几何Heronian平均,提出了毕达哥拉斯模糊几何Heronian平均（PFGHM）算子;基于对数函数设计了新的毕达哥拉斯模糊交叉熵用于衡量信息之间的差异性;构造了基于PFGHM算子和交叉熵的毕达哥拉斯决策模型,并通过高校引进人才团队的选择实例验证模型的可靠性。     

求解PCB钻孔机刀具路径规划的交叉熵方法

工作刀具路径的优化程度是PCB钻孔机的重要性能指标,对其进行很好的优化有助于提高PCB设备的加工质量和加工速度.首先对刀具路径进行建模,然后对交叉熵算法进行描述并应用交叉熵方法对刀具路径进行求解.实验结果表明,选择交叉熵方法对环境进行建模简单、有效,在求解刀具路径规划方面具有一定的优势.     

Fuzzy Modeling and Control for Conical Magnetic Bearings Using Linear Matrix Inequality

A general nonlinear model with six degree-of-freedom rotor dynamics and electromagnetic force equations for conical magnetic bearings is developed. For simplicity, a T–S (Takagi–Sugeno) fuzzy model for the nonlinear magnetic bearings assumed no rotor eccentricity is first derived, and a fuzzy control design based on the T–S fuzzy model is then proposed for the high speed and high accuracy control of the complex magnetic bearing systems. The suggested fuzzy control design approach for nonlinear magnetic bearings can be cast into a linear matrix inequality (LMI) problem via robust performance analysis, and the LMI problem can be solved efficiently using the convex optimization techniques. Computer simulations are presented for illustrating the performance of the control strategy considering simultaneous rotor rotation tracking and gap deviations regulation.     

Smooth curve extraction by mean field annealing

In this paper, we attack the figure — ground discrimination problem from a combinatorial optimization perspective. In general, the solutions proposed in the past solved this problem only partially: either the mathematical model encoding the figure — ground problem was too simple or the optimization methods that were used were not efficient enough or they could not guarantee to find the global minimum of the cost function describing the figure — ground model. The method that we devised and which is described in this paper is tailored around the following contributions. First, we suggest a mathematical model encoding the figure — ground discrimination problem that makes explicit a definition of shape (or figure) based on cocircularity, smoothness, proximity, and contrast. This model consists of building a cost function on the basis of image element interactions. Moreover, this cost function fits the constraints of aninteracting spin system, which in turn is a well suited physical model to solve hard combinatorial optimization problems. Second, we suggest a combinatorial optimization method for solving the figure — ground problem, namely mean field annealing which combines the mean field approximation and annealing. Mean field annealing may well be viewed as a deterministic approximation of stochastic methods such as simulated annealing. We describe in detail the theoretical bases of this method, derive a computational model, and provide a practical algorithm. Finally, some experimental results are shown for both synthetic and real images.This research has been sponsored in part by Commissariat à l'Energie Atomique, and in part by the ORASIS project (PRC Communications Homme/Machine).     

How Inductive Inference Strategies Discover Their Errors

Several well-known inductive inference strategies change the actual hypothesis only when they discover that it "provably misclassifies" an example seen so far. This notion is made mathematically precise, and its general power is characterized. In spite of its strength, it is shown that this approach is not of universal power. Consequently, hypotheses are considered which "unprovably misclassify" examples, and the properties of this approach are studied. Among others, it turns out that this type is of the same power as monotonic identification. Then it is shown that universal power can be achieved only when an unbounded number of alternations of these dual types of hypotheses is allowed. Finally, a universal method is presented, enabling an inductive inference strategy to verify the incorrectness of any of its incorrect intermediate hypotheses.     

Geometric analysis of collaborative optimization

Instead of the past mathematical analyses, an intuitive geometric analysis of the collaborative optimization (CO) algorithm is presented in this paper, which reveals some geometric properties of CO and gives a direct geometric interpretation of the reason for the reported computational difficulties in CO. The analysis shows that if the system-level optimum point at one iteration is outside the feasible region of the original optimization problem, at the next iteration, the system-level optimization problem may be infeasible due to the system-level consistency equality constraints. One way to solve the problem of the infeasibility is to relax the system-level consistency equality constraints using inequality constraints. However it is a delicate job to determine a rational relaxed tolerance because feasibility and consistency have conflicting requirements for the tolerance, that is, the more relaxed the better for feasibility while the stricter the better for consistency. Based on the geometric analysis, a method of variable relaxed tolerance is put forward to solve this problem. In this method, an adaptive adjustment of the tolerance is made at each iteration according to the quantified inconsistency between two subsystems. In the last section, the capabilities and limitations of the proposed method are illustrated by three examples.     

Hybridizing the cross-entropy method: An application to the max-cut problem

Cross-entropy has been recently proposed as a heuristic method for solving combinatorial optimization problems. We briefly review this methodology and then suggest a hybrid version with the goal of improving its performance. In the context of the well-known max-cut problem, we compare an implementation of the original cross-entropy method with our proposed version. The suggested changes are not particular to the max-cut problem and could be considered for future applications to other combinatorial optimization problems.     

An inequality governing nonlinear H∞ control

This note gives necessary and sufficient conditions for solving a reasonable version of the nonlinear H^∞ control problem. The most objectionable hypothesis is elegant and holds in the linear case, but every possibly may not be forced for nonlinear systems. What we discover in distinction to Isidori and Astolfi (1992) and Ball et al. (1993) is that the key formula is not a (nonlinear) Riccati partial differential inequality, but a much more complicated inequality mixing partial derivatives and an approximation theoretic construction called the best approximation operator. This Chebeshev-Riccati inequality when specialized to the linear case gives the famous solution to the H^∞ control problem found in Doyle et al. (1989). While complicated the Chebeshev-Riccati inequality is (modulo a considerable number of hypotheses behind it) a solution to the nonlinear H^∞ control problem. It should serve as a rational basis for discovering new formulas and compromises. We follow the conventions of Ball et al. (1993) and this note adds directly to that paper.     

Hexahedral mesh smoothing via local element regularization and global mesh optimization

The quality of finite element meshes is one of the key factors that affect the accuracy and reliability of finite element analysis results. In order to improve the quality of hexahedral meshes, we present a novel hexahedral mesh smoothing algorithm which combines a local regularization for each hexahedral mesh, using dual element based geometric transformation, with a global optimization operator for all hexahedral meshes. The global optimization operator is composed of three main terms, including the volumetric Laplacian operator of hexahedral meshes and the geometric constraints of surface meshes which keep the volumetric details and the surface details, and another is the transformed node displacements condition which maintains the regularity of all elements. The global optimization operator is formulated as a quadratic optimization problem, which is easily solved by solving a sparse linear system. Several experimental results are presented to demonstrate that our method obtains higher quality results than other state-of-the-art approaches.