An evolutionary algorithm based on Nash Dominance for Equilibrium Problems with Equilibrium Constraints

This paper introduces an evolutionary algorithm for the solution of a class of hierarchical (“leader-follower”) games known as Equilibrium Problems with Equilibrium Constraints (EPECs). In one manifestation of such games, players at the upper level who assume the role of leaders, are assumed to act non cooperatively to maximize individual payoffs. At the same time, each leader's payoffs are constrained not only by their competitor's actions but also by the behaviour of the followers at the lower level which manifests in the form of an equilibrium constraint. By a redefinition of the selection criteria used in evolutionary methods, the paper demonstrates that the solution for such games can be found via a simple modification to a standard evolutionary multiobjective algorithm. We give a proposed algorithm (NDEMO) and illustrate it with numerical examples drawn from both the transportation systems management literature and the electricity generation industry underlying the applicability of NDEMO in multidisciplinary contexts.     

处理动态优化问题中控制变量路径约束的方法

目前国际上对动态优化问题中的状态变量路径约束已有一些研究,但专门处理控制变量路径约束的方法却鲜见报道. 本文首先介绍两种分别基于三角函数变换、约束算子截断来处理控制变量路径约束的方法,然后提出一种基于光滑化的二次罚函数方法. 光滑化罚函数方法不仅能够处理控制变量路径约束,而且还能同时处理关于状态变量的路径约束. 最后使用目前流行的控制变量参数化 (Control variable parameterization, CVP)策略对最终获得的、不再含控制变量路径约束的动态优化问题求解. 实例测试一展现了三种方法各自的特点;实例测试二表明了光滑罚函数方法的有效性和优越性.     

一种求解强凸优化问题的最优随机算法

随机梯度下降(SGD)算法是处理大规模数据的有效方法之一.黑箱方法SGD在强凸条件下能达到最优的O(1/T)收敛速率,但对于求解L1+L2正则化学习问题的结构优化算法,如COMID(composite objective mirror descent)仅具有O(lnT/T)的收敛速率.提出一种能够保证稀疏性基于COMID的加权算法,证明了其不仅具有O(1/T)的收敛速率,还具有on-the-fly计算的优点,从而减少了计算代价.实验结果表明了理论分析的正确性和所提算法的有效性.     

A Numerical Method for Solving Stochastic Optimal Control Problems with Linear Control

We introduce a numerical method to solve stochastic optimal control problems which are linear in the control. We facilitate the idea of solving two-point boundary value problems with spline functions in order to solve the resulting dynamic programming equation. We then show how to effectively reduce the dimension in the proposed algorithm, which improves computational time and memory constraints. An example, motivated as an invest problem with uncertain cost, is provided, and the effectiveness of our method demonstrated.     

A Mixed Finite Element Scheme for Optimal Control Problems with Pointwise State Constraints

In this paper, we propose a mixed variational scheme for optimal control problems with point-wise state constraints, the main idea is to reformulate the optimal control problems to a constrained minimization problem involving only the state, which is characterized by a fourth order variational inequality. Then mixed form based on this fourth order variational inequality is formulated and a direct numerical algorithm is proposed without the optimality conditions of underlying optimal control problems. The a priori and a posteriori error estimates are proved for the mixed finite element scheme. Numerical experiments confirm the efficiency of the new strategy.     

Convex Control Systems and Convex Optimal Control Problems With Constraints

This note discusses the concepts of convex control systems and convex optimal control problems. We study control systems governed by ordinary differential equations in the presence of state and target constraints. Our note is devoted to the following main question: under which additional assumptions is a "sophisticated" constrained optimal control problem equivalent to a "simple" convex minimization problem in a related Hilbert space. We determine some classes of convex control systems and show that, for suitable cost functionals and constraints, optimal control problems for these classes of systems correspond to convex optimization problems. The latter can be reliably solved using standard numerical algorithms and effective regularization schemes. In particular, we propose a conceptual computational approach based on gradient-type methods and proximal point techniques.     

Optimal Control of Nonlinear Stochastic Systems under Constraints: An Approximate Determination Method

An approximate method is developed for determining the optimal control for nonlinear stochastic systems under mixed equality- and inequality-type constraints on the parameters of the system, control functions, and phase coordinate in the presence of random parameters and additive and multiplicative noises. The method is based on the reduction of the initial stochastic problem to a deterministic problem for the cumulants of a random process described by stochastic differential equations.     

Reduced Basis Method and Error Estimation for Parametrized Optimal Control Problems with Control Constraints

We propose a Reduced Basis method for the solution of parametrized optimal control problems with control constraints for which we extend the method proposed in Dedè, L. (SIAM J. Sci. Comput. 32:997, 2010) for the unconstrained problem. The case of a linear-quadratic optimal control problem is considered with the primal equation represented by a linear parabolic partial differential equation. The standard offline–online decomposition of the Reduced Basis method is employed with the Finite Element approximation as the “truth” one for the offline step. An error estimate is derived and an heuristic indicator is proposed to evaluate the Reduced Basis error on the optimal control problem at the online step; also, the indicator is used at the offline step in a Greedy algorithm to build the Reduced Basis space. We solve numerical tests in the two-dimensional case with applications to heat conduction and environmental optimal control problems.     

Dual Stochastic Dominance and Quantile Risk Measures

Following the seminal work by Markowitz, the portfolio selection problem is usually modeled as a bicriteria optimization problem where a reasonable trade–off between expected rate of return and risk is sought. In the classical Markowitz model, the risk is measured with variance. Several other risk measures have been later considered thus creating the entire family of mean–risk (Markowitz type) models. In this paper, we analyze mean–risk models using quantiles and tail characteristics of the distribution. Value at risk (VAR), defined as the maximum loss at a specified confidence level, is a widely used quantile risk measure. The corresponding second order quantile measure, called the worst conditional expectation or Tail VAR, represents the mean shortfall at a specified confidence level. It has more attractive theoretical properties and it leads to LP solvable portfolio optimization models in the case of discrete random variables, i.e., in the case of returns defined by their realizations under the specified scenarios. We show that the mean–risk models using the worst conditional expectation or some of its extensions are in harmony with the stochastic dominance order. For this purpose, we exploit duality relations of convex analysis to develop the quantile model of stochastic dominance for general distributions.     

Iterative Path Integral Approach to Nonlinear Stochastic Optimal Control Under Compound Poisson Noise

Nonlinear stochastic optimal control theory has played an important role in many fields. In this theory, uncertainties of dynamics have usually been represented by Brownian motion, which is Gaussian white noise. However, there are many stochastic phenomena whose probability density has a long tail, which suggests the necessity to study the effect of non‐Gaussianity. This paper employs Lévy processes, which cause outliers with a significantly higher probability than Brownian motion, to describe such uncertainties. In general, the optimal control law is obtained by solving the Hamilton–Jacobi–Bellman equation. This paper shows that the path‐integral approach combined with the policy iteration method is efficiently applicable to solve the Hamilton–Jacobi–Bellman equation in the Lévy problem setting. Finally, numerical simulations illustrate the usefulness of this method.     

A Computational Method for Stochastic Optimal Control Problems in Financial Mathematics

Principle of optimality or dynamic programming leads to derivation of a partial differential equation (PDE) for solving optimal control problems, namely the Hamilton‐Jacobi‐Bellman (HJB) equation. In general, this equation cannot be solved analytically; thus many computing strategies have been developed for optimal control problems. Many problems in financial mathematics involve the solution of stochastic optimal control (SOC) problems. In this work, the variational iteration method (VIM) is applied for solving SOC problems. In fact, solutions for the value function and the corresponding optimal strategies are obtained numerically. We solve a stochastic linear regulator problem to investigate the applicability and simplicity of the presented method and prove its convergence. In particular, for Merton's portfolio selection model as a problem of portfolio optimization, the proposed numerical method is applied for the first time and its usefulness is demonstrated. For the nonlinear case, we investigate its convergence using Banach's fixed point theorem. The numerical results confirm the simplicity and efficiency of our method.     

A Neural Network Approach for Solving Optimal Control Problems with Inequality Constraints and Some Applications

In this paper, a class of nonlinear optimal control problems with inequality constraints is considered. Based on Karush–Kuhn–Tucker optimality conditions of nonlinear optimization problems and by constructing an error function, we define an unconstrained minimization problem. In the minimization problem, we use trial solutions for the state, Lagrange multipliers, and control functions where these trial solutions are constructed by using two-layered perceptron. We then minimize the error function using a dynamic optimization method where weights and biases associated with all neurons are unknown. The stability and convergence analysis of the dynamic optimization scheme is also studied. Substituting the optimal values of the weights and biases in the trial solutions, we obtain the optimal solution of the original problem. Several examples are given to show the efficiency of the method. We also provide two applicable examples in robotic engineering.     

Partially Observed Stochastic Shortest Path Problems With Approximate Solution by Neurodynamic Programming

We analyze a class of Markov decision processes with imperfect state information that evolve on an infinite time horizon and have a total cost criterion. In particular, we are interested in problems with stochastic shortest path structure, assuming the following: 1) the existence of a policy that guarantees termination with probability one and 2) the property that any policy that fails to guarantee termination has infinite expected cost from some initial state. We also assume that termination is perfectly recognized. In this paper, we clarify and expand upon arguments (given in an earlier paper) for establishing the existence, uniqueness, and characterization of stationary optimal policies, and the convergence of value and policy iteration. We also present an illustrative example, involving the search for a partially observed target that moves randomly on a grid, and we develop a simulation-based algorithm (based on neurodynamic programming techniques) for computing policies that approximately minimize the expected number of stages to complete the search.     

Stochastic Dominance for Measure Based Uncertain Decision Making

Our interest is in the problem of comparing alternatives with uncertain payoffs when the uncertainty is represented using a measure. We first describe various aspects of the use of a measure to represent uncertainty. We recall that probability is a special well‐understood example of measure‐based uncertainty. We note that stochastic dominance provides a well‐established method for comparing alternatives in the case of probabilistic uncertainty. Inspired by this we develop an extension of the use of stochastic dominance for comparing uncertainty profiles to the case where the uncertainty is represented by a measure. We refer to this as measure based stochastic dominance. Do to the fact that in most cases a stochastic dominance relationship does not exist between alternatives this requires us to consider the use of surrogates for measure based stochastic dominance to compare alternatives. Here we investigate a class of surrogates for measure based stochastic dominance that we call Measure Weighted Means (MWM). As we see these MWM are numeric values consistent with measure based stochastic dominance.     

Time-optimized 3D Path Smoothing with Kinematic Constraints

International Journal of Control, Automation and Systems - This paper proposes a novel time-optimized smoothing algorithm, that optimizes a path planner output, considering three dimensional (3D)...     

带不等式路径约束最优控制问题的惩罚函数法

控制变量参数化（Control variable parameterization,CVP）方法是目前求解流程工业中最优操作问题的主流数值方法,但如果问题中包含路径约束,特别是不等式路径约束时,CVP方法则需要考虑专门的处理手段.为了克服该缺点,本文提出一种基于L1精确惩罚函数的方法,能够有效处理关于控制变量、状态变量、甚至控制变量/状态变量复杂耦合形式下的不等式路径约束.此外,为了能使用基于梯度的成熟优化算法,本文还引进了最新出现的光滑化技巧对非光滑的惩罚项进行磨光.最终得到了能高效处理不等式路径约束的改进型CVP架构,并给出相应数值算法.经典的带不等式路径约束最优控制问题上的测试结果及与国外文献报道的比较研究表明：本文所提出的改进型CVP 架构及相应算法在精度和效率上兼有良好表现.     

基于动态规划的无人机航路优化问题研究

无人机航路规划往往指无人机在初始位置、终止位置和一些目标任务结点确定之后的航迹优化问题;在无人机飞行任务执行过程中,无人机需要在参考飞行航线的约束下,根据局部地形、地貌、障碍、威胁等信息以及飞机本身机动能力的限制,实时的计算出飞行航路,并跟随该航迹完成飞行任务;在利用Dynapath(动态路径)算法进行参考航线优化的过程中,考虑了飞机机动性能的限制;在此基础上,详细讨论了参考航线由多航段组成时航路点的处理方法并进行了计算机仿真,仿真结果表明该方法可以比较理想地进行航路点的处理,得到比较良好的最优航迹。     

Stochastic Bounded Consensus Tracking of Second-Order Multi-Agent Systems with Measurement Noises and Sampled-Data

This paper investigates the stochastic bounded consensus tracking problems of second-order multi-agent systems, where the control input of an agent can only use the information measured at the sampling instants from its neighbors or the virtual leader with a time-varying reference state, and the measurements are corrupted by random noises. The probability limit theory, the algebra graph theory, and some other techniques are employed to derive the necessary and sufficient condition guaranteeing the mean square bounded consensus tracking. It turns out that the maximum allowable sampling period depends on not only the network topology but also the constant feedback gains. Furthermore, the effects of the sampling period on tracking performance, including the tracking speed and the static tracking error, are also analyzed. The results show that reducing the sampling period can accelerate the tracking speed and decrease the static tracking error. Simulations are provided to demonstrate the effectiveness of the theoretical results.     

人口发展的双线性最优控制

本文根据我国的实际情况,建立了人口发展过程的动态模型,定义了各种函数关系,指出了连续模型和离散模型之间的转换关系.以育龄妇女平均生育率作为控制量,找到了为实现理想的人口发展过程的最优控制所应满足的必要条件,从而得到了关于控制量受限制情况下,双线性最优控制的计算方法.     

一种具有端点方向约束的快速航迹规划方法

传统A＊算法在进行具有端点方向约束的航迹规划时,由于缺乏方向引导而导致效率低下。针对该问题,提出一种动态引导A＊算法。在原有A＊算法的基础上引入动态变化的引导点,给出引导点的设置策略,构造新的代价函数,设计引导点的切换方法。实验结果表明,在进行具有端点方向约束的航迹规划时,通过设置合适的引导点参数,该方法可以规划出从起始点的特定方向出发、并沿指定方向到达目标的飞行航迹,且与传统A＊算法相比,其规划速度能够提高lO倍以上。