An Exact Penalty Function Method for Continuous Inequality Constrained Optimal Control Problem

In this paper, we consider a class of optimal control problems subject to equality terminal state constraints and continuous state and control inequality constraints. By using the control parametrization technique and a time scaling transformation, the constrained optimal control problem is approximated by a sequence of optimal parameter selection problems with equality terminal state constraints and continuous state inequality constraints. Each of these constrained optimal parameter selection problems can be regarded as an optimization problem subject to equality constraints and continuous inequality constraints. On this basis, an exact penalty function method is used to devise a computational method to solve these optimization problems with equality constraints and continuous inequality constraints. The main idea is to augment the exact penalty function constructed from the equality constraints and continuous inequality constraints to the objective function, forming a new one. This gives rise to a sequence of unconstrained optimization problems. It is shown that, for sufficiently large penalty parameter value, any local minimizer of the unconstrained optimization problem is a local minimizer of the optimization problem with equality constraints and continuous inequality constraints. The convergent properties of the optimal parameter selection problems with equality constraints and continuous inequality constraints to the original optimal control problem are also discussed. For illustration, three examples are solved showing the effectiveness and applicability of the approach proposed.     

A Variational Approach to Perturbation Feedback Control for Optimal Control Problems with Terminal Constraints and Free Terminal Time

Set-Valued and Variational Analysis - Using the method of characteristics, for an optimal control problem with terminal constraints and free terminal time, we construct a family of extremals along...     

Optimal Control Problems via Exact Penalty Functions

The nonsmoothness is viewed by many people as at least an undesirable (if not unavoidable) property. Our aim here is to show that recent developments in Nonsmooth Analysis (especially in Exact Penalization Theory) allow one to treat successfully even some quite smooth problems by tools of Nonsmooth Analysis and Nondifferentiable Optimization. Our approach is illustrated by one Classical Control Problem of finding optimal parameters in a system described by ordinary differential equations.     

An Affine Scaling Interior Trust Region Method via Optimal Path for Solving Monotone Variational Inequality Problem with Linear Constraints

Based on a differentiable merit function proposed by Taji et al. in ＂Math. Prog. Stud., 58, 1993, 369-383＂, the authors propose an affine scaling interior trust region strategy via optimal path to modify Newton method for the strictly monotone variational inequality problem subject to linear equality and inequality constraints. By using the eigensystem decomposition and affine scaling mapping, the authors form an affine scaling optimal curvilinear path very easily in order to approximately solve the trust region subproblem. Theoretical analysis is given which shows that the proposed algorithm is globally convergent and has a local quadratic convergence rate under some reasonable conditions.     

Partial Exact Penalty for Mathematical Programs with Equilibrium Constraints

It is well known that mathematical programs with equilibrium constraints (MPEC) violate the standard constraint qualifications such as Mangasarian–Fromovitz constraint qualification (MFCQ) and hence the usual Karush–Kuhn–Tucker conditions cannot be used as stationary conditions unless relatively strong assumptions are satisfied. This observation has led to a number of weaker stationary conditions, with Mordukhovich stationary (M-stationary) condition being the strongest among the weaker conditions. In nonlinear programming, it is known that MFCQ leads to an exact penalization. In this paper we show that MPEC GMFCQ, an MPEC variant of MFCQ, leads to a partial exact penalty where all the constraints except a simple linear complementarity constraint are moved to the objective function. The partial exact penalty function, however, is nonsmooth. By smoothing the partial exact penalty function, we design an algorithm which is shown to be globally convergent to an M-stationary point under an extended version of the MPEC GMFCQ.     

Necessary Conditions for Free End-Time, Measurably Time Dependent Optimal Control Problems with State Constraints

Recently, necessary conditions have been derived for fixed-time optimal control problems with state constraints, formulated in terms of a differential inclusion, under very weak hypotheses on the data. These allow the multifunction describing admissible velocities to be unbounded and possibly nonconvex valued. This paper extends the earlier necessary conditions, to allow for free end-times. A notable feature of the new free end-time necessary conditions is that they cover problems with measurably time dependent data. For such problems, standard analytical techniques for deriving free-time necessary conditions, which depend on a transformation of the time variable, no longer work. Instead, we use variational methods based on the calculus of 'essential values".     

化不等式约束问题为无约束的广义可微限域精确罚函数方法

本文给出了广义可微精确罚函数的概念及一类所谓广义限域可微精确罚函数.本文预先选定罚因子,将不等式约束问题化为单一的无约束问题,并给出了具全局收敛性的算法.本文的罚函数构造简单,假设条件少而且算法的构造与收敛性结果是独特的.     

Optimality Conditions and Exact Penalty for Mathematical Programs with Switching Constraints

Journal of Optimization Theory and Applications - In this paper, we mainly study the Abadie constraint qualification (ACQ) and the strong ACQ of a convex multifunction. To characterize the general...     

弹性力学平面问题的精确元法

本文在阶梯折算法[1]和精确解析法[2]的基础上,提出构造有限元的新方法——精确元法.该方法不用一般变分原理,可适用于任意变系数正定和非正定偏微分方程.利用该方法得到弹性力学平面问题的一个非协调任意四边形单元.它具有八个自由度.由于没有采用雅可比变换,该单元可以蜕化为三角形单元,在工程中使用起来较为方便.文中给出收敛性证明.文末给出算例,位移和应力均给出较好的结果,在单元的节点上有较好的数值精度.     

Global Method for Monotone Variational Inequality Problems with Inequality Constraints

We consider optimization methods for monotone variational inequality problems with nonlinear inequality constraints. First, we study the mixed complementarity problem based on the original problem. Then, a merit function for the mixed complementarity problem is proposed, and some desirable properties of the merit function are obtained. Through the merit function, the original variational inequality problem is reformulated as simple bounded minimization. Under certain assumptions, we show that any stationary point of the optimization problem is a solution of the problem considered. Finally, we propose a descent method for the variational inequality problem and prove its global convergence.     

随机线性二次最优控制:从离散到连续时间模型

在一般情形下,分析了离散时间LQ问题与连续时间情形两者之间的自然联系.首先回顾了连续时间和离散时间随机LQ问题及对应Riccati微分/差分方程的相关结论.接下来在假设Riccati微分方程有解的前提下,证明了离散化步长足够小时,Riccati差分方程有解.然后针对连续和离散时间模型,采用配对问题最优控制的反馈形式,分别构造了一个辅助反馈控制,并证明该控制可驱使对应模型的性能指标逼近于配对问题的值函数,以此得到了关于两个模型之间联系的初步结论.最后藉由前述结论以及控制问题的特性,揭晓了连续时间和离散时间模型之间的自然联系,并给出了Riccati差分方程和微分方程的解之间的误差估计.由此联系,可构造相应离散系统和LQ问题,以适当的阶估计连续时间LQ问题的解,抑或为离散时间模型构造一个近似最优控制.无论哪种思路,都旨在降低直接求解原问题的难度和复杂性.     

An Optimal Control Method for Nonlinear Inverse Diffusion Coefficient Problem

This paper investigates the solution of a parameter identification problem associated with the two-dimensional heat equation with variable diffusion coefficient. The singularity of the diffusion coefficient results in a nonlinear inverse problem which makes theoretical analysis rather difficult. Using an optimal control method, we formulate the problem as a minimization problem and prove the existence and uniqueness of the solution in weighted Sobolev spaces. The necessary conditions for the existence of the minimizer are also given. The results can be extended to more general parabolic equations with singular coefficients.     

An Optimal Control Problem for Flows with Discontinuities

In this paper, we study a design problem for a duct flow with a shock. The presence of the shock causes numerical difficulties. Good shock-capturing schemes with low continuity properties often cannot be combined successfully with efficient optimization methods requiring smooth functions. A remedy studied in this paper is to introduce the shock location as an explicit variable. This allows one to fit the shock and yields a problem with sufficiently smooth functions. We prove the existence of optimal solutions, Fréchet differentiability, and the existence of Lagrange multipliers. In the second part, we introduce and investigate the discrete problem and study the relations between the optimality conditions for the infinite-dimensional problem and the discretized one. This reveals important information for the numerical solution of the problem. Numerical examples are given to demonstrate the theoretical findings.     

An Algorithm for Continuous Type Optimal Location Problem

In this paper, we extend the ordinary discrete type facility location problems to continuous type ones. Unlike the discrete type facility location problem in which the objective function isn't everywhere differentiable, the objective function in the continuous type facility location problem is strictly convex and continuously differentiable. An algorithm without line search for solving the continuous type facility location problems is proposed and its global convergence, linear convergence rate is proved. Numerical experiments illustrate that the algorithm suggested in this paper have smaller amount of computation, quicker convergence rate than the gradient method and conjugate direction method in some sense.     

Some Exact Penalty Results for Nonlinear Programs and Mathematical Programs with Equilibrium Constraints

Recently, some exact penalty results for nonlinear programs and mathematical programs with equilibrium constraints were proved by Luo, Pang, and Ralph (Ref. 1). In this paper, we show that those results remain valid under some other mild conditions. One of these conditions, called strong convexity with order , is discussed in detail.     

关于一类等式约束优化的简单光滑精确罚函数

本文给出了一类等式约束优化的简单光滑精确罚函数,该精确罚函数有别于传统罚函数,它是光滑的和简单的,即在该精确罚函数表达式中,不含有目标函数的梯度.     

状态约束下的抛物控制问题的罚函数方法

该文讨论了一类状态变量约束下由发展方程导出的最优控制系统，通过原问题的扰动，得到了状态变量与控制变量分离的最优性条件．