Higher order variational integrators in optimal control theory

Higher order variational integrators are analyzed and applied to optimal control problems posed with mechanical systems. First, we derive two different kinds of high order variational integrators based on different dimensions of the underlying approximation space. While the first well-known integrator is equivalent to a symplectic partitioned Runge-Kutta method, the second integrator, denoted as symplectic Galerkin integrator, yields a method which in general, cannot be written as a standard symplectic Runge-Kutta scheme [1]. Furthermore, we use these integrators for the discretization of optimal control problems. By analyzing the adjoint systems of the optimal control problem and its discretized counterpart, we prove that for these particular integrators optimization and discretization commute [2]. This property guarantees that the accuracy is preserved for the adjoint system which is also referred to as the Covector Mapping Principle. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Pseudospectral Chebyshev Optimal Control of Constrained Nonlinear Dynamical Systems

A pseudospectral method for generating optimal trajectories of linear and nonlinear constrained dynamic systems is proposed. The method consists of representing the solution of the optimal control problem by an mth degree interpolating polynomial, using Chebyshev nodes, and then discretizing the problem using a cell-averaging technique. The optimal control problem is thereby transformed into an algebraic nonlinear programming problem. Due to its dynamic nature, the proposed method avoids many of the numerical difficulties typically encountered in solving standard optimal control problems. Furthermore, for discontinuous optimal control problems, we develop and implement a Chebyshev smoothing procedure which extracts the piecewise smooth solution from the oscillatory solution near the points of discontinuities. Numerical examples are provided, which confirm the convergence of the proposed method. Moreover, a comparison is made with optimal solutions obtained by closed-form analysis and/or other numerical methods in the literature.     

A globally convergent semi-smooth Newton method for control-state constrained DAE optimal control problems

We investigate a semi-smooth Newton method for the numerical solution of optimal control problems subject to differential-algebraic equations (DAEs) and mixed control-state constraints. The necessary conditions are stated in terms of a local minimum principle. By use of the Fischer-Burmeister function the local minimum principle is transformed into an equivalent nonlinear and semi-smooth equation in appropriate Banach spaces. This nonlinear and semi-smooth equation is solved by a semi-smooth Newton method. We extend known local and global convergence results for ODE optimal control problems to the DAE optimal control problems under consideration. Special emphasis is laid on the calculation of Newton steps which are given by a linear DAE boundary value problem. Regularity conditions which ensure the existence of solutions are provided. A regularization strategy for inconsistent boundary value problems is suggested. Numerical illustrations for the optimal control of a pendulum and for the optimal control of discretized Navier-Stokes equations conclude the article.     

A fully distributed ADMM-based dispatch approach for virtual power plant problems

In this paper, a fully distributed approach is proposed for a class of virtual power plant (VPP) problems. By characterizing two specific VPP problems, we first give a comprehensive VPP formulation that maximizes the economic benefit subjected to the power balance constraint, line transmission limits and local constraints of all distributed energy resources (DERs). Then, utilizing the alternating direction method of multipliers and consensus optimization, a distributed VPP dispatch algorithm is developed for the general VPP problem. In particular, Theorem 1 is derived to show the convergence of the algorithm. The proposed algorithm is completely distributed without requiring a centralized controller, and each DER is regarded as an agent by implementing local computation and only communicates information with its neighbors to cooperatively find the globally optimal solution. The algorithm brings some advantages, such as the privacy protection and more scalability than centralized control methods. Furthermore, a new variant of the algorithm is presented for improving the convergence rate. Finally, several case studies are used to illustrate the efficiency and effectiveness of the proposed algorithms.     

最优控制问题参数化研究—基于勒让德正交多项式逼近

使用勒让德正交多项式逼近方法,将Lagrange型最优控制问题转化为非线性规划问题.采用序列二次规划方法对此非线性规划问题的求解,并对多项式逼近和非线性规划求解后得到的解是否收敛给出了证明和实例分析.     

On the Newton Interior-Point Method for Nonlinear Programming Problems

Interior-point methods have been developed largely for nonlinear programming problems. In this paper, we generalize the global Newton interior-point method introduced in Ref. 1 and we establish a global convergence theory for it, under the same assumptions as those stated in Ref. 1. The generalized algorithm gives the possibility of choosing different descent directions for a merit function so that difficulties due to small steplength for the perturbed Newton direction can be avoided. The particular choice of the perturbation enables us to interpret the generalized method as an inexact Newton method. Also, we suggest a more general criterion for backtracking, which is useful when the perturbed Newton system is not solved exactly. We include numerical experimentation on discrete optimal control problems.     

A globally convergent Newton method for solving strongly monotone variational inequalities

Variational inequality problems have been used to formulate and study equilibrium problems, which arise in many fields including economics, operations research and regional sciences. For solving variational inequality problems, various iterative methods such as projection methods and the nonlinear Jacobi method have been developed. These methods are convergent to a solution under certain conditions, but their rates of convergence are typically linear. In this paper we propose to modify the Newton method for variational inequality problems by using a certain differentiable merit function to determine a suitable step length. The purpose of introducing this merit function is to provide some measure of the discrepancy between the solution and the current iterate. It is then shown that, under the strong monotonicity assumption, the method is globally convergent and, under some additional assumptions, the rate of convergence is quadratic. Limited computational experience indicates the high efficiency of the proposed method.     

Stability in stochastic programming with recourse-estimated parameters

In this paper, stability of the optimal solution of stochastic programs with recourse with respect to parameters of the given distribution of random coefficients is studied. Provided that the set of admissible solutions is defined by equality constraints only, asymptotical normality of the optimal solution follows by standard methods. If nonnegativity constraints are taken into account the problem is solved under assumption of strict complementarity known from the theory of nonlinear programming (Theorem 1). The general results are applied to the simple recourse problem with random right-hand sides under various assumptions on the underlying distribution (Theorems 2–4).     

The application of optimal control methodology to nonlinear programming problems

Dynamic programming techniques have proven to be more successful than alternative nonlinear programming algorithms for solving many discrete-time optimal control problems. The reason for this is that, because of the stagewise decomposition which characterizes dynamic programming, the computational burden grows approximately linearly with the numbern of decision times, whereas the burden for other methods tends to grow faster (e.g.,n ³ for Newton's method). The idea motivating the present study is that the advantages of dynamic programming can be brought to bear on classical nonlinear programming problems if only they can somehow be rephrased as optimal control problems.As shown herein, it is indeed the case that many prominent problems in the nonlinear programming literature can be viewed as optimal control problems, and for these problems, modern dynamic programming methodology is competitive with respect to processing time. The mechanism behind this success is that such methodology achieves quadratic convergence without requiring solution of large systems of linear equations.     

An interior point potential reduction method for constrained equations

We study the problem of solving a constrained system of nonlinear equations by a combination of the classical damped Newton method for (unconstrained) smooth equations and the recent interior point potential reduction methods for linear programs, linear and nonlinear complementarity problems. In general, constrained equations provide a unified formulation for many mathematical programming problems, including complementarity problems of various kinds and the Karush-Kuhn-Tucker systems of variational inequalities and nonlinear programs. Combining ideas from the damped Newton and interior point methods, we present an iterative algorithm for solving a constrained system of equations and investigate its convergence properties. Specialization of the algorithm and its convergence analysis to complementarity problems of various kinds and the Karush-Kuhn-Tucker systems of variational inequalities are discussed in detail. We also report the computational results of the implementation of the algorithm for solving several classes of convex programs. The work of this author was based on research supported by the National Science Foundation under grants DDM-9104078 and CCR-9213739 and the Office of Naval Research under grant N00014-93-1-0228. The work of this author was based on research supported by the National Science Foundation under grant DMI-9496178 and the Office of Naval Research under grants N00014-93-1-0234 and N00014-94-1-0340.     

On the convergence of circle packings to the Riemann map

Rodin and Sullivan (1987) proved Thurston’s conjecture that a scheme based on the Circle Packing Theorem converges to the Riemann mapping, thereby providing a refreshing geometric view of Riemann’s Mapping Theorem. We now present a new proof of the Rodin–Sullivan theorem. This proof is based on the argument principle, and has the following virtues. 1. It applies to more general packings. The Rodin–Sullivan paper deals with packings based on the hexagonal combinatorics. Later, quantitative estimates were found, which also worked for bounded valence packings. Here, the bounded valence assumption is unnecessary and irrelevant. 2. Our method is rather elementary, and accessible to non-experts. In particular, quasiconformal maps are not needed. Consequently, this gives an independent proof of Riemann’s Conformal Mapping Theorem. (The Rodin–Sullivan proof uses results that rely on Riemann’s Mapping Theorem.) 3. Our approach gives the convergence of the first and second derivatives, without significant additional difficulties. While previous work has established the convergence of the first two derivatives for bounded valence packings, now the bounded valence assumption is unnecessary. Oblatum 15-V-1995 & 13-XI-1995     

The multiplier method of Hestenes and Powell applied to convex programming

For nonlinear programming problems with equality constraints, Hestenes and Powell have independently proposed a dual method of solution in which squares of the constraint functions are added as penalties to the Lagrangian, and a certain simple rule is used for updating the Lagrange multipliers after each cycle. Powell has essentially shown that the rate of convergence is linear if one starts with a sufficiently high penalty factor and sufficiently near to a local solution satisfying the usual second-order sufficient conditions for optimality. This paper furnishes the corresponding method for inequality-constrained problems. Global convergence to an optimal solution is established in the convex case for an arbitrary penalty factor and without the requirement that an exact minimum be calculated at each cycle. Furthermore, the Lagrange multipliers are shown to converge, even though the optimal multipliers may not be unique.This work was supported in part by the Air Force Office of Scientific Research under Grant No. AF-AFOSR-72-2269.     

Semidiscrete Ritz-Galerkin approximation of nonlinear parabolic boundary control problems—Strong convergence of optimal controls

A class of optimal control problems for a parabolic equation with nonlinear boundary condition and constraints on the control and the state is considered. Associated approximate problems are established, where the equation of state is defined by a semidiscrete Ritz-Galerkin method. Moreover, we are able to allow for the discretization of admissible controls. We show the convergence of the approximate controls to the solution of the exact control problem, as the discretization parameter tends toward zero. This result holds true under the assumption of a certain sufficient second-order optimality condition.Dedicated to the 60th birthday of Lothar von Wolfersdorf     

An extended Gauss–Seidel method for a class of multi-valued complementarity problems

The complementarity problem is one of the basic topics in nonlinear analysis; however, the methods for solving complementarity problems are usually developed for problems with single-valued mappings. In this paper we examine a class of complementarity problems with multi-valued mappings and propose an extension of the Gauss–Seidel algorithm for finding its solution. Its convergence is proved under off-diagonal antitonicity assumptions. Applications to Walrasian type equilibrium problems and to nonlinear input–output problems are also given. In this work, the authors were supported by Brescia University grant PRIN - 2006: “Oligopolistic models and order monotonicity properties”, the third author was also supported by the joint RFBR–NNSF grant, project No. 07-01-92101.     

Computing exact solution to nonlinear integer programming: Convergent Lagrangian and objective level cut method

In this paper, we propose a convergent Lagrangian and objective level cut method for computing exact solution to two classes of nonlinear integer programming problems: separable nonlinear integer programming and polynomial zero-one programming. The method exposes an optimal solution to the convex hull of a revised perturbation function by successively reshaping or re-confining the perturbation function. The objective level cut is used to eliminate the duality gap and thus to guarantee the convergence of the Lagrangian method on a revised domain. Computational results are reported for a variety of nonlinear integer programming problems and demonstrate that the proposed method is promising in solving medium-size nonlinear integer programming problems.     

Two-Grid Algorithm of $H^{1}$-Galerkin Mixed Finite Element Methods for Semilinear Parabolic Integro-Differential Equations

In this paper, we present a two-grid discretization scheme for semilinear parabolic integro-differential equations by $H^{1}$-Galerkin mixed finite element methods. We use the lowest order Raviart-Thomas mixed finite elements and continuous linear finite element for spatial discretization, and backward Euler scheme for temporal discretization. Firstly, a priori error estimates and some superclose properties are derived. Secondly, a two-grid scheme is presented and its convergence is discussed. In the proposed two-grid scheme, the solution of the nonlinear system on a fine grid is reduced to the solution of the nonlinear system on a much coarser grid and the solution of two symmetric and positive definite linear algebraic equations on the fine grid and the resulting solution still maintains optimal accuracy. Finally, a numerical experiment is implemented to verify theoretical results of the proposed scheme. The theoretical and numerical results show that the two-grid method achieves the same convergence property as the one-grid method with the choice $h=H^2$.     

A Discretizing Levenberg-Marquardt Scheme for Solving Nonlinear Ill-Posed Integral Equations

To reduce the computational cost, we propose a regularizing modified Levenberg-Marquardt scheme via multiscale Galerkin method for solving nonlinear ill-posed problems. Convergence results for the regularizing modified Levenberg-Marquardt scheme for the solution of nonlinear ill-posed problems have been proved. Based on these results, we propose a modified heuristic parameter choice rule to terminate the regularizing modified Levenberg-Marquardt scheme. By imposing certain conditions on the noise, we derive optimal convergence rates on the approximate solution under special source conditions. Numerical results are presented to illustrate the performance of the regularizing modified Levenberg-Marquardt scheme under the modified heuristic parameter choice.     

The Lagrange-Newton method for nonlinear optimal control problems

We investigate local convergence of the Lagrange-Newton method for nonlinear optimal control problems subject to control constraints including the situation where the terminal state is fixed. Sufficient conditions for local quadratic convergence of the method based on stability results for the solutions of nonlinear control problems are discussed.     

Fractional-Order Legendre Functions and Their Application to Solve Fractional Optimal Control of Systems Described by Integro-differential Equations

In this paper, we introduce a set of functions called fractional-order Legendre functions (FLFs) to obtain the numerical solution of optimal control problems subject to the linear and nonlinear fractional integro-differential equations. We consider the properties of these functions to construct the operational matrix of the fractional integration. Also, we achieved a general formulation for operational matrix of multiplication of these functions to solve the nonlinear problems for the first time. Then by using these matrices the mentioned fractional optimal control problem is reduced to a system of algebraic equations. In fact the functions of the problem are approximated by fractional-order Legendre functions with unknown coefficients in the constraint equations, performance index and conditions. Thus, a fractional optimal control problem converts to an optimization problem, which can then be solved numerically. The convergence of the method is discussed and finally, some numerical examples are presented to show the efficiency and accuracy of the method.     

An Objective Penalty Function of Bilevel Programming

Penalty methods are very efficient in finding an optimal solution to constrained optimization problems. In this paper, we present an objective penalty function with two penalty parameters for inequality constrained bilevel programming under the convexity assumption to the lower level problem. Under some conditions, an optimal solution to a bilevel programming defined by the objective penalty function is proved to be an optimal solution to the original bilevel programming. Moreover, based on the objective penalty function, an algorithm is developed to obtain an optimal solution to the original bilevel programming, with its convergence proved under some conditions.