Convergence of gradient methods for optimal control problems

A general convergence theorem for gradient algorithms in normed spaces is given and is applied to the unconstrained optimal control problem. A further application is given to time-lag systems of neutral type.This work was completed while the author held a Science Research Council Postdoctoral Fellowship at Loughborough University of Technology, Loughborough, Leicestershire, England.     

On conditions for the convergence of algorithms in optimal control problems

We obtain sufficient conditions for the convergence of iteration algorithms in Banach spaces. These conditions imply the weak convergence of subsequences to the set of solutions. Bibliography: 4 titles. Translated fromObchyslyuval'na ta Prykladna Matematyka, No. 81 1997, pp. 70–71.     

Summary and comparison of gradient-restoration algorithms for optimal control problems

This paper considers the problem of minimizing a functionalI which depends on the statex(t), the controlu(t), and the parameter . Here,I is a scalar,x ann-vector,u anm-vector, and ap-vector. At the initial point, the state is prescribed. At the final point, the state and the parameter are required to satisfyq scalar relations. Along the interval of integration, the state, the control, and the parameter are required to satisfyn scalar differential equations.Four types of gradient-restoration algorithms are considered, and their relative efficiency (in terms of number of iterations for convergence) is evaluated. The algorithms being considered are as follows: sequential gradient-restoration algorithm, complete restoration (SGRA-CR), sequential gradient-restoration algorithm, incomplete restoration (SGRA-IR), combined gradient-restoration algorithm, no restoration (CGRA-NR), and combined gradient-restoration algorithm, incomplete restoration (CGRA-IR).Evaluation of these algorithms is accomplished through six numerical examples. The results indicate that (i) the inclusion of a restoration phase is necessary for rapid convergence and (ii) while SGRA-CR is the most desirable algorithm if feasibility of the suboptimal solutions is required, rapidity of convergence to the optimal solution can be increased if one employs algorithms with incomplete restoration, in particular, CGRA-IR.This research was supported by the Office of Scientific Research, Office of Aerospace Research, United States Air Force, Grant No. AF-AFOSR-72-2185.     

Supplementary optimality properties of gradient-restoration algorithms for optimal control problems

In this paper, sequential gradient-restoration algorithms for optimal control problems are considered, and attention is focused on the gradient phase. It is shown that the Lagrange multipliers associated with the gradient phase not only solve the auxiliary minimization problem of the gradient phase, but are also endowed with a supplementary optimality property: they minimize the error in the optimality conditions, subject to the multiplier differential equations and boundary conditions, for given state, control, and parameter.Dedicated to R. BellmanThis work was supported by the National Science Foundation, Grant No. ENG-79-18667.     

Convergence of the gradient projection method in optimal control problems

The article examines the convergence of the popular gradient projection method for optimal control problems with fixed time and a free right-hand end point. General conditions are derived that substantiate weak convergence of the sequence of control iterations to the set of extremum controls satisfying Pontryagin’s maximum principle. Under stronger assumptions we prove strong convergence of the sequence in the Banach L ₁-norm. __________ Translated from Nelineinaya Dinamika i Upravlenie, No. 3, pp. 139–148, 2003.     

Parallel algorithms for solving nonlinear two-point boundary-value problems which arise in optimal control

This paper describes a collection of parallel optimal control algorithms which are suitable for implementation on an advanced computer with the facility for large-scale parallel processing. Specifically, a parallel nongradient algorithm and a parallel variablemetric algorithm are used to search for the initial costate vector that defines the solution to the optimal control problem. To avoid the computational problems sometimes associated with simultaneous forward integration of both the state and costate equations, a parallel shooting procedure based upon partitioning of the integration interval is considered. To further speed computations, parallel integration methods are proposed. Application of this all-parallel procedure to a forced Van der Pol system indicates that convergence time is significantly less than that required by highly efficient serial procedures.This research was supported in part by the Air Force Office of Scientific Research, Air Force Systems Command, USAF, under Grant No. AFOSR-77-3418.     

First-order strong variation algorithms for optimal control problems with terminal inequality constraints

Journal of Optimization Theory and Applications - This paper presents eight algorithms for solving optimal control problems with general constraints on the control and inequality constraints on the...     

Recent advances in vehicle routing exact algorithms

The capacitated vehicle routing problem (CVRP) is the problem in which a set of identical vehicles located at a central depot is to be optimally routed to supply customers with known demands subject to vehicle capacity constraints. This paper provides a review of the most recent developments that had a major impact in the current state-of-the-art of exact algorithms for the CVRP. The most important mathematical formulations for the problem together with various CVRP relaxations are reviewed. The paper also describes the recent exact methods for the CVRP and reports a comparison of their computational performances.      

Function-space quasi-Newton algorithms for optimal control problems with bounded controls and singular arcs

Two existing function-space quasi-Newton algorithms, the Davidon algorithm and the projected gradient algorithm, are modified so that they may handle directly control-variable inequality constraints. A third quasi-Newton-type algorithm, developed by Broyden, is extended to optimal control problems. The Broyden algorithm is further modified so that it may handle directly control-variable inequality constraints. From a computational viewpoint, dyadic operator implementation of quasi-Newton methods is shown to be superior to the integral kernel representation. The quasi-Newton methods, along with the steepest descent method and two conjugate gradient algorithms, are simulated on three relatively simple (yet representative) bounded control problems, two of which possess singular subarcs. Overall, the Broyden algorithm was found to be superior. The most notable result of the simulations was the clear superiority of the Broyden and Davidon algorithms in producing a sharp singular control subarc.This research was supported by the National Science Foundation under Grant Nos. GK-30115 and ENG 74-21618 and by the National Aeronautics and Space Administration under Contract No. NAS 9-12872.     

Strongly optimal algorithms and optimal information in estimation problems

This paper studies some aspects of information-based complexity theory applied to estimation, identification, and prediction problems. Particular emphasis is given to constructive aspects of optimal algorithms and optimal information, taking into account the characteristics of certain types of problems. Special attention is devoted to the investigation of strongly optimal algorithms and optimal information in the linear case. Two main results are obtained for the class of problems considered. First, central algorithms are proved to be strongly optimal. Second, a simple solution is given to a particular case of optimal information, called optimal sampling design, which is of great interest in system and identification theory.     

A priori error estimates for nonstationary optimal control problems with gradient state constraints

We are concerned with the discretization of semilinear parabolic optimization problems subject to constraints on the first derivative of the solution of the PDE. Constructing sequences of feasible controls for the dG(0)-cG(1) discretization of the problem and relying on a quadratric growth condtion, we show convergence orders as temporal and spatial mesh tends to zero. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On parabolic distributed optimal control problems with restrictions on the gradient

Convex optimal control problems for parabolic distributed parameter systems with an integral constraint for the gradient of the state are considered. The objective function is a general convex integral functional in which the control constraints are incorporated. Necessary and sufficient conditions for optimality are derived.     

Tuning distributed control algorithms for optimal functioning

In this paper, we present a model which characterizes distributed computing algorithms. The goals of this model are to offer an abstract representation of asynchronous and heterogeneous distributed systems, to present a mechanism for specifying externally observable behaviours of distributed processes and to provide rules for combining these processes into networks with desired properties (good functioning, fairness...). Once these good properties are found, the determination of the optimal rules are studied.Subsequently, the model is applied to three classical distributed computing problems: namely the dining philosophers problem, the mutual exclusion problem and the deadlock problem, (generalizing results of our previous publications [1], [2]). The property of fairness has a special position that we discuss.     

First-order strong variation algorithms for optimal control

Journal of Optimization Theory and Applications - This paper presents two demonstrably convergent, first-order, differential dynamic programming algorithms for the solution of optimal control...     

A numerical study of augmented penalty function algorithms for terminally constrained optimal control problems

Three augmented penalty function algorithms are tested and compared with an ordinary penalty function algorithm for two demonstration optimal control problems. Although the augmented penalty function is quite helpful in solving control problems with terminal state constraints, the convergence can be improved significantly by providing systematic increases in the penalty constant.     

On an inexact gradient method using Proper Orthogonal Decomposition for parabolic optimal control problems

This paper is devoted to a numerical solution technique for linear quadratic parabolic optimal control problems using the model order reduction technique of Proper Orthogonal Decomposition (POD). The proposed technique is an inexact gradient descent method where the step size is determined with a line-search algorithm evaluating the state and adjoint equations with POD. The gradient is evaluated with a Finite Element method which allows for a recently developed a posteriori error estimation technique to rate the error in the control. The method is compared to another algorithm presented by Tröltzsch and Volkwein (Comput. Optim. Appl. 42(1):43–63 2009).     

Genetic algorithms applied to the solution of hybrid optimal control problems in astrodynamics

Many space mission planning problems may be formulated as hybrid optimal control problems, i.e. problems that include both continuous-valued variables and categorical (binary) variables. There may be thousands to millions of possible solutions; a current practice is to pre-prune the categorical state space to limit the number of possible missions to a number that may be evaluated via total enumeration. Of course this risks pruning away the optimal solution. The method developed here avoids the need for pre-pruning by incorporating a new solution approach using nested genetic algorithms; an outer-loop genetic algorithm that optimizes the categorical variable sequence and an inner-loop genetic algorithm that can use either a shape-based approximation or a Lambert problem solver to quickly locate near-optimal solutions and return the cost to the outer-loop genetic algorithm. This solution technique is tested on three asteroid tour missions of increasing complexity and is shown to yield near-optimal, and possibly optimal, missions in many fewer evaluations than total enumeration would require.     

Achieving optimal convergence order for FEM in control constrained optimal control problems

We propose a new heuristic approach to overcome convergence order restrictions implied by the low regularity of the optimal control due to the activity interface of control constraints. Aligning the mesh with the interface yields an improved approximation of the control. Utility of the approach is demonstrated by numerical experiments. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)