Conditionally optimal algorithms and estimation of reduced order models

The paper presents some extensions of the optimality results obtained in previous work on algorithms used in the field of system identification in the light of information-based complexity. In particular, a class of conditional algorithms is defined by means of a restriction on the space of solution elements and a corresponding conditional worst case error is introduced. We define conditional central algorithms and show their optimality. A conditional central algorithm is then constructed by modifying a projection algorithm and obtaining in this way a conditional projection algorithm. This algorithm is shown to enjoy local optimality properties with reference to the problem element space within the class of conditionally correct algorithms. Finally, it is shown how these results can be used to handle the problem of reduced order model estimation.     

Recent advances in gradient algorithms for optimal control problems

This paper summarizes recent advances in the area of gradient algorithms for optimal control problems, with particular emphasis on the work performed by the staff of the Aero-Astronautics Group of Rice University. The following basic problem is considered: minimize a functionalI which depends on the statex(t), the controlu(t), and the parameter π. Here,I is a scalar,x ann-vector,u anm-vector, and π ap-vector. At the initial point, the state is prescribed. At the final point, the statex and the parameter π 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. First, the sequential gradient-restoration algorithm and the combined gradient-restoration algorithm are presented. The descent properties of these algorithms are studied, and schemes to determine the optimum stepsize are discussed. Both of the above algorithms require the solution of a linear, two-point boundary-value problem at each iteration. Hence, a discussion of integration techniques is given. Next, a family of gradient-restoration algorithms is introduced. Not only does this family include the previous two algorithms as particular cases, but it allows one to generate several additional algorithms, namely, those with alternate restoration and optional restoration. Then, two modifications of the sequential gradient-restoration algorithm are presented in an effort to accelerate terminal convergence. In the first modification, the quadratic constraint imposed on the variations of the control is modified by the inclusion of a positive-definite weighting matrix (the matrix of the second derivatives of the Hamiltonian with respect to the control). The second modification is a conjugate-gradient extension of the sequential gradient-restoration algorithm. Next, the addition of a nondifferential constraint, to be satisfied everywhere along the interval of integration, is considered. In theory, this seems to be only a minor modification of the basic problem. In practice, the change is considerable in that it enlarges dramatically the number and variety of problems of optimal control which can be treated by gradient-restoration algorithms. Indeed, by suitable transformations, almost every known problem of optimal control theory can be brought into this scheme. This statement applies, for instance, to the following situations: (i) problems with control equality constraints, (ii) problems with state equality constraints, (iii) problems with equality constraints on the time rate of change of the state, (iv) problems with control inequality constraints, (v) problems with state inequality constraints, and (vi) problems with inequality constraints on the time rate of change of the state. Finally, the simultaneous presence of nondifferential constraints and multiple subarcs is considered. The possibility that the analytical form of the functions under consideration might change from one subarc to another is taken into account. The resulting formulation is particularly relevant to those problems of optimal control involving bounds on the control or the state or the time derivative of the state. For these problems, one might be unwilling to accept the simplistic view of a continuous extremal arc. Indeed, one might want to take the more realistic view of an extremal arc composed of several subarcs, some internal to the boundary being considered and some lying on the boundary. The paper ends with a section dealing with transformation techniques. This section illustrates several analytical devices by means of which a great number of problems of optimal control can be reduced to one of the formulations presented here. In particular, the following topics are treated: (i) time normalization, (ii) free initial state, (iii) bounded control, and (iv) bounded state.     

Statistical problems of selection and synthesis of optimal algorithms

In this paper one investigates a probability model of a mixed algorithm and one formulates fundamental problems regarding the construction of an accuracy-optimal algorithm in the class of algorithms of bounded complexity. It is shown that different variants of the problem reduce to linear programming problems.We shall use the following notations negation - & conjunction - V disjunction - implication - equivalence - _M the indicator of the set M - the set of integers - ⁺ the set of natural numbers - pr₁ the first projection of an element from the Cartesian product of two sets - Pr₂ the second projection of an element from the Cartesian product of two sets - P the probability measure on a fundamental probability space - P the distribution of the vector of the data - E the symbol for mathematical expectation - E the symbol for the conditional mathematical expectation Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 166, pp. 72–90, 1988.     

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.     

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.     

Almost optimal solution of initial-value problems by randomized and quantum algorithms

We establish essentially optimal bounds on the complexity of initial-value problems in the randomized and quantum settings. For this purpose we define a sequence of new algorithms whose error/cost properties improve from step to step. These algorithms yield new upper complexity bounds, which differ from known lower bounds by only an arbitrarily small positive parameter in the exponent, and a logarithmic factor. In both the randomized and quantum settings, initial-value problems turn out to be essentially as difficult as scalar integration.     

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.     

Linear problems (with extended range) have linear optimal algorithms

LetF ₁ andF ₂ be normed linear spaces andS:F ₀ F ₂ a linear operator on a balanced subsetF ₀ ofF ₁. IfN denotes a finite dimensional linear information operator onF ₀, it is known that there need not be alinear algorithm:N(F ₄) F ₂ which is optimal in the sense that (N(f)) –S(f is minimized. We show that the linear problem defined byS andN can be regarded as having a linear optimal algorithm if we allow the range of to be extended in a natural way. The result depends upon imbeddingF ₂ isometrically in the space of continuous functions on a compact Hausdorff spaceX. This is done by making use of a consequence of the classical Banach-Alaoglu theorem.     

Strongly nonlinear unilateral problems

This paper will be concerned with the existence of the solutions to the strongly nonlinear unilateral (and bilateral) problems. Furthermore, some other properties of the solutions are proved.     

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.     

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.     

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.     

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...     

Hybrid estimation algorithms

The optimal, mean-square estimate of the state of a hybrid system is difficult to determine because the equations of state evolution are nonlinear and non-Gaussian. When there is a direct, albeit noisy, measurement of the modal state, it is possible to derive a useful approximation to the optimal estimator. This simplified algorithm is tested on a target tracking problem, and is seen to be superior to the conventional extended Kalman filter.This research was partially supported by a grant from the Hughes Aircraft Company and by the MICRO Program of the State of California under Project No. 91-156.     

Extending a class of continuous estimation of distribution algorithms to dynamic problems

In this paper, a class of continuous Estimation of Distribution Algorithms (EDAs) based on Gaussian models is analyzed to investigate their potential for solving dynamic optimization problems where the global optima may change dramatically during time. Experimental results on a number of dynamic problems show that the proposed strategy for dynamic optimization can significantly improve the performance of the original EDAs and the optimal solutions can be consistently located.     

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.     

A posteriori error estimation and adaptivity for elliptic optimal control problems with state constraints

In this paper optimal control problems governed by elliptic semilinear equations and subject to pointwise state constraints are considered. These problems are discretized using finite element methods and a posteriori error estimates are derived assessing the error with respect to the cost functional. These estimates are used to obtain quantitative information on the discretization error as well as for guiding an adaptive algorithm for local mesh refinement. Numerical examples illustrate the behavior of the method.     

Strongly polynomial and fully combinatorial algorithms for bisubmodular function minimization

Bisubmodular functions are a natural “directed”, or “signed”, extension of submodular functions with several applications. Recently Fujishige and Iwata showed how to extend the Iwata, Fleischer, and Fujishige (IFF) algorithm for submodular function minimization (SFM) to bisubmodular function minimization (BSFM). However, they were able to extend only the weakly polynomial version of IFF to BSFM. Here we investigate the difficulty that prevented them from also extending the strongly polynomial version of IFF to BSFM, and we show a way around the difficulty. This new method gives a somewhat simpler strongly polynomial SFM algorithm, as well as the first combinatorial strongly polynomial algorithm for BSFM. This further leads to extending Iwata’s fully combinatorial version of IFF to BSFM. The research of S. T. McCormick was supported by an NSERC Operating Grant. The research of S. Fujishige was supported by a Grant-in-Aid of the Ministry of Education, Culture, Science and Technology of Japan.     

Correct algebras over estimation algorithms in the set of regular recognition problems with nonoverlapping classes

Algebras over estimation algorithms in the set of regular problems with nonoverlapping classes are considered. A correctness criterion for the arbitrary degree algebraic closure of the model of estimation algorithms in the classification problems of this type is proposed; this criterion can be efficiently verified. An estimate of the minimal degree of the algebraic closure that is sufficient for constructing a correct classifier in an arbitrary regular problem with nonoverlapping classes is found.