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Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

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NASA NTRS1976

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.

Edge, E. R.↗
NASA NTRS1981

Recent developments in quasi-Newton methods for structural analysis and synthesis

Unlike the Newton-Raphson method, quasi-Newton methods by virture of the updates and step length control procedures are globally convergent and hence better suited for the solution of nonlinear problems of structural analysis and synthesis. Extension of quasi-Newton algorithms to large scale problems has led to the development of sparse update algorithms and to economical strategies for evaluating sparse Hessians. Ill-conditioning problems have led to the development of self-scaled variable metric and conjugate gradient algorithms, as well as the use of the singular perturbation theory. This paper emphasizes the effectiveness of such quasi-Newton algorithms for nonlinear structural analysis and synthesis.

Kamat, M. P.↗
NASA NTRS1976

A quasi-Newton approach to optimization problems with probability density constraints

A quasi-Newton method is presented for minimizing a nonlinear function while constraining the variables to be nonnegative and sum to one. The nonnegativity constraints were eliminated by working with the squares of the variables and the resulting problem was solved using Tapia's general theory of quasi-Newton methods for constrained optimization. A user's guide for a computer program implementing this algorithm is provided.

Tapia, R. A.↗
NASA NTRS1978

Quasi-Newton Methods

The problem to be solved is formulated precisely and the introduction of quasi-Newton methods is motivated by considering the classical Newton and secant methods and their properties. Three highly successful quasi-Newton methods are surveyed: Broyden's method for the solution of general nonlinear equations, and the Davidon-Fletcher-Powell and Broyden-Fletcher-Goldfarb-Shanno procedures for unconstrained minimization. Finally, the properties of these methods are compared to those of Newton's method and UHMLE in potential applications to maximum-likelihood estimation of parameters in mixture distributions.

Walker, H. F.↗
NASA NTRS1973

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

Two existing function space algorithms, Davidon and projected gradient, are modified so that they may handle directly control variable inequality constraints. A third quasi-Newton type algorithm developed by C. G. Broyden is extended to optimal control problems. The Broyden algorithm is further modified so that it also may handle directly control variable inequality constraints. These methods along with a pure gradient and two conjugate gradient algorithms are simulated on three relatively simple yet representative bounded control problems, two of which have singular subarcs. Overall the Broyden algorithm was found to be superior. The most notable result of the study was the clear superiority of the Broyden and Davidon algorithms in producing a sharp interior control subarc.

Edge, E. R.↗
NASA NTRS1974

Shuttle ascent trajectory optimization with function space quasi-Newton techniques

A Space Shuttle ascent trajectory optimization problem from lift-off to orbital insertion is solved with a function space version of a quasi-Newton parameter optimization method developed by Broyden. The problem includes five parameter and one bounded-function controls, two state-variable constraints, and four terminal conditions. The bounded controls are treated directly, while the remaining constraints are adjoined to the performance index (maximum payload) with penalty functions. The problem is formulated as a four-phase variational problem (liftoff, pitch-over, gravity-turn, linear tangent steering), and the appropriate gradients are developed by first variation theory. A projection operator is introduced to aid in the interpretation of the algorithm with mixed parameter and function controls.

Edge, E. R.↗
NASA NTRS1981

A quasi-Newton procedure for identifying pilot-related parameters of the optimal control model

The development and application of a quasi-Newton gradient search procedure for identifying independent pilot related parameters of the optimal control model for pilot/vehicle systems is reported. A sensitivity analysis procedure which determines whether a given model parameter is required to match a specific experimental result, and which experimentally induced parameter changes are required to account for behavioral and performance differences, is described. Application of the identification scheme to training effects in a manual control task is described.

Levison, W. H.↗
NASA NTRS1983

A quasi-Newton versus a homotopy method for nonlinear structural analysis

The globally convergent quasi-Newton minimization algorithm and the homotopy algorithms are discussed in detail and their effectiveness in solving certain classes of highly nonlinear problems of structural analysis is demonstrated. The application of the double dogleg strategy controls the directions and step-lengths of the quasi-Newtonian algorithm and overcomes the problem of nonpositive definite Hessians being produced during the iteration process. The algorithms are applied to a centrally loaded clamped beam, the snap-through of a shallow arch, and a shallow reticulated dome.

Kamat, M. P.↗
NASA NTRS1982

Minimization versus homotopy algorithms

The relative merits and demerits of the minimization techniques are assessed using globally convergent quasi-Newton algorithms on the one hand and the homotopy algorithms on the other hand for the solution of problems of nonlinear structural analysis. Like the homotopy algorithms, the globally convergent quasi-Newton algorithms are equally suited for the solution of the nonlinear equations of structural analysis directly without having to pose the problem as an equivalent minimization problem. In the close neighborhood of the limit and bifurcation points quasi-Newton algorithms experience difficulties. Homotopy algorithms are robust for practically all types of nonlinear problems but are computationally not as cost effective since they provide an extremely accurate prediction of the response by calculating it as a large number of points. Globally convergent algorithms can perform well with very approximate Hessians, while homotopy algorithms require extremely accurate Hessians. While quasi-Newton algorithms can be very easily structured to exploit sparsity and symmetry, homotopy algorithms are not presently so structured and would require special modifications for exploitation of such features without sacrificing robustness and global convergence.

Kamat, M. P.↗
NASA NTRS1972

Davidon-Broyden rank-one minimization methods in Hilbert space with application to optimal control problems

The Davidon-Broyden class of rank one, quasi-Newton minimization methods is extended from Euclidean spaces to infinite-dimensional, real Hilbert spaces. For several techniques of choosing the step size, conditions are found which assure convergence of the associated iterates to the location of the minimum of a positive definite quadratic functional. For those techniques, convergence is achieved without the problem of the computation of a one-dimensional minimum at each iteration. The application of this class of minimization methods for the direct computation of the solution of an optimal control problem is outlined. The performance of various members of the class are compared by solving a sample optimal control problem. Finally, the sample problem is solved by other known gradient methods, and the results are compared with those obtained with the rank one quasi-Newton methods.

Straeter, T. A.↗
NASA NTRS1979

Optimization of multiple flyby trajectories

A procedure has been developed which minimizes total delta-V (instantaneous velocity change) for a multiple flyby trajectory with constraints on flyby altitude and orientation. The solution is found by varying the locations of maneuver points between each flyby to minimize the delta-Vs at the maneuver points. Each trajectory segment connecting consecutive maneuver points is found by solving an N-body analog to Lambert's problem. Multiconic techniques are used for trajectory propagation and for computation of the state transition matrix. The constrained parameter optimization problem is converted to an unconstrained problem by means of penalty functions and then solved with a quasi-Newton algorithm utilizing analytic first derivatives. This procedure has been successfully applied to Galileo satellite tour trajectories.

Damario, L. A.↗
NASA NTRS1982

Methodology for multiaircraft minimum noise impact landing trajectories

The problem of optimizing landing trajectories with respect to noise impact index is addressed. In contrast with previous work, multilandings are considered. In order to make the problem tractable the trajectories are specified in functional form with certain parameters left free for selection by the optimization methods. Constraint of aircraft dynamic behavior, trajectory separation, pilot workload, passenger comfort, and maximum noise intensity all enter into the determination of what is an allowable trajectory. A version of the quasi-Newton iterative procedure is used to determine the optimum parameter values. The results show improvement in noise impact to the airport considered and the potential for even greater improvement at many airports.

Cook, G.↗
NASA NTRS1983

Improved method for transonic airfoil design-by-optimization

An improved method for use of optimization techniques in transonic airfoil design is demonstrated. FLO6QNM incorporates a modified quasi-Newton optimization package, and is shown to be more reliable and efficient than the method developed previously at NASA-Ames, which used the COPES/CONMIN optimization program. The design codes are compared on a series of test cases with known solutions, and the effects of problem scaling, proximity of initial point to solution, and objective function precision are studied. In contrast to the older method, well-converged solutions are shown to be attainable in the context of engineering design using computational fluid dynamics tools, a new result. The improvements are due to better performance by the optimization routine and to the use of problem-adaptive finite difference step sizes for gradient evaluation.

Kennelly, R. A., Jr.↗
NASA NTRS1983

Finite element analysis of moisture effects in graphite-epoxy composites

A numerical analysis procedure is developed for the micro-mechanical investigation of moisture effects on graphite-epoxy composites. The procedure employs a nonlinear constitutive relationship in a finite element formulation. A quasi-Newton method is used to solve the resulting nonlinear algebraic equations. The results presented indicate that the stresses at the graphite-epoxy interface have a strong dependence on moisture content, fiber spacing, and applied load.

Tobing, F. E. L.↗
NASA NTRS1983

Improved method for transonic airfoil design-by-optimization

An improved method for use of optimization techniques in transonic airfoil design is demonstrated. FLO6QNM incorporates a modified quasi-Newton optimization package, and is shown to be more reliable and efficient than the method developed previously at NASA-Ames, which used the COPES/CONMIN optimization problem. The design codes are compared on a series of test cases with known solutions, and the effects of problem scaling, proximity of initial point to solution, and objective function precision are studied. In contrast to the older method, well-converged solutions are shown to be attainable in the context of engineering design using computational fluid dynamics tools, a new result. The improvements are due to better performance by the optimization routine and to the use of problem-adaptive finite difference step sizes for gradient evaluation.

Kennelly, R. A., Jr.↗
NASA NTRS1985

Algorithm Solves Constrained and Unconstrained Optimization Problems

Is quasi-Newton iteration utilizing Broyden/Fletcher/Goldfarb/Shanno update on inverse Hessian matrix. Capable of solving constrained optimization unconstrained optimization and constraints only problems with one to five independent variables from one to five constraint functions and one dependent function optimized.

Denson, M. A.↗