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NASA NTRS1991

The nonconvex multi-dimensional Riemann problem for Hamilton-Jacobi equations

Simple inequalities are presented for the viscosity solution of a Hamilton-Jacobi equation in N space dimensions when neither the initial data nor the Hamiltonian need be convex (or concave). The initial data are uniformly Lipschitz and can be written as the sum of a convex function in a group of variables and a concave function in the remaining variables, therefore including the nonconvex Riemann problem. The inequalities become equalities wherever a 'maxmin' equals a 'minmax', and thus a representation formula for this problem is obtained, generalizing the classical Hopi formulas.

Bardi, Martino↗

NASA NTRS2024

A Convexification-Based Outer-Approximation Method for Convex and Nonconvex MINLP

The advancement of domain reduction techniques has significantly enhanced the performance of solvers in mathematical programming. This paper delves into the impact of integrating convexification and domain reduction techniques within the Outer-Approximation method. We propose a refined convexification-based Outer-Approximation method alongside a Branch-and-Bound method for both convex and nonconvex Mixed-Integer Nonlinear Programming problems. These methods have been developed and incorporated into the open-source Mixed-Integer Nonlinear Decomposition Toolbox for Pyomo-MindtPy. Comprehensive benchmark tests were conducted, validating the effectiveness and reliability of our proposed algorithms. These tests highlight the improvements achieved by incorporating convexification and domain reduction techniques into the Outer-Approximation and Branch-and-Bound methods.

Optimization↗

NASA NTRS1989

The nonconvex multi-dimensional Riemann problem for Hamilton-Jacobi equations

Simple inequalities for the Riemann problem for a Hamilton-Jacobi equation in N space dimension when neither the initial data nor the Hamiltonian need be convex (or concave) are presented. The initial data is globally continuous, affine in each orthant, with a possible jump in normal derivative across each coordinate plane, x sub i = 0. The inequalities become equalities wherever a maxmin equals a minmax and thus an exact closed form solution to this problem is then obtained.

Osher, Stanley↗

NASA NTRS2008

Robust Path Planning and Feedback Design Under Stochastic Uncertainty

Autonomous vehicles require optimal path planning algorithms to achieve mission goals while avoiding obstacles and being robust to uncertainties. The uncertainties arise from exogenous disturbances, modeling errors, and sensor noise, which can be characterized via stochastic models. Previous work defined a notion of robustness in a stochastic setting by using the concept of chance constraints. This requires that mission constraint violation can occur with a probability less than a prescribed value.In this paper we describe a novel method for optimal chance constrained path planning with feedback design. The approach optimizes both the reference trajectory to be followed and the feedback controller used to reject uncertainty. Our method extends recent results in constrained control synthesis based on convex optimization to solve control problems with nonconvex constraints. This extension is essential for path planning problems, which inherently have nonconvex obstacle avoidance constraints. Unlike previous approaches to chance constrained path planning, the new approach optimizes the feedback gain as wellas the reference trajectory.The key idea is to couple a fast, nonconvex solver that does not take into account uncertainty, with existing robust approaches that apply only to convex feasible regions. By alternating between robust and nonrobust solutions, the new algorithm guarantees convergence to a global optimum. We apply the new method to an unmanned aircraft and show simulation results that demonstrate the efficacy of the approach.

autonomuys vehicles↗

NASA NTRS2012

Lossless Convexification of Control Constraints for a Class of Nonlinear Optimal Control Problems

In this paper we consider a class of optimal control problems that have continuous-time nonlinear dynamics and nonconvex control constraints. We propose a convex relaxation of the nonconvex control constraints, and prove that the optimal solution to the relaxed problem is the globally optimal solution to the original problem with nonconvex control constraints. This lossless convexification enables a computationally simpler problem to be solved instead of the original problem. We demonstrate the approach in simulation with a planetary soft landing problem involving a nonlinear gravity field.

planetary soft landing↗

NASA NTRS1989

Recent experience in simultaneous control-structure optimization

To show the feasibility of simultaneous optimization as design procedure, low order problems were used in conjunction with simple control formulations. The numerical results indicate that simultaneous optimization is not only feasible, but also advantageous. Such advantages come at the expense of introducing complexities beyond those encountered in structure optimization alone, or control optimization alone. Examples include: larger design parameter space, optimization may combine continuous and combinatoric variables, and the combined objective function may be nonconvex. Future extensions to include large order problems, more complex objective functions and constraints, and more sophisticated control formulations will require further research to ensure that the additional complexities do not outweigh the advantages of simultaneous optimization. Some areas requiring more efficient tools than currently available include: multiobjective criteria and nonconvex optimization. Efficient techniques to deal with optimization over combinatoric and continuous variables, and with truncation issues for structure and control parameters of both the model space as well as the design space need to be developed.

Salama, M.↗

NASA NTRS1992

Design of structure/control systems with transient response constraints exhibiting relative minima

Structural optimization problems involving dynamic behavior constraints often exhibit nonconvex design spaces. The direct application of a global optimization algorithm requires a large number of function evaluations which in term require a large number of dynamic structural analyses. This work presents a strategy aimed at finding the global optimum for problems with transient dynamic behavior constraints based on approximation concepts. The method consists of generating and solving a sequence of approximate problems using a global optimizer. The approximations are explicit and capture the inherent nonconvexity of the exact functions. A simple example problem is presented.

Sepulveda, A. E.↗

NASA NTRS2013

Mixed-Strategy Chance Constrained Optimal Control

This paper presents a novel chance constrained optimal control (CCOC) algorithm that chooses a control action probabilistically. A CCOC problem is to find a control input that minimizes the expected cost while guaranteeing that the probability of violating a set of constraints is below a user-specified threshold. We show that a probabilistic control approach, which we refer to as a mixed control strategy, enables us to obtain a cost that is better than what deterministic control strategies can achieve when the CCOC problem is nonconvex. The resulting mixed-strategy CCOC problem turns out to be a convexification of the original nonconvex CCOC problem. Furthermore, we also show that a mixed control strategy only needs to "mix" up to two deterministic control actions in order to achieve optimality. Building upon an iterative dual optimization, the proposed algorithm quickly converges to the optimal mixed control strategy with a user-specified tolerance.

Ono, Masahiro↗

NASA NTRS1973

A convergence theory for a class of nonlinear programming problems.

A recent convergence theory of Elkin concerning methods for unconstrained minimization is extended to a certain class of nonlinear programming problems. As in Elkin's original approach, the analysis of a variety of step-length algorithms is treated entirely separately from that of several direction algorithms. This allows for their combination into many different methods for solving the constrained problem. These include some of the methods of Rosen and Zoutendijk. We also extend the results of Topkis and Veinott to nonconvex sets and drop their requirement of the uniform feasibility of a subsequence of the search directions.

Rauch, S. W.↗

NASA NTRS1983

Structural optimization with dynamic behavior constraints

The minimum weight optimum design of damped linearly elastic structural systems subjected to periodic loading with behavior constraints on maximum deflections and side constraints on design variables is addressed. Attention is focused on the two major impediments to an optimal solution: (1) the time parametric nature of the behavior constraints; and (2) the severe nonconvexity of the design space. A solution method based on upper bound approximations for the behavior constraints and an innovative mathematical programming scheme for seeking the optimal frequency subspace is set forth. Numerical results for several test problems illustrate the effectiveness of the method reported.

Mills-Curran, W. C.↗

NASA NTRS1984

Classical and neo-classical cruise-dash optimization

Cruise-dash flight performance is analyzed in the context of singular perturbations. Attention is given to the problem of determining an atmospheric flight path between given end points which minimizes a linear combination of time and fuel. It is shown that nonconvexity in the fuel-flow vs. airspeed graph has important consequences in optimum-cruise problems with time restrictions. Certain velocity regions are nonoptimal for cruise-dash and optimal cruise-dash sometimes requires time-shared operation between two altitude-airspeed points. Calculations are presented illustrating the occurrence of time-shared operation between two altitude-airspeed combinations for optimal cruise-dash.

Cliff, E. M.↗

NASA NTRS1985

Optimal symmetric flight studies

Several topics in optimal symmetric flight of airbreathing vehicles are examined. In one study, an approximation scheme designed for onboard real-time energy management of climb-dash is developed and calculations for a high-performance aircraft presented. In another, a vehicle model intermediate in complexity between energy and point-mass models is explored and some quirks in optimal flight characteristics peculiar to the model uncovered. In yet another study, energy-modelling procedures are re-examined with a view to stretching the range of validity of zeroth-order approximation by special choice of state variables. In a final study, time-fuel tradeoffs in cruise-dash are examined for the consequences of nonconvexities appearing in the classical steady cruise-dash model. Two appendices provide retrospective looks at two early publications on energy modelling and related optimal control theory.

Weston, A. R.↗

NASA NTRS1987

Collision detection for spacecraft proximity operations

Collision Detection for Spacecraft Proximity Operations This thesis describes the development of a new collision detection algorithm to be used when two spacecraft are operating in the same vicinity. The two spacecraft are modelled as unions of convex polyhedra, where the polyhedron resulting from the union may be either convex or nonconvex. The relative motion of the two spacecraft is assumed to be such that one vehicle is moving with constant linear and angular velocity with respect to the other. The algorithm determines if a collision is possible and, if so, predicts the time when the collision will take place. The theoretical basis for the new collision detection algorithm is the C-function formulation of the configuration space approach recently introduced by researchers in robotics. Three different types of C-functions are defined that model the contacts between the vertices, edges, and faces of the polyhedra representing the two spacecraft. These C-functions are used to formulate three "collision" conditions. The first of these conditions limits the points representing potential collisions to the zeros of the C-functions. The new algorithm is fundamentally a search for the smallest zero of any C-function that satisfies the second and third collision conditions. The C-functions are shown to be transcendental functions of time for the assumed trajectory of the moving spacecraft. The zeros of these functions cannot be expressed in dosed form. Therefore, numerical search procedures are developed to find aLl of the zeros of a C-function in specified bounded intervals of time. These bounded intervals of time are found by examining the second and third collision conditions. The capabilities of the new algorithm are demonstrated for several example cases. These include examples of collisions determined by zeros of each of the three different types of C-functions. In addition to predicting the time of first contact of the polyhedra, the algorithm identifies the features of the two polyhedra that are touching at this time. The new collision detection algorithm is the first such algorithm that is capable of solving the collision detection problem exactly for the case where the moving object has constant linear and angular velocities. This is a significant improvement on previous collision detection algorithms described in the literature. In particular, the ability to handle constant angular velocity represents a more realistic type of rotational motion than those which have been used in other algorithms.

Robin M Vaughan↗

NASA NTRS1988

Radar investigation of asteroids and planetary satellites

The aim is to make radar reconnaissance of near-Earth asteroids, mainbelt ateroids, the Galilean satellites, the Martian satellites, and the largest Saturnian satellites, using the Arecibo 13-cm and the Goldstone 3.5-cm systems. Measurements of echo strength, polarization, and delay/Doppler distribution of echo power provide information about dimensions, spin vector, large-scale topography, cm-to-m-scale morphology, and surface bulk density. The observations also yield refined estimates of target orbital elements. Radar signatures were measured for 31 mainbelt asteroids and 16 near-Earth asteroids since this task began eight years ago. The dispersion in asteroid radar albedoes and circular polarization ratios is extreme, revealing huge differences in surface morphologies, bulk densities, and metal concentration. For the most part, correction between radar signature and VIS/IR class is not high. Many near-Earth asteroids have extremely irregular, nonconvex shapes, but some have polar silhouettes that appear only slightly noncircular. The signatures of 1627 Ivar, 1986 DA, and the approximately 180-km mainbelt asteroid 216 Kleopatra suggest bifurcated shapes. Observational milestones during 1987 and 1988 are noted.

Ostro, Steven J.↗

NASA NTRS1990

Radar images of asteroid 1627 Ivar

Radar echoes from the near-earth asteroid 1627 Ivar, whose orbit crosses the earth's, reveal it to be about twice as long as it is wide, with a maximum dimension no less than 7 km and probably within 20 percent of 12 km. The surface is fairly smooth at centimeter-to-meter scales but appears irregular and nonconvex at kilometer scales.

Ostro, S. J.↗

NASA NTRS1991

Collision detection for spacecraft proximity operations

A new collision detection algorithm has been developed for use when two spacecraft are operating in the same vicinity. The two spacecraft are modeled as unions of convex polyhedra, where the resulting polyhedron many be either convex or nonconvex. The relative motion of the two spacecraft is assumed to be such that one vehicle is moving with constant linear and angular velocity with respect to the other. Contacts between the vertices, faces, and edges of the polyhedra representing the two spacecraft are shown to occur when the value of one or more of a set of functions is zero. The collision detection algorithm is then formulated as a search for the zeros (roots) of these functions. Special properties of the functions for the assumed relative trajectory are exploited to expedite the zero search. The new algorithm is the first algorithm that can solve the collision detection problem exactly for relative motion with constant angular velocity. This is a significant improvement over models of rotational motion used in previous collision detection algorithms.

Vaughan, Robin M.↗

NASA NTRS1991

New displacement-based methods for optimal truss topology design

Two alternate methods for maximum stiffness truss topology design are presented. The ground structure approach is used, and the problem is formulated in terms of displacements and bar areas. This large, nonconvex optimization problem can be solved by a simultaneous analysis and design approach. Alternatively, an equivalent, unconstrained, and convex problem in the displacements only can be formulated, and this problem can be solved by a nonsmooth, steepest descent algorithm. In both methods, the explicit solving of the equilibrium equations and the assembly of the global stiffness matrix are circumvented. A large number of examples have been studied, showing the attractive features of topology design as well as exposing interesting features of optimal topologies.

Bendsoe, Martin P.↗

NASA NTRS1995

Radar Observations of Asteroids 7 Iris, 9 Metis, 12 Victoria, 216 Kleopatra, and 654 Zelinda

We report 13-cm wavelength radar observations of the main-belt asteroids 7 Iris, 9 Metis, 12 Victoria, 216 Kleopatra, and 654 Zelinda obtained at Arecibo between 1980 and 1989. The echoes are highly polarized yet broadly distributed in Doppler frequency, indicating that our targets are smooth on decimeter scales but very rough on some scale(s) larger than about I m. The echo spectra are generally consistent with existing size, shape, and spin information based on radiometric, lightcurve, and occultation data. All of our targets possess distinctive radar signatures that reveal large- scale topography. Reflectivity spikes within narrow ranges of rotation phase suggest large flat regions on Iris, Metis, and Zelinda, while bimodal spectra imply nonconvex, possibly bifurcated shapes for Kleopatra and Victoria. Kleopatra has the highest radar albedo yet measured for a main-belt asteroid, indicating a high metal concentration and making Kleopatra the best main-belt candidate for a core remnant of a differentiated and subsequently disrupted parent body. Upon completion of the Arecibo telescope upgrade, there will be several opportunities per year to resolve main-belt asteroids with hundreds of delay-Doppler cells, which can be inverted to provide estimates of both three-dimensional shape and radar scattering properties.

Mitchell, David L.↗