Showing papers on "Nonlinear programming published in 1992"

Global Optimization: Deterministic Approaches

Abstract: Contents: Some Important Classes of Global Optimization Problems.- Outer Approximation.- Concavity Cut.- Branch and Bound.- Cutting Methods.- Successive Approximation Methods.- Successive Partition Methods.- Decomposition of Large Scale Problems.- Special Problems of Concave Minimization.- D.C. Programming.- Lipschitz and Continuous Optimization.

Functional optimization for fair surface design

Abstract: This paper presents a simple-to-use mechanism for the creation of complex smoothly shaped surfaces of any genus or topology. The work described here is the result of research into the fairness of curves and surfaces specified through geometric interpolarity constraints. Constraints consist of positions and, optionally, surface normals and surface curvatures. The outcome of our investigation is a recommendation for the use of nonlinear optimization techniques that minimize a fairness functional based on the variation of curvature. The approach produces very high quality surfaces with predictable, intuitive behavior, while generating, where possible, simple shapes, such as cylinders, spheres, or tori which are commonly used in geometric modeling. From a designer''s point of view, this approach allows the specification of a desired surface in the most natural way. Though computationally intense, the techniques described have now become practical because of the wide availability of very fast work stations. As the processing power available on each desk-top further increases, the techniques described here will become real-time and interactive.

A new reformulation-linearization technique for bilinear programming problems

Abstract: This paper is concerned with the development of an algorithm for general bilinear programming problems. Such problems find numerous applications in economics and game theory, location theory, nonlinear multi-commodity network flows, dynamic assignment and production, and various risk management problems. The proposed approach develops a new Reformulation-Linearization Technique (RLT) for this problem, and imbeds it within a provably convergent branch-and-bound algorithm. The method first reformulates the problem by constructing a set of nonnegative variable factors using the problem constraints, and suitably multiplies combinations of these factors with the original problem constraints to generate additional valid nonlinear constraints. The resulting nonlinear program is subsequently linearized by defining a new set of variables, one for each nonlinear term. This “RLT” process yields a linear programming problem whose optimal value provides a tight lower bound on the optimal value to the bilinear programming problem. Various implementation schemes and constraint generation procedures are investigated for the purpose of further tightening the resulting linearization. The lower bound thus produced theoretically dominates, and practically is far tighter, than that obtained by using convex envelopes over hyper-rectangles. In fact, for some special cases, this process is shown to yield an exact linear programming representation. For the associated branch-and-bound algorithm, various admissible branching schemes are discussed, including one in which branching is performed by partitioning the intervals for only one set of variables x or y, whichever are fewer in number. Computational experience is provided to demonstrate the viability of the algorithm. For a large number of test problems from the literature, the initial bounding linear program itself solves the underlying bilinear programming problem.

Discrete approximations to optimal trajectories using direct transcription and nonlinear programming

Abstract: A recently developed method for solving optimal trajectory problems uses a piecewise-polynomial representation of the state and control variables, enforces the equations of motion via a collocation procedure, and thus approximates the original calculus-of-variations problem with a nonlinear-programming problem, which is solved numerically. This paper identifies this method as a direct transcription method and proceeds to investigate the relationship between the original optimal-control problem and the nonlinear-programming problem. The discretized adjoint equation of the collocation method is found to have deficient accuracy, and an alternate scheme which discretizes the equations of motion using an explicit Runge-Kutta parallel-shooting approach is developed. Both methods are applied to finite-thrust spacecraft trajectory problems, including a low-thrust escape spiral, a three-burn rendezvous, and a low-thrust transfer to the moon.

Lagrange programming neural networks

Abstract: A class of neural networks appropriate for general nonlinear programming, i.e., problems including both equality and inequality constraints, is analyzed in detail. The methodology is based on the Lagrange multiplier theory in optimization and seeks to provide solutions satisfying the necessary conditions of optimality. The equilibrium point of the network satisfies the Kuhn-Tucker condition for the problem. No explicit restriction is imposed on the form of the cost function apart from some general regularity and convexity conditions. The stability of the neural networks is analyzed in detail. The transient behavior of the network is simulated and the validity of the approach is verified for a practical problem, maximum entropy image restoration. >

Theory of algorithms for unconstrained optimization

Abstract: A few months ago, while preparing a lecture to an audience that included engineers and numerical analysts, I asked myself the question: from the point of view of a user of nonlinear optimization routines, how interesting and practical is the body of theoretical analysis developed in this field? To make the question a bit more precise, I decided to select the best optimization methods known to date – those methods that deserve to be in a subroutine library – and for each method ask: what do we know about the behaviour of this method, as implemented in practice? To make my task more tractable, I decided to consider only algorithms for unconstrained optimization.

Exciting trajectories for the identification of base inertial parameters of robots

Abstract: A common way to identify the inertial parameters of robots is to use a linear model in relation to the parameters and standard least-squares (LS) techniques. This article presents a method to generate exciting identification trajectories in order to minimize the effect of noise and error modeling on the LS solution. Using nonlinear optimization techniques, the condition number of a matrix W obtained from the energy model is minimized, and the scaling of its terms is carried out. An example of a three-degree-of-freedom robot is presented.

Entropic Proximal Mappings with Applications to Nonlinear Programming

Abstract: We introduce a family of new transforms based on imitating the proximal mapping of Moreau and the associated Moreau-Yosida proximal approximation of a function. The transforms are constructed in terms of the AÂ†-divergence functional a generalization of the relative entropy and of Bregman's measure of distance. An analogue of Moreau's theorem associated with these entropy-like distances is proved. We show that the resulting Entropic Proximal Maps share properties similar to the proximal mapping and provide a fairly general framework for constructing approximation and smoothing schemes for optimization problems. Applications of the results to the construction of generalized augmented Lagrangians for nonlinear programs and the minimax problem are presented.

A parameter optimization approach for the optimal control of large-scale musculoskeletal systems.

Abstract: This paper describes a computational method for solving optimal control problems involving large-scale, nonlinear, dynamical systems. Central to the approach is the idea that any optimal control problem can be converted into a standard nonlinear programming problem by parameterizing each control history using a set of nodal points, which then become the variables in the resulting parameter optimization problem. A key feature of the method is that it dispenses with the need to solve the two-point, boundary-value problem derived from the necessary conditions of optimal control theory. Gradient-based methods for solving such problems do not always converge due to computational errors introduced by the highly nonlinear characteristics of the costate variables. Instead, by converting the optimal control problem into a parameter optimization problem, any number of well-developed and proven nonlinear programming algorithms can be used to compute the near-optimal control trajectories. The utility of the parameter optimization approach for solving general optimal control problems for human movement is demonstrated by applying it to a detailed optimal control model for maximum-height human jumping. The validity of the near-optimal control solution is established by comparing it to a solution of the two-point, boundary-value problem derived on the basis of a bang-bang optimal control algorithm. Quantitative comparisons between model and experiment further show that the parameter optimization solution reproduces the major features of a maximum-height, countermovement jump (i.e., trajectories of body-segmental displacements, vertical and fore-aft ground reaction forces, displacement, velocity, and acceleration of the whole-body center of mass, pattern of lower-extremity muscular activity, jump height, and total ground contact time).

Optimal apportionment of reliability and redundancy in series systems under multiple objectives

Abstract: A multiobjective reliability apportionment problem for a series system with time-dependent reliability is presented. The resulting mathematical programming formulation determines the optimal level of component reliability and the number of redundant components at each stage. The problem is a multiobjective, nonlinear, mixed-integer mathematical programming problem, subject to several design constraints. Sequential unconstrained minimization techniques in conjunction with heuristic algorithms are used to find an optimum solution. A generalization of the problem in view of inherent vagueness in the objective and the constraint functions results in an ill-structured reliability apportionment problem. This multiobjective fuzzy optimization problem is solved using nonlinear programming. The computational procedure is illustrated through a numerical example. The fuzzy optimization techniques can be useful during initial stages of the conceptual design of engineering systems where the design goals and design constraints have not been clearly identified or stated, and for decision making problems in ill-structured situations. >

Decomposition and nondifferentiable optimization with the projective algorithm

Abstract: This paper deals with an application of a variant of Karmarkar's projective algorithm for linear programming to the solution of a generic nondifferentiable minimization problem. This problem is closely related to the Dantzig-Wolfe decomposition technique used in large-scale convex programming. The proposed method is based on a column generation technique defining a sequence of primal linear programming maximization problems. Associated with each problem one defines a weighted potential function which is minimized using a variant of the projective algorithm. When a point close to the minimum of the potential function is reached, a corresponding point in the dual space is constructed, which is close to the analytic center of a polytope containing the solution set of the nondifferentiable optimization problem. An admissible cut of the polytope, corresponding to a new supporting hyperplane of the epigraph of the function to minimize, is then generated at this approximate analytic center. In the primal space this new cut translates into a new column for the associated linear programming problem. The algorithm has performed well on a set of convex nondifferentiable programming problems.

Linear and quadratic programming neural network analysis

Abstract: Neural networks for linear and quadratic programming are analyzed. The network proposed by M.P. Kennedy and L.O. Chua (IEEE Trans. Circuits Syst., vol.35, pp.554-562, May 1988) is justified from the viewpoint of optimization theory and the technique is extended to solve optimization problems, such as the least-squares problem. For quadratic programming, the network converges either to an equilibrium or to an exact solution, depending on whether the problem has constraints or not. The results also suggest an analytical approach to solve the linear system Bx=b without calculating the matrix inverse. The results are directly applicable to optimization problems with C/sup 2/ convex objective functions and linear constraints. The dynamics and applicability of the networks are demonstrated by simulation. The distance between the equilibria of the networks and the problem solutions can be controlled by the appropriate choice of a network parameter. >

Method for Optimal Actuator and Sensor Placement for Large Flexible Structures

Abstract: A method of finding the optimal sensor and actuator locations for the control of flexible structures is presented. The method is based on the orthogonal projection of structural modes into the intersection subspace of the controllable and observable subspaces corresponding to an actuator/sensor pair. The controllability and observability grammians are then used to weight the projections to reflect the degrees of controllabili ty and observability. This method produces a three-dimensio nal design space wherein sets of optimal actuators and sensors may be selected. A novel parameter is introduced that is potentially useful for studying the problem of the number of actuators and sensors, in addition to their optimal locations. UPPOSE a specific number of actuators and sensors is given and they are placed at specific locations on a flexible structure such that the effectiveness of the chosen actuator and sensor locations could be analyzed. If it turns out that the a priori chosen number and locations for the actuators and sensors are not sufficiently effective, the question naturally arises as to how the locations could be changed to improve the system. Furthermore, it is possible that the a priori number of actuators and/or sensors used is insufficient or redundant. Thus, there is clearly a need for a computationall y feasible technique that is capable of determining an optimal set of locations and the minimal number of actuators and sensors. In general, there will be many more candidate locations (perhaps an order of magnitude more) than the number of actuators and sensors actually available. If the number of actuators and sensors is known a priori, all possible combina- tions could be evaluated, and in principle, the global optimum could then be found. Unfortunately, the number of possible combinations increases factorially, and therefore an exhaus- tive search for a global optimal is usually computationally infeasible, while nonlinear programming based techniques typically produce local minimums. In the past, various definitions of the degree of controllabil- ity and observability have been used in guiding the search for optimal actuator and sensor locations. Among these, the de- gree of controllabili ty defined by scalar measures of recovery regions appears useful for the purpose of actuator and sensor placement.1'3 A second approach4 uses the projection magni- tudes of eigenvectors into the input and output matrices to define gross measures of modal controllability and observabil- ity. However, only little attention is given to the development of a systematic search strategy for actually solving for an optimal set of actuators and sensors, and most attention has been directed toward defining what constitutes most suitable actuator and sensor locations. In this paper, the problem of defining and obtaining the optimal actuator and sensor locations is addressed. A method that is based on the controllability and observability of an actuator/sensor pair is introduced. An outline of the present paper follows. First, the model of actuator and sensor loca- tions for a linear, second-order dynamical system is presented. The basic assumption is that we are given a set of significant modes whose control is desired via feedback. In the next section, controllable, observable, and their intersection sub- spaces are presented, which forms the basis of the method presented in the sequel. The following section presents the main results of this paper. A cost function that is based on the weighted projection of structural modes into the intersection subspace of the controllable and observable subspaces is intro- duced, and a simple interpretation in terms of balanced coor- dinates is given. A novel method for selecting optimal sensor and actuator locations based on the preceding cost function is outlined. The weighted projections of the structural modes can be viewed as a scalar field in three-dimensi onal design space wherein a designer can easily select a set of actuators and sensors based on his or her own criteria without resorting to elaborate nonlinear programming strategies. The method also allows for the comparison of many actuator and sensor candi- date locations since the computational effort depends only on the product of the number of actuator and sensor location candidates rather than combinatorially based search strategies whpse computational effort is in the order of factorials. In the next section, the method of finding optimal locations is ap- plied to an existing laboratory structure to demonstrate the algorithm. Finally, a few concluding remarks are given.

Genetic algorithms in optimization problems with discrete and integer design variables

Abstract: The paper describes an implementation of genetic search methods in the optimal design of structural systems with a mix of continuous, integer and discrete design variables. Design variable representation schemes for such mixed variables are proposed and the performance of each is evaluated in the context of structural design problems. The approach is proposed as an alternative to the branch-and-bound techniques that are used in conjunction with nonlinear programming methods. The methodology is inherently equipped with a better chance of locating the global optimum than the conventional gradient based methods.

Time-optimal slewing of flexible spacecraft

Abstract: The tune optimal slewing problem of flexible spacecraft is considered. The system is discretized by the assumed modes method, and the problem is solved for a linearized model in reduced state space by parameter optimization. Optimality is verified by the maximal principle. The linear solution is further used to obtain time optimal solutions for the non-linear problem.

Optimization-availability-based design of water-distribution networks

Abstract: A practical measure for water‐distribution system reliability, based on hydraulic availability is presented and incorporated in an optimal design procedure for component sizing. The measure combines hydraulic and mechanical availability in a form that defines the proportion of the time that the system will satisfactorily fulfill its function. However, rather than a simple discrete failure relationship with absolute failure if pressure heads fall below a prescribed minimum the hydraulic availability is modeled with continuous increasing acceptability as higher pressures occur. Availability is considered in a nonlinear optimization model that is. reduced in complexity by linking the optimizer with a network solver to implicitly solve the hydraulic constraints. The results of the model application show an increasing marginal cost for higher levels of availability, and the optimal designs tend to follow the engineering rules of thumb for system design.

Preconditioners for indefinite systems arising in optimization

Abstract: Methods are discussed for the solution of sparse linear equations $Ky = z$, where K is symmetric and indefinite. Since exact solutions are not always required, direct and iterative methods are both...

Reliability optimization of systems by a surrogate-constraints algorithm

Abstract: A method for solving the problem of optimizing both, redundancy (number of redundant components) and component reliability in each stage of a system under multiple constraints is presented. A mixed-integer nonlinear programming formulation and the surrogate dual method are used. The solution of the surrogate dual problem is not always feasible in the original problem, that is, a 'surrogate gap' exists. Two countermeasures to surrogate gaps are considered: (1) modifying the original problem to tighten the constraints, with the modification being continued until the solution of the surrogate dual problem of the modified problem becomes feasible in the original problem, and (2) decreasing component reliabilities in the vertical direction to the tangential plane of the objective function. The method applies to reliability optimization problems for general systems, enabling complex systems such as communication networks to be treated. Some computational results are shown and compared with other approaches; they show the efficiency of the method. >

A decomposition approach to nonlinear multi-area generation scheduling with tie-line constraints using expert systems

Abstract: The authors propose a rigorous method to treat multiarea generation scheduling with tie line limits. An expert system was used for obtaining the initial solution. As the generation scheduling problem involves unit commitment and economic dispatch, the method adopts an iterative procedure to deal with these two phases. The hourly load demand and the area power generation will cause the tie flows to change. To maintain the operation security in every area, the spinning reserve should comply with the area power generation rather than its load demand. After economic dispatch, it is necessary to adjust the unit commitment in each area for preserving the spinning reserve requirements. Heuristics were used to modify the generation unit combinations. The objective is to find an economic generation schedule for a multiarea system. The interchange transactions among areas represent the transportation problem, embedded within the nonlinear optimization process. The equivalent system concept is adopted, and the transmission losses are included in this study. A four-area system with each area consisting of 26 units was used to test the efficiency of the proposed algorithm. >

Linear multiplicative programming

Abstract: An algorithm for solving a linear multiplicative programming problem (referred to as LMP) is proposed. LMP minimizes the product of two linear functions subject to general linear constraints. The product of two linear functions is a typical non-convex function, so that it can have multiple local minima. It is shown, however, that LMP can be solved efficiently by the combination of the parametric simplex method and any standard convex minimization procedure. The computational results indicate that the amount of computation is not much different from that of solving linear programs of the same size. In addition, the method proposed for LMP can be extended to a convex multiplicative programming problem (CMP), which minimizes the product of two convex functions under convex constraints.

Analyzing and Optimizing Multibody Systems

Abstract: Optimization of holonomic as well as non-holonomic multibody systems is presented as a nonlinear programming problem that can be solved with general-purpose optimization codes. The adjoint variable approach is used for calculating design derivatives of a rather general integral type performance measure with respect to design parameters. The resulting equations are solved by numerical integration backward in time. A multi-step integration algorithm with order and step-size control is adapted for this application by including an interpolation scheme. Numerical experiments and a comparison to the common approach of approximating the gradient of the performance measure by finite differences show that high efficiency, accuracy, and reliability are achievable.

Application of sparse nonlinear programming to trajectory optimization

Abstract: The most effective numerical techniques for the solution of trajectory optimization and optimal control problems combine a nonlinear iteration procedure with some type of parametric approximation to the trajectory dynamics. Early methods attempted to parameterize the dynamics using a small number of variables because the iterative search procedures could not successfully solve larger problems. With the development of more robust nonlinear programming algorithms, it is now feasible and desirable to consider formulations of the trajectory optimization problem incorporating a large number of variables and constraints. The purpose of this paper is to address the manner in which a trajectory is parameterized and the design of the nonlinear programming algorithm to effectively deal with this formulation.

Systems Analysis in Ground-Water Planning and Management

Abstract: The objective of this paper is to review the state of the art of systems analysis and optimization techniques developed in the field of water resources for the planning and management of a ground-water system. The areas reviewed include the following: ground-water management models, inverse solution techniques for parameter identification, and optimal experimental design methods. Emphasis is placed upon ground-water supply management models, as opposed to models used for ground-water quality management. The techniques that have been used in the optimization of ground-water management include: linear programming, mixed-integer and quadratic programming, differential dynamic programming, nonlinear programming, and simulation. The inverse problem of parameter identification pertains the optimal determination of model parameters using historical input and output observations. Because of data limitation in both quantity and quality, the inverse problem is inherently ill posed. This paper summarizes recent advances made in the inverse procedures and methods developed to alleviate the problems of instability and nonuniqueness of the identified parameters. The optimal experimental design problem addresses the issue of data requirements and optimal sampling strategies for the purpose of parameter identification. A criterion must be established for the optimal design of a pumping test. The fundamental concept of optimal experimental design and various criteria used for optimization are reviewed.

Sensitivity analysis for variational inequalities

Abstract: In this paper we study the behavior of the local solutions of perturbed variational inequalities, governed by perturbations to both the variational inequality function and the feasible region. Assuming appropriate second-order and regularity conditions, we show that the perturbed local solution set is Lipschitz continuous and directionally differentiable. Even when the directional differentiability is not guaranteed, we are still able to describe and characterize first-order information concerning the perturbed local solution set. We also discuss relations to nonlinear programming sensitivity analysis.