Showing papers on "Nonlinear programming published in 1988"

Neural networks for nonlinear programming

Abstract: The dynamics of the modified canonical nonlinear programming circuit are studied and how to guarantee the stability of the network's solution. By considering the total cocontent function, the solution of the canonical nonlinear programming circuit is reconciled with the problem being modeled. In addition, it is shown how the circuit can be realized using a neural network, thereby extending the results of D.W. Tank and J.J. Hopefield (ibid., vol.CAS-33, p.533-41, May 1986) to the general nonlinear programming problem. >

Accelerating the convergence of the back-propagation method

Abstract: The utility of the back-propagation method in establishing suitable weights in a distributed adaptive network has been demonstrated repeatedly. Unfortunately, in many applications, the number of iterations required before convergence can be large. Modifications to the back-propagation algorithm described by Rumelhart et al. (1986) can greatly accelerate convergence. The modifications consist of three changes:1) instead of updating the network weights after each pattern is presented to the network, the network is updated only after the entire repertoire of patterns to be learned has been presented to the network, at which time the algebraic sums of all the weight changes are applied:2) instead of keeping ź, the "learning rate" (i.e., the multiplier on the step size) constant, it is varied dynamically so that the algorithm utilizes a near-optimum ź, as determined by the local optimization topography; and3) the momentum factor ź is set to zero when, as signified by a failure of a step to reduce the total error, the information inherent in prior steps is more likely to be misleading than beneficial. Only after the network takes a useful step, i.e., one that reduces the total error, does ź again assume a non-zero value. Considering the selection of weights in neural nets as a problem in classical nonlinear optimization theory, the rationale for algorithms seeking only those weights that produce the globally minimum error is reviewed and rejected.

The mathematics of nonlinear programming

Abstract: Unconstrained Optimization via Calculus. Convex Sets and Convex Functions.- Iterative Methods for Unconstrained Optimization.- Least Squares Optimization.- Convex Programming and the Karush-Kuhn-Tucker Conditions.- Penalty Methods.-Optimization with Equality Constraints.

Sensitivity Analysis for Equilibrium Network Flow

Abstract: Direct application of existing sensitivity analysis methods for nonlinear programming problems or for variational inequalities to nonlinear programming or variational inequality formulations of the equilibrium traffic assignment problem is not feasible, since, in general, the solution to the equilibrium traffic assignment problem does not satisfy the uniqueness conditions required by the sensitivity analysis methods. This paper presents an approach for sensitivity analysis of equilibrium traffic assignment problems in which an equivalent restricted problem is developed which has the desired uniqueness properties; the existing methods are applied to this restricted problem to calculate the derivatives of the equilibrium arc flows with respect to perturbations of the cost functions and of the trip table. These derivatives are then shown to be equivalent to the derivatives of the original unrestricted equilibrium traffic assignment problem; therefore, the method yields the desired sensitivity analysis results.

Learning Algorithms for Connectionist Networks: Applied Gradient Methods of Nonlinear Optimization

Abstract: The problem of learning using connectionist networks, in which network connection strengths are modified systematically so that the response of the network increasingly approximates the desired response can be structured as an optimization problem. The widely used back propagation method of connectionist learning [19, 21, 18] is set in the context of nonlinear optimization. In this framework, the issues of stability, convergence and parallelism are considered. As a form of gradient descent with fixed step size, back propagation is known to be unstable, which is illustrated using Rosenbrock's function. This is contrasted with stable methods which involve a line search in the gradient direction. The convergence criterion for connectionist problems involving binary functions is discussed relative to the behavior of gradient descent in the vicinity of local minima. A minimax criterion is compared with the least squares criterion. The contribution of the momentum term [19, 18] to more rapid convergence is interpreted relative to the geometry of the weight space. It is shown that in plateau regions of relatively constant gradient, the momentum term acts to increase the step size by a factor of 1/1-μ, where μ is the momentum term. In valley regions with steep sides, the momentum constant acts to focus the search direction toward the local minimum by averaging oscillations in the gradient. Comments University of Pennsylvania Department of Computer and Information Science Technical Report No. MSCIS-88-62. This technical report is available at ScholarlyCommons: http://repository.upenn.edu/cis_reports/597 LEARNING ALGORITHMS FOR CONNECTIONIST NETWORKS: APPLIED GRADIENT METHODS OF NONLINEAR OPTIMIZATION

Circuit optimization: the state of the art

Abstract: The authors review the current state of the art in circuit optimization, emphasizing techniques suitable for modern microwave CAD (computer-aided design). The main thrust in the field is currently the solution of realistic design and modeling problems, addressing such concepts as physical tolerances and model uncertainties. A unified hierarchical treatment of circuit models forms the basis of the presentation. It exposes tolerance phenomena at different parameter/response levels. The concepts of design centering, tolerance assignment, and postproduction tuning in relation to yield enhancement and cost reduction suitable for integrated circuits are discussed. Suitable techniques for optimization oriented worst-case and statistical design are reviewed. A generalized l/sub p/ centering algorithm is proposed and discussed. Multicircuit optimization directed at both CAD and robust device modeling is formalized. Tuning is addressed in some detail, both at the design stage and for production alignment. State-of-the-art gradient-based nonlinear optimization methods are reviewed with emphasis given to recent, but well tested, advances in minimax, l/sub 1/, and l/sub 2/ optimization. >

Contaminated groundwater remediation design using simulation, optimization, and sensitivity theory: 1. Model development

Abstract: The problem of designing contaminated groundwater remediation systems using hydraulic control is addressed. Two nonlinear optimization formulations are proposed which model the design process for the location and pump rates of injection and extraction wells in an aquifer cleanup system. The formulations are designed to find a pumping system which (1) removes the most contaminant over a fixed time period and (2) reduces contaminant concentration to specified levels by the end of a fixed time period at least cost. The formulations employ a two-dimensional Galerkin finite element simulation model of steady state groundwater flow and transient convective-dispersive transport. To make the optimization problems computationally tractable sensitivity theory is used to derive a general relationship for computing the derivatives of an arbitrary function of the simulation outputs with respect to model inputs. This relationship is then applied to the convective-dispersive transport equation.

First- and second-order epi-differentiability in nonlinear programming

Abstract: Problems are considered in which an objective function expressible as a max of finitely many C2 functions, or more generally as the composition of a piecewise linear-quadratic function with a C2 mapping, is minimized subject to finitely many C2 constraints. The essential objective function in such a problem, which is the sum of the given objective and the indicator of the constraints, is shown to be twice epi-differentiable at any point where the active constraints (if any) satisfy the Mangasarian-Fromovitz qualification. The epi-derivatives are defined by taking epigraphical limits of classical firstand second-order difference quotients instead of pointwise limits, and they reveal properties of local geometric approximation that have not previously been observed.

On the identification of active constraints

Abstract: Nondegeneracy conditions that guarantee that the optimal active constraints are identified in a finite number of iterations are studied. Results of this type have only been established for a few algorithms, and then under restrictive hypothesis. The main result is a characterization of those algorithms that identify the optimal constraints in a finite number of iterations. This result is obtained with a nondegeneracy assumption which is equivalent, in the standard nonlinear programming problem, to the assumption that there is a set of strictly complementary Lagrange multipliers. As an important consequence of the authors’ results the way that this characterization applies to gradient projection and sequential quadratic programming algorithms is shown.

Economic-emission load dispatch through goal programming techniques

Abstract: The economic-emission load dispatch problem which accounts for minimization of both cost and emission is a multiple, conflicting-objective function problem. Goal programming techniques are most suitable for such type of problems. Here, the economic-emission load dispatch problem is solved through linear and nonlinear goal programming algorithms. The application and validity of the proposed algorithms are demonstrated for a sample system having six generators. >

Sensitivity analysis of nonlinear programs and differentiability properties of metric projections

Abstract: This paper is concerned with a study of differentiability properties of the optimal value function and an associated optimal solution of a parametrized nonlinear program. Second order analysis is presented essentially under the Mangasarian-Fromovitz constraint qualification when the corresponding vector of Lagrange multipliers is not necessarily unique. It is shown that under certain regularity conditions the optimal value function possesses second order directional derivatives and the optimal solution mapping is directionally differentiable. The results obtained are applied to an investigation of metric projections in finite-dimensional spaces.

A Heuristic for Minimizing the Number of Tool Switches on a Flexible Machine

Abstract: This paper addresses the problem of scheduling N jobs on a single machine equipped with an automatic tool interchange mechanism. We consider the case where the total number of tools required to process all N jobs is greater than the capacity of the tool magazine, and where processing times and switching times are independent. The underlying problem is to find the job sequence and tool replacement policy that minimizes the total number of switches. This is equivalent to minimizing the makespan. Two industrial applications of the model are cited. The problem is formulated as a nonlinear integer program and solved with a dual-based relaxation heuristic designed to quickly find good local solutions. An example is given to highlight the computations and a series of test cases is examined to gauge the performance of the proposed methodology. The results demonstrate that in almost all cases global optimality is obtained, but in notably less time than current techniques admit. This points up the practica...

Nonlinear programming extensions to rational function approximation methods for unsteady aerodynamic forces

Abstract: The approximation of unsteady generalized aerodynamic forces in the equations of motion of a flexible aircraft are discussed. Two methods of formulating these approximations are extended to include the same flexibility in constraining the approximations and the same methodology in optimizing nonlinear parameters as another currently used extended least-squares method. Optimal selection of nonlinear parameters is made in each of the three methods by use of the same nonlinear, nongradient optimizer. The objective of the nonlinear optimization is to obtain rational approximations to the unsteady aerodynamics whose state-space realization is lower order than that required when no optimization of the nonlinear terms is performed. The free linear parameters are determined using the least-squares matrix techniques of a Lagrange multiplier formulation of an objective function which incorporates selected linear equality constraints. State-space mathematical models resulting from different approaches are described and results are presented that show comparative evaluations from application of each of the extended methods to a numerical example.

Game theory approach for the integrated design of structures and controls

Abstract: The problem of design of actively controlled structures subject to constraints on the damping parameters of the closed-loop system is formulated as a multiobjeetive optimization problem. The structural weight and the controlled system energy are considered as objective functions for minimization with cross-sectional areas of members as design variables. A computational procedure is developed for solving the multiobjeetive optimization problem using cooperative game theory. The feasibility of the procedure is demonstrated through the design of two truss structures.

Centered newton method for mathematical programming

Abstract: The purpose of this paper is to introduce a generic class of algorithms for solving a system of nonlinear equations, Linear Programming problems, Quadratic Programming problems, Nonlinear Programming problems and general complementarity problems. The algorithms were obtained by modifying the standard Newton-Raphson method applied to a system of nonlinear equations in the complementarity conditions so that it is biased towards ‘center curve’ passing through the solutions. The search direction of the methods is a positive combination of the Newton direction and a ‘centering’ direction which is also given by applying the Newton method to a projected system of the complementarity equations. These two directions guide the generated sequence of the approximations towards the solution and the center variety respectively. A class of ‘penalized norms’ and ‘guiding cones’ is also introduced for choosing step lengths in bivariate search.

Determining optimal reorder intervals in capacitated production-distribution systems

Abstract: The problem of determining consistent and realistic reorder intervals in complex production-distribution environments was formulated as a large scale, nonlinear, integer programming problem by Maxwell and Muckstadt (Maxwell, W. L., J. A. Muckstadt. 1985. Establishing consistent and realistic reorder intervals in production-distribution systems. Oper. Res. 33(6, November–December) 1316–1341.). They show how the special structure of the problem permits its solution by a standard network flow algorithm. In this paper, we review the Maxwell-Muckstadt model, provide necessary and sufficient conditions that characterize the solution, and show that the optimal partition of nodes in the production-distribution network is invariant to an arbitrary scaling of the set-up and holding cost parameters. We consider two capacitated versions of the model: one with a single constrained work center, and the other with multiple constrained work centers. For single constraint problems, the invariance corollary provides a simp...

Contaminated groundwater remediation design using simulation, optimization, and sensitivity theory: 2. Analysis of a field site

Abstract: An optimization approach to designing contaminated ground water aquifer remediation systems is described and used to analyze alternate hypothetical remediation strategies at a Superfund site in Woburn, Massachusetts. The methodology combines two-dimensional convective-dispersive transport simulation, nonlinear optimization, and sensitivity theory. Remediation strategies are generated based on different design criteria as represented by two alternate optimization formulations. It is demonstrated that field scale simulation models can be successfully incorporated into a nonlinear optimization frame-work to solve important design problems. Through the use of sensitivity theory for the transport simulation model it is possible to solve field scale problems with at least an order of magnitude less computational effort than when using perturbation methods.

Multicriteria Truss Optimization

Abstract: The origin of structural optimization can be traced back several centuries (Ref. 1), but it is only during the last two decades or so, with the advent of modern computers, that it has evolved into a mature discipline in engineering. The literature published in this field is extensive and it can be reasonably discussed here only by referring to some recently written articles and textbooks found in Refs. 2–4. The major part of the articles deal with such numerical optimization techniques in finite-dimensional problems as optimality criteria or mathematical programming methods, but considerable efforts have also been made in applying the control theory approach to distributed parameter structural systems. The finite element method is commonly used in analyzing load supporting structures and there is usually a finite-dimensional optimization problem associated with it. In this chapter truss design problems, which by nature belong to this class, are considered. Various mainly nonlinear programming approaches have been developed to numerically solve scalar problems where the number of design variables and constraints is constantly increasing.

Accelerating the convergence of the diagonalization and projection algorithms for finite-dimensional variational inequalities

Abstract: This paper presents an acceleration step for the linearly convergent diagonalization and projection algorithms for finite-dimensional variational inequalities which is reminiscent of a PARTAN step in nonlinear programming. After establishing the convergence of this technique for both algorithms, several numerical examples are presented to illustrate the sometimes dramatic savings in computation time which this simple acceleration step yields.

Global Regularity Theorems

Abstract: For a system of nondifferentiable convex inequalities and linear equalities, the best bound is given for the quotient of the distance of an infeasible point and the norm of the residual vector. Applications to stability theory are then obtained.

Introduction to operations research

Abstract: Using a style of presentation that makes even the more difficult topics easy to understand, this text covers all the important quantitative models of operations research. The formulation of problems in mathematical terms is explained, together with the solution of the resulting models, and the interpretation of results. Coverage is given of linear programming, network flows, integer programming, nonlinear programming, dynamic programming, queueing models, inventory models and discrete-event simulation. The manner in which the topics are developed leads students to discover the underlying concepts for themselves, with a minimum of both notation and unnecessary jargon. An appendix provides notational conventions for matrix algebra.

State-of-the-art in parallel nonlinear optimization

Abstract: This survey is concerned with variants of nonlinear optimization methods designed for implementation on parallel computers. First, we consider a variety of methods for unconstrained minimization. We consider a particular type of parallelism (simultaneous function and gradient evaluations), and we concentrate on the main sources of inspiration: conjugate directions, homogeneous functions, variable-metric updates, and multi-dimensional searches. The computational process for solving small and medium-size constrained optimization problems is usually based on unconstrained optimization. This provides a straightforward opportunity for the introduction of parallelism. In the present survey, however, we focus on promising approaches for solving large, well-structured constrained problems: dualization of problems with separable objective and constraint functions, and decomposition of hierarchical problems with linking variables (typical for Bender's decomposition in the linear case). Finally, we outline the key issues in future computational studies of parallel nonlinear optimization algorithms.

Directional behaviour of optimal solutions in nonlinear mathematical programming

Abstract: This paper is concerned with Holder, Lipschitz and differential properties of the optimal solutions of a nonlinear mathematical programming problem with perturbations in some fixed direction. These properties are obtained with virtually minimal regularity conditions using an approach based on duality and stability. The Holder property is used to obtain the directional derivative for the optimal value function.

Minimum time trajectory planner for the discrete dynamic robot model with dynamic constraints

Abstract: A minimum time trajectory planner is proposed for a manipulator arm. A totally discrete approach is adopted, in contrast to other models which use continuous-time but resort to discretization in the computation. The Neuman and Tourassis discrete-dynamic robot model is used to model the robot dynamics. The proposed trajectory planner includes joint-torque constraints to fully utilize the joint actuators. Realistic constraints such as the joint-jerk and joint-velocity constraints are incorporated into the model. The nonlinear optimization problem associated with the planner is partially linearized, which enables the iterative method of approximate programming to be used in solving the problem. Numerical examples for a two-link revolute arm are presented to demonstrate the use of the proposed trajectory planner. It is numerically verified that the convergence of the iterative algorithm is quadratic, and the trajectory planner therefore is computationally efficient. The use of a near-minimum time-cost function is also shown to yield a solution close to that obtained with the true minimum time-cost function. >

Introduction to Optimization

Abstract: Preface Introduction PART 1 - UNCONSTRAINED OPTIMIZATION: Introduction to Unconstrained Optimization Techniques One-Dimensional Optimization Multi-Dimensional Optimization PART 2 - CONSTRAINED OPTIMIZATION: Linear Programming The Simplex Method for Linear Programming Further Details of the Simplex Method Duality and Parametric Programming How to Apply Linear Programming Examples of Linear Programming Problems PART 3 - CONSTRAINED OPTIMIZATION: NON-LINEAR AND DISCRETE: Non-Linear Programming Integer Programming Dynamic Programming References Subject Index.

Multilevel Decomposition Approach to the Preliminary Sizing of a Transport Aircraft Wing

Abstract: A multilevel/multidisciplinary optimization scheme for sizing an aircraft wing structure is described. A methodology using nonlinear programming in application to a very large engineering problem is presented. This capability is due to the decomposition approach. Over 1300 design variables are considered for this nonlinear optimization task. In addition, a mathematical link is established coupling the detail of structural sizing to the overall system performance objective, such as fuel consumption. The scheme is implemented as a three level system analyzing aircraft mission performance at the top level, the total aircraft structure as the middle level, and individual stiffened wing skin cover panels at the bottom level. Numerical show effectiveness of the method and its good convergence characteristics.