Showing papers on "Nonlinear programming published in 1990"

Further developments in LP-based optimal power flow

Abstract: The authors describe developments that have transformed the LP (linear programming) approach into a truly general-purpose OPF (optimal power flow) solver, with computational and other advantages over even recent nonlinear programming (NLP) methods. it is pointed out that the nonseparable loss-minimization problem can now be solved, giving the same results as NLP on power systems of any size and type. Coupled formulations, where for instance voltages and VAr become constraints on MW scheduling, are handled. Former limitations on the modeling of generator cost curves have been eliminated. In addition, the approach accommodates a large variety of power system operating limits, including the very important category of contingency constraints. All of the reported enhancements are fully implemented in the production OPF software described here, and most have already been utilized within the industry. >

Nonlinear switched capacitor 'neural' networks for optimization problems

Abstract: A systematic approach is presented for the design of analog neural nonlinear programming solvers using switched-capacitor (SC) integrated circuit techniques. The method is based on formulating a dynamic gradient system whose state evolves in time toward the solution point of the corresponding programming problem. A neuron cell for the linear and the quadratic problem suitable for monolithic implementation is introduced. The design of this neuron and its corresponding synapses using SC techniques is considered in detail. An SC circuit architecture based on a reduced set of basic building blocks with high modularity is presented. Simulation results using a mixed-mode simulator (DIANA) and experimental results from breadboard prototypes are included, illustrating the validity of the proposed techniques. >

Convex separable optimization is not much harder than linear optimization

Abstract: The polynomiality of nonlinear separable convex (concave) optimization problems, on linear constraints with a matrix with “small” subdeterminants, and the polynomiality of such integer problems, provided the inteter linear version of such problems ins polynomial, is proven. This paper presents a general-purpose algorithm for converting procedures that solves linear programming problems. The conversion is polynomial for constraint matrices with polynomially bounded subdeterminants. Among the important corollaries of the algorithm is the extension of the polynomial solvability of integer linear programming problems with totally unimodular constraint matrix, to integer-separable convex programming. An algorithm for finding a e-accurate optimal continuous solution to the nonlinear problem that is polynomial in log(1/e) and the input size and the largest subdeterminant of the constraint matrix is also presented. These developments are based on proximity results between the continuous and integral optimal solutions for problems with any nonlinear separable convex objective function. The practical feature of our algorithm is that is does not demand an explicit representation of the nonlinear function, only a polynomial number of function evaluations on a prespecified grid.

A comparative study of algorithms for matrix balancing

Abstract: The problem of adjusting the entries of a large matrix to satisfy prior consistency requirements occurs in economics, urban planning, statistics, demography, and stochastic modeling; these problems are called Matrix Balancing Problems. We describe five applications of matrix balancing and compare the algorithmic and computational performance of balancing procedures that represent the two primary approaches for matrix balancing-matrix scaling and nonlinear optimization. The algorithms we study are the RAS algorithm, a diagonal similarity scaling algorithm, and a truncated Newton algorithm for network optimization. We present results from computational experiments with large-scale problems based on producing consistent estimates of Social Accounting Matrices for developing countries.

A trust region algorithm for equality constrained optimization

Abstract: A trust region algorithm for equality constrained optimization is proposed that employs a differentiable exact penalty function. Under certain conditions global convergence and local superlinear convergence results are proved.

Sensitivity analysis based heuristic algorithms for mathematical programs with variational inequality constraints

Abstract: In this paper we consider heuristic algorithms for a special case of the generalized bilevel mathematical programming problem in which one of the levels is represented as a variational inequality problem. Such problems arise in network design and economic planning. We obtain derivative information needed to implement these algorithms for such bilevel problems from the theory of sensitivity analysis for variational inequalities. We provide computational results for several numerical examples.

Structural damage detection based on static and modal analysis

Abstract: The present paper describes an approach for structural damage assessment that has its basis in methods of system identification. Response of a damaged structure differs from predictions obtained from an analytical model of the original structure, where the analytical model is typically a finite-element representation. The out- put error approach of system identification is employed to determine changes in the analytical model necessary to minimize differences between the measured and predicted response. Structural damage is represented by changes in element stiffness matrices resulting from variations in geometry or material properties of the structure during damage. Measurements of static deflections and vibration modes are used in the identification procedure. The identification methodology is implemented for representative structural systems. Principal shortcomings in the proposed approach and methods to circumvent these problems are also discussed. of obtaining poor results. These difficulties are clearly evi- denced by the results obtained. Smith and Hendricks8 follow a similar approach using two different identification methods to identify the stiffness matrix based on the minimum deviation approach and using eigenmodes as experimental data. Similar difficulties are re- ported in their work. The entries of the stiffness matrix corre- sponding to the damaged members do show considerable vari- ations. However, entries corresponding to undamaged members are also affected, thereby making the damage detec- tion process more uncertain. The analysis of changes in the stiffness matrix is typically cumbersome, may not always yield correct answers, and does not permit the determination of the extent of damage. This paper presents an approach that is designed to circum- vent the problems just discussed. The output error method or structural identification9 is used, wherein the analytical model is refined to minimize the difference between the predicted and measured response of the structure. Iterative nonlinear pro- gramming methods are employed to determine a solution to the unconstrained optimization problem. Damage is repre- sented by reduction in the elastic extensional and shear moduli of the element, and those are designated as the design varia- bles of the problem. The use of static structural displacements as the measured response is a departure from the standard practice of using eigenmodes alone for the identification pro- blem. Numerical evidence clearly indicates that when eigen- modes alone are used for identification, the location and ex- tent of damage predicted by the optimization approach is dependent on the number of modes used to match the measured and the predicted response. Higher modes are diffi- cult to determine and measure, and the use of static displace- ments obtained by a loading that simulates higher modes is proposed as a solution to this problem. The paper also presents an implementation of the proposed damage assessment strategies, with special focus on problems of practical significance. In this context, the use of incomplete modal or static displacement information in the identification problem is discussed. Further, the approach of treating the modulus of each structural element as an independent design variable results in a large dimensionality problem. This results in significant computational costs when using a gradient-based nonlinear programming algorithm for function minimization. The use of a reduced set of dominant design variables and the construction of equivalent reduced-order models for damage assessment are explored with some success.

Trust region algorithms for optimization with nonlinear equality and inequality constraints

Abstract: We consider the general nonlinear optimization problem defined as, minimize a nonlinear real-valued function of several variables, subject to a set of nonlinear equality and inequality constraints. This class of problems arise in many real life applications, for example in engineering design, chemical equilibrium, simulation and data fitting. In this research, we present algorithms that use the trust region technique to solve these problems. First, we develop an algorithm for solving the nonlinear equality constrained optimization, then we generalize the algorithm to handle the inclusion of nonlinear inequality constraints in the problem. The algorithms use the successive quadratic programming (SQP) approach and trust region technique. We define a model subproblem which minimizes a quadratic approximation of the Lagrangian subject to modified relaxed linearizations of the problem nonlinear constraints and a trust region constraint. Inequality constraints are handled by a compromise between an active set strategy and IQP subproblem solution technique. An analysis which describes the local convergence properties of our algorithms is presented. The algorithms are implemented and the model minimization is done approximately by using the dogleg approach. Numerical results are presented and compared with the results of a popular line search method. Some examples are presented in which the ability of our method to use directions of negative curvature results in greater reliability. Results of the numerical experiments indicate that our method is very robust and reasonably efficient.

Optimal finite-thrust spacecraft trajectories using collocation and nonlinear programming

Abstract: A new method is described for the determination of optimal spacecraft trajectories in an inverse-square field using finite, fixed thrust. The method employs a recently developed optimization technique which uses a piecewise polynomial representation for the state and controls, and collocation, thus converting the optimal control problem into a nonlinear programming problem, which is solved numerically. This technique has been modified to provide efficient handling of those portions of the trajectory which can be determined analytically, i.e., the coast arcs. Among the problems that have been solved using this method are optimal rendezvous and transfer (including multirevolution cases) and optimal multiburn orbit insertion from hyperbolic approach.

Optimal Reliability‐Based Design of Pumping and Distribution Systems

Abstract: A reliability-based optimization model for water-distribution systems has been developed. The model is aimed at the following goals: (1) Design of the pipe network including the number, location, and size of pumps and tanks; (2) design of the pumping system using a reliability-based procedure considering both hydraulic failures of the entire network and mechanical failure of the pumping system; and (3) determination of the optimal operation of the pumps. The optimization problem is a large mixed-integer, nonlinear programming problem that is solved using a heuristic algorithm consisting of a master problem and a subproblem. The master problem is a pure 0–1 integer programming model, and the subproblem is a large nonlinear programming model solved in an optimal control framework. The conservation of flow and energy constraints are solved implicitly for each iteration of the nonlinear optimization procedure using a hydraulic simulation model, and the reliability constraints are also solved implicitly using a reliability model. The nonlinear programming problem is solved using a generalized reduced gradient code.

Oceanographic Experiment Design by Simulated Annealing

Abstract: The oceanographic experiment design problem is discussed in the context of several simple examples drawn from acoustic tomography. The optimization of an objective function—chosen to characterize the array design— is carried out using the technique of simulated annealing. A detailed description of this method and its implementation for the examples above, is provided. Although computationally intensive, when it is carefully implemented, simulated annealing is found to give superior results to more traditional methods of nonlinear optimization.

Mixed-integer nonlinear programming techniques for the synthesis of engineering systems

Abstract: The development of new mixed-integer nonlinear programming (MINLP) algorithms, coupled with advances in computers and software, is opening promising possibilities to rigorously model, optimize, and automate the synthesis of engineering systems. A general overview of the MINLP approach and algorithms will be presented in this paper with the aim of gaining a basic understanding of these techniques. Strengths and weaknesses will be discussed, as well as difficulties and challenges which still need to be overcome. In particular, it will be shown how proper problem representations, effective modeling schemes, and solution strategies can play a crucial role in the successful application of these techniques. The application of MINLP algorithms in synthesis will be illustrated with several examples.

Model for real-time optimal flood control operation of a reservoir system

Abstract: A methodology and model have been developed for the real-time optimal flood operation of river-reservoir systems. This methodology is based upon combining a nonlinear programming model with a flood-routing simulation model within an optimal control framework. The generalized reduced gradient code GRG2 is used to perform the nonlinear optimization and the simulator is the U.S. National Wheather Service DWOPER code. Application of the model is illustrated through a case study of Lake Travis on the Lower Colorado River in Texas.

High-quality approximation of eigenvalues in structural optimization

Abstract: A new function for approximating natural frequency constraints during structural optimization is presented. The Rayleigh Quotient Approximation (RQA) presented here icreases the quality of the approximate frequency constraint by approximating the modal strain and kinetic energies instead of the frequency eigenvalue itself

A penalty approach for nonlinear optimization with discrete design variables

Abstract: A penalty approach for the solution of nonlinear discrete optimization problems is proposed. In general, the penalty approach is used for converting a constrained op-timization problem into a sequence of unconstrained problems. The objective function for the unconstrained problem at each step of the sequential optimization includes terms that introduce penalty depending on the degree of constraint violation. In addition to the penalty terms for constraint violation, the proposed approach intro-duces penalty terms to reflect the requirement that the design variables take discrete values. A variable magnitude penalty term in the form of a sine function is introduced and implemented with the extended interior penalty method of the optimization package NEWSUMT-A. The performance of the proposed method is investigated by several numer-ical examples.

Robust error‐in‐variables estimation using nonlinear programming techniques

Abstract: For systems described by algebraic or differential equation models where all variables are subject to error, the error-in-variables method (EVM) for parameter estimation has been shown to be superior to standard least-squares techniques. Previous EVM algorithms were developed assuming linear (or linearized) model equations. Unfortunately, many chemical engineering processes operate in strongly nonlinear regions where linear approximations may be inaccurate. In this paper, new algorithms using nonlinear programming techniques for the error-in-variables methods are proposed. In addition, a method for discerning when these methods are necessary is discussed. The proposed algorithms are compared to the least-squares method and traditional error-in-variable approaches. Improved parameter estimates for several steady-state nonlinear processes are demonstrated.

Signal processing via least squares error modeling

Abstract: The signal model presently considered is composed of a linear combination of basis signals chosen to reflect the basic nature believed to characterize the data being modeled. The basis signals are dependent on a set of real parameters selected to ensure that the signal model best approximates the data in a least-square-error (LSE) sense. In the nonlinear programming algorithms presented for computing the optimum parameter selection, the emphasis is placed on computational efficiency considerations. The development is formulated in a vector-space setting and uses such fundamental vector-space concepts as inner products, the range- and null-space matrices, orthogonal vectors, and the generalized Gramm-Schmidt orthogonalization procedure. A running set of representative signal-processing examples are presented to illustrate the theoretical concepts as well as point out the utility of LSE modeling. These examples include the modeling of empirical data as a sum of complex exponentials and sinusoids, linear prediction, linear recursive identification, and direction finding. >

Fourier-Based Optimal Control of Nonlinear Dynamic Systems

Abstract: A method for generating near optimal trajectories of linear and nonlinear dynamic systems, represented by deterministic, lumped-parameter models, is proposed. The method is based on a Fourier series approximation of each generalized coordinate that converts the optimal control problem into an algebraic nonlinear programming problem. The results of computer simulation studies compare favorably to optimal solutions obtained by closed-form analyses and/or by other numerical schemes

Matching moments to phase distributions: nonlinear programming approaches

Abstract: We present a nonlinear programming (NLP) approach to the problem of matching three moments to phase distributions. We first discuss the formulation and implementation of a general NLP problem and then consider NLP problems for searching over two families of phase distributions: mixtures of two Erlang distributions and real-parametered continuous Coxian distributions. Restricting the search to select from a subset of phase distributions allows us to greatly simplify the NLP problem, resulting in more efficient and predictable search procedures. Conversely, the restriction also reduces the variety of distributions the search algorithm can select. Tradeoffs between the formulations and possible refinements and extensions are discussed.

FEM and evolution strategies in the optimal design of electromagnetic devices

Abstract: The application of evolution strategies to the optimal design of electromagnetic devices is investigated The corresponding field analysis is performed by the finite element method The strategies involve a simplified simulation of biological evolution They are especially advantageous in the case of strongly nonlinear optimization problems The three strategies used are the (1+1), the

A sequential quadratic programming algorithm using an incomplete solution of the subproblem

Abstract: We analyze sequential quadratic programming (SQP) methods to solve nonlinear constrained optimization problems that are more flexible in their definition than standard SQP methods. The type of flexibility introduced is motivated by the necessity to deviate from the standard approach when solving large problems. Specifically we no longer require a minimizes of the QP subproblem to be determined or particular Lagrange multiplier estimates to be used. Our main focus is on an SQP algorithm that uses a particular augmented Lagrangian merit function. New results are derived for this algorithm under weaker conditions than previously assumed; in particular, it is not assumed that the iterates lie on a compact set.

Large scale nonlinear deterministic and stochastic optimization: formulations involving simulation of subsurface contamination

Abstract: Once subsurface water supplies become contaminated, designing cost-effective and reliable remediation schemes becomes a difficult task. The combination of finite element simulation of groundwater contaminant transport with nonlinear optimization is one approach to determine the best well selection and optimal fluid withdrawal and injection rates to contain and remove the contaminated water. Both deterministic and stochastic programming problems have been formulated and solved. These tend to be large scale problems, owing to the simulation component which serves as a portion of the constraint set. The overall problem of combined groundwater process simulation and nonlinear optimization is discussed along with example problems. Because the contaminant transport simulation models give highly uncertain results, quantifying their uncertainty and incorporating reliability into the remediation design results in a class of large stochastic nonlinear problems. The reliability problem is beginning to be addressed, and some strategies and formulations involving chance constraints and Monte Carlo methods are presented.