Showing papers on "Nonlinear programming published in 1968"

Nonlinear Programming: Sequential Unconstrained Minimization Techniques

Abstract: : This report gives the most comprehensive and detailed treatment to date of some of the most powerful mathematical programming techniques currently known--sequential unconstrained methods for constrained minimization problems in Euclidean n-space--giving many new results not published elsewhere. It provides a fresh presentation of nonlinear programming theory, a detailed review of other unconstrained methods, and a development of the latest algorithms for unconstrained minimization. (Author)

Programming with linear fractional functionals

Abstract: Charnes and Cooper [1] showed that a linear programming problem with a linear fractional objective function could be solved by solving at most two ordinary linear programming problems. In addition, they showed that where it is known a priori that the denominator of the objective function has a unique sign in the feasible region, only one problem need be solved. In the present note it is shown that if a finite solution to the problem exists, only one linear programming problem must be solved. This is because the denominator cannot have two different signs in the feasible region, except in ways which are not of practical importance.

Nonlinear Programming: Theory, Algorithms and Applications

Abstract: BASICS. The Nature of Optimization Problems. Analytical Background. Factorable Functions. UNCONSTRAINED PROBLEMS. Unconstrained Optimization Models. Minimizing a Function of a Single Variable. General Convergence Theory for Unconstrained Minimization Algorithms. Newton's Method With Variations. Conjugate Direction Algorithms. Quasi-Newton Methods. OPTIMALITY CONDITIONS FOR CONSTRAINED PROBLEMS. First- and Second-Order Optimality Conditions. Applications of Optimality Conditions. LINEARLY CONSTRAINED PROBLEMS. Models with Linear Constraints. Variable-Reduction Algorithms. NONLINEARLY CONSTRAINED PROBLEMS. Models with Nonlinear Constraints. Direct Algorithms for Nonlinearly Constrained Problems. Sequential Unconstrained Minimization Techniques. Sequential Constraint Linearization Techniques. OTHER TOPICS. Obtaining Global Solutions. Geometric Programming. References. Author and Subject Indexes.

Nonlinear systems;: The parameter analysis and design

Abstract: Book on nonlinear systems analysis and design covering parameter mapping, symmetrical, transient and forced oscillations and stability analysis

Selected applications of nonlinear programming

Abstract: : The report presents selected applications of nonlinear programming in some detail. The first chapter, which is a general introduction to nonlinear programming, contains definitions, classifications of problems, mathematical characteristics, and solution procedures. The remaining chapters deal with various problems and their nonlinear programming models.

Optimum Control of Reactive Power Flow

Abstract: The general problem of minimizing the operating cost of a power system by proper selection of the active and reactive productions is formulated as a nonlinear programming problem in accordance with previous work by Carpentier of Electricitede France. This general problem is particularized to the minimization of transmission line losses by suitable selection of the reactive productions and transformer tap settings. An efficient computational procedure based on the Newton-Raphson method for solving the power-flow equations and on the dual (Lagrangian) variables of the Kuhn and Tucker theorem is discussed. This minimization procedure has been applied successfully to a 500-node system studied by the Bonneville Power Administration for which an effective power-flow program had been developed previously. The dual variables associated with the primary (electrical) variables are obtained in the course of the computation, and their engineering significance for power system design and tariffication is emphasized. The procedure has been extended to the general case of combined active and reactive optimization.

Linear Programming as an Aid in Planning Kilovar Requirements

Abstract: In the planning and design of a high-voltage transmission network it is sometimes desirable to install controllable kilovar ar capacity at several locations to support bus voltages during emergencies. This problem arises when transmission line or generation outages cause bus voltage magnitudes to decrease below desirable limits. The problem of selecting where and how much kilovar capacity is required has many feasible solutions which satisfy the conditions imposed. A method for locating that solution with the minimum total installed capacity is presented. This nonlinear programming problem will be solved by using a linear approximation, solving the linear programming problem, and then correcting the linear approximation through the use of differences between the linear and nonlinear results. The method is illustrated by an application to a portion of a large high-voltage network.

An Iterative Chebyshev Approximation Method for Network Design

Abstract: One of the most important problems of computeraided network design is the optimization of network characteristics by iterative calculation. In this paper, the problem of realizing a network whose transmission characteristics approximate a given function in Chebyshev sense is treated as a nonlinear programming problem, and a method of solving this problem by successively solving linear programming problems, which are derived by locally linearizing the original nonlinear programming problem, is proposed. An improvement of the method for reducing the computation time is also considered and is proved to be practical and very effective by many design examples.

Systems Reliability Subject to Multiple Nonlinear Constraints

Abstract: A simple computational procedure has been developed for maximizing reliability of multistage parallel systems subject to multiple nonlinear constraints. It appears that the procedure can be applied to a variety of optimization problems with separable objective and multiple constraint functions.

Computer Models in Advertising Media Selection

Abstract: There is a clear dichotomy between the optimizing and the nonoptimizing approaches. The four main methods using the optimizing approach are: (1) linear programming, (2) nonlinear programming, (3) i...

Large Signal Dynamics of Load-Frequency Control Systems and Their Optimization Using Nonlinear Programming: II

Abstract: The first part of this paper deals with the response of a large load-frequency control system (power system) when subjected to a major disturbance. If the system becomes unstable, upon the occurrence of a disturbance, incidents like the great blackout of 1965 in the Eastern United States and Canada can occur. This paper differs from much previous work in the respect that the assumption that the system be linear is not made. A numerical method is presented to solve for the nonlinear system. The linear case can be treated as a degenerate case of the more difficult problem. Since it is possible to find an exact solution for a linear system, examples are presented to show good agreement between the exact solution and the solution obtained by the numerical method.

Geometric programming: duality in quadratic programming and lp-approximation.

Abstract: : The duality theory of geometric programming as developed by Duffin, Peterson and Zener is based on abstract properties shared by certain classical inequalities, such as Cauchy's arithmetic-geometric mean inequality and Holder's inequality. Inequalities with these abstract properties have been termed 'geometric inequalities.' In this paper we establish a new geometric inequality and use it to extend the 'refined duality theory' for 'posynomial' geometric programs. This extended duality theory treats both 'quadratically-constrained quadratic programs' and 'l sub p-constrained l sub p-approximation (regression) problems' through a rather novel and unified formulation of these two classes of programs. This work generalizes some of the work of others on linearly-constrained quadratic programs, and provides to the best of our knowledge the first explicit formulation of duality for constrained approximation problems. Other people have developed duality theories for a larger class of programs, namely all convex programs, but those theories (when applied to the programs considered here) are not nearly as strong as the theory developed here. This theory has virtually all of the desirable features of its analog for posynomial programs, and its proof provides useful computational procedures. (Author)

Time Domain Approximations and an Active Network Realization of Transfer Functions Derived from Ideal Filters

Abstract: Rational polynomial Laplace transform approximations for truncated versions of \sin (kt + b)/(kt + b) are produced by use of a nonlinear programming technique designed to help avoid the usual false minimum problems. An economical network synthesis technique for realization of the nonminimum phase functions thus produced is also shown.

Application of the Created Response Surface Technique to Structural Optimization

Abstract: : The Created Response Surface Technique was applied to some problems of structural optimization with multi-load conditions. The constrained optimization problem was converted to an unconstrained problem by the use of penalty functions that varied inversely with the distance of the design point from a constraint. Response surfaces also were introduced for optimization with discrete variables. The Rosenbrock Method of unconstrained optimization was used. The original method was modified to take advantage of the interaction between the optimization procedure and the response of the structure to change in discrete sections. This, when coupled with the r extrapolation procedure, resulted in considerable savings in computing time. To quantitatively evaluate the effectiveness of the techniques presented, a pilot computer program for the optimization of small-scale structures was developed. To minimize the effort involved in the development of the program a relatively inefficient analysis module was selected from a previously coded computer program. This analysis was limited to approximately 50 degrees of freedom and originally contained only an axial force member in the element library. This was considered sufficient for initial research. Two problems were investigated, that of the 3-bar truss studied by Schmit and Mallett and that of the 25-bar truss studied by Fox and Schmit. Fox and Schmit included buckling constraints and used tube diameter and thickness as the design variables. The cross-sectional areas of the circular tubes are used here as the design variables. The Euler criteria are included as stress limits for each member.

Non-linear programming method for limit analysis of rotationally symmetric shells∗☆

Abstract: A numerical method is suggested to compute lower and upper bounds on the yield-point load, for the limit analysis of a rotationally symmetric sandwich shell under central loading. The distributions of stresses and velocities are established so that a non-linear programming technique may be used in a reasonable amount of computer time. Significant bounds have been obtained for configurations for which standard methods of analysis had been inadequate.

Solution of a nonlinear programming problem

Abstract: : The article presents an approximate method for solution of the problem of maximization of a nonlinear separable function with two-sided limitations on the unknowns and the fixed common lower boundary of the modulus of the difference of the unknowns. The possibility of applying this method to some problems of air traffic control in high-density zones is shown. Appropriate data can be stored in the operative memory of a 'Ural-4' computer. (Author)

A review and discussion of some methods for system identification and modeling.

Abstract: Mathematically testing the validity of a theoretical model with an observed physical system is an important step in understanding and utilizing such a system. Perhaps even more useful is the generation of computational techniques which use input-output data from physical systems to automatically construct mathematical models which, in some sense, provide the ‘best’ descritpions of the real systems. This paper briefly discusses a few of the more recent mathematical techniques available for model generation and testing. A new identification method based upon convex and linear programming is discussed in detail and a number of examples indicating its applicability are given. The linear programming method is basically an approximation to a convex programming problem, the solution of which determines the coefficients of the differential equation describing the observed system data. A number of extensions of the identification method indicate some of its most useful properties. The order of the assumed model differential equation can be larger than that of the unknown system and the identification process will either assign zero values to the superfluous coefficients of the model or pole-zero cancellations will occur in the factored form of the laplace transform of the model transfer function. ‘Best’ lowest order models may be selected automatically. Liner constraints among the coefficients of the model differential equation may be used to restrict the allowable ranges of the coefficients. Multiple sets of data for a single system may be used simultaneously in the indentification process. Multiple input-output systems or systems described by difference equations or with transportation lag can also be identified. Coefficients of time varying and/or nonlinear models may be determined.

A Loading Problem in Process Type Production

Abstract: Given a set {di} of continuous rates of demand for n products, the associated costs, unit processing times, and set-up times, and given N different types of processors with possibly more than one processor per type, it is desired to determine the single-stage production pattern under the constraint that only EMQ's are manufactured. An LP is formulated to which we apply the decomposition principle and generate a nonlinear programming problem in integer variables as a subproblem to be generated at each iteration. This subproblem is solved by dynamic programming approach.

Duality in discrete programming. iii. nonlinear objective function and constraints.

Abstract: : The results of 'Duality in Discrete Programming,' are extended in this paper (which, however, is self-contained) to the case of dual programs with nonlinear objective function and nonlinear constraints. The symmetric dual nonlinear programs studied by Dantzig, Eisenberg and Cottle, as well as the dual nonlinear programs formulated by Wolfe, Mangasarian and Huard are generalized by allowing some of the variables to be constrained to belong to arbitrary sets of real numbers, and dropping the requirement that the objective function and the constraints be convex (concave) in these variables. The basic properties established for the above problems are shown to carry over, with some qualification, to their generalized counterparts which encompass, among others, various types of mixed-integer nonlinear programs. (Author)

Non-Linear Dynamic Programming of Hydraulic Fracturing Models

Abstract: Nonlinear programing techniques are applied to hydraulic fracturing models resulting in the design of an optimum fracture treatment The objective function is chosen as the discounted present worth of the well and is optimized with respect to 8 decision variables The region of feasibility is restricted by both upper and lower bounds on each decision variable and by constraints placed on the interactions of the various variables within the models The possibility of applying venture analysis to fracture treatment is also discussed This involves the assignment of a probability distribution to uncertain parameters in the model This allows the estimate of a probability curve showing the probability of achieving various values of the objective function

Duality in discrete programming: iv. applications.

Abstract: : The paper develops a method for solving integer and mixed-integer nonlinear programs whose objective and constraint-functions are differentiable and concave (convex) on the set of nonnegative variables (i.e. on the domain obtained by disregarding the integrity conditions). (Author)

The discrete maximum principle with applications to management science.

Abstract: : A general version of the discrete control system is formulated which includes difference inequalities and the possibility of inequality as well as equality state space constraints. The authors apply the well-known Kuhn-Tucker theorem to derive necessary conditions that the primal and adjoint functions must satisfy in order that they be a solution to the optimum control problem. These necessary conditions are valid only if the constraint qualification holds. One of the necessary conditions is the Pontryagin maximum principle. A special version of the general problem is stated, and applied to a separable non-linear programming problem. Finally, production control examples are solved by means of the theory. (Author)