Showing papers on "Nonlinear programming published in 1969"

Nonlinear Programming

Abstract: Part 1 (if): Assume that Z is closed. We must show that if A is closed for all k → x x , k → y y , where ( k A ∈ ) k y x , then ( ) A ∈ y x . By the definition of Z being closed, we know that all points arbitrarily close to Z are in Z. Let k → x x , k → y y , and ( k A ∈ ) k y x . Now, for any ε > 0, there exists an N such that for all k ≥ N we have || || k ε − < x x , || || k ε − < y y which implies that ( ) , x y is arbitrarily close to Z, so ( ) , x y ∈ Z and ( ) A ∈ y x . Thus, A is closed.

Theory of optimal control and mathematical programming

Abstract: Book on theory of optimal control and mathematical programming covering linear, nonlinear, quadratic programmings, etc

Elements of large-scale mathematical programming

Abstract: : A unifying framework of concepts central to the optimization of large structured systems is developed and used in the organization of the literature. The basic concepts are divided in two groups, (1) problem manipulations, in which a given problem is restated in an alternative form more amenable to solution, and (2) solution strategies which reduce an optimization problem to a related sequence of simpler problems that can be solved by specialized methods.

Nonlinear Programming Solutions for Load-Flow, Minimum-Loss, and Economic Dispatching Problems

Abstract: A unified approach to load-flow, minimum-loss, and economic dispatching problems is presented. A load-flow solution is shown to coincide with the minimum of a function of the power system equations. An unconstrained minimization method, developed by Fletcher-Powell, is used to solve the load-flow problem. The method always finds a solution or indicates the nonexistence of a solution. Its performance is highly independent of the reference- slack bus position and requires no acceleration factors. Several constrained minimization techniques that solve the minimum-loss and economic dispatching problems are investigated. These include the Fiacco-McCormick, Lootsma, and Zangwill methods. The technique finally recommended is shown to be an extension of the method used to solve the load-flow problem. The approved IEEE test systems, and other systems whose response to conventional methods was known, have been solved.

Convergence Conditions for Nonlinear Programming Algorithms

Abstract: Conditions which are necessary and sufficient for convergence of a nonlinear programming algorithm are stated. It is also shown that the convergence conditions can be easily applied to most programming algorithms. As examples, algorithms by Arrow, Hurwicz and Uzawa; Cauchy; Frank and Wolfe; and Newton-Raphson are proven to converge by direct application of the convergence conditions. Also the Topkis-Veinott convergence conditions for feasible direction algorithms are shown to be a special case of the conditions stated in this paper.

Combined Use of the Powell and Fletcher - Powell Nonlinear Programming Methods for Optimal Load Flows

Abstract: The feasibility solving optimal load flow problems by means of nonlinear programming techniques has been previously demonstrated. Recently, a new method by Powell for constrained minimization has claimed a better performance than existing methods. An investigation was carried out to test Powell's method on optimal load flow problems and results obtained confirm expectations. Powell's method and its accompanying Fletcher-Powell method, which performs the actual minimizations, are presented. The power system optimal load flow problem is discussed, and its equations are presented in the form required by the nonlinear programming approach. Finally, a numerical example on the IEEE 30- bus standard test system is presented in which an economic dispatching is accomplished with the new method. Computation time is small enough to permit the application of the method for dispatching at practical intervals of time.

An approximative algorithm for the fixed charge problem

Abstract: This paper describes an approximate solution method for solving the fixed charge problem. This heuristic approach is applied to a set of test problems to explore the margin of error. The results indicate that the proposed fixed charge simplex algorithm is capable of finding optimal or near optimal solutions to moderate sized fixed charge problems. In the absence of an exact method, this heuristic should prove useful in solving this fundamental nonlinear programming problem.

Nonzero-sum stochastic games

Abstract: : The paper extends the basic work that has been done on zero-sum stochastic games to those that are nonzero-sum. Appropriately defined equilibrium points are shown to exist for both the case where the players seek to maximize the total value of their discounted period rewards and the case where they wish to maximize their average reward per period. For the latter case, conditions required on the structure of the Markov chains are less stringent than those imposed in previous work on zero-sum stochastic games, extensions to n-person games and underlying semi-Markov processes are discussed, and finding an equilibrium point is shown to be equivalent to solving a certain nonlinear programming problem.

Constrained Nonlinear Optimization by Heuristic Programming

Abstract: This paper develops a new algorithm for solving the general nonlinear programming problem that melds the flexible simplex search of Nelder and Mead with various additional rules to take care of equality and/or inequality constraints. The set of violated inequalities and equalities is lumped into one inequality constraint loosely satisfied during the early progress of the optimization and more closely satisfied during its final stages. To permit this type of search, the method sets up a special tolerance criterion, a function that does not depend on either the values of the objective function or the values of the constraints. The new algorithm has solved successfully a number of problems that have been proposed in the literature as test problems. Finally, to indicate the algorithm's capabilities, the paper describes an example composed of a linear objective function of twenty-four variables subject to fourteen nonlinear equalities and thirty inequalities.

Application of non‐linear programming to optimum grillage design with non‐convex sets of variables

Abstract: The application of non-linear programming methods for the optimum design of statically indeterminate structures is discussed, with special emphasis on the design of elastic grillages loaded laterally and in plane. Some features of SUMT (sequential unconstrained minimization technique) are demonstrated by means of numerous examples of varying complexity. The Variable Metric method of search is discussed and compared to Powell's Direct Method. It is shown that non-convex sets of design variables are often encountered in structural problems of the grillage type. SUMT may still be used, but the choice of starting value and initial response factor decisively influences the chance of finding the global optimum. It is demonstrated that a fully stressed design may not necessarily correspond to the minimum weight design. Optimum design of grillages which are simultaneously subjected to lateral and in-plane loads may be performed efficiently by means of non-linear programming.

On Second Order Necessary Conditions of Optimality

Abstract: Second order necessary conditions of optimality with straightforward application to nonlinear programming of optimal control problems

Applications of mathematical control theory to accounting and budgeting: ii. the continuous joint trading madel.

Abstract: : The paper applies the mathematical control theory to the accounting network flows, where the flow rates are constrained by linear inequalities. The optimal control policy is of the 'generalized bang-bang' variety which is obtained by solving at each instant in time a linear programming problem whose objective function parameters are determined by the 'switching function' which is derived from the Hamiltonian function. The interpretation of the adjoint variables of the control problem and the dual evaluators of the linear programming problem demonstrates an interesting interaction of the cross section phase of the problem, which is characterized by linear programming, and the dynamic phase of the problem, which is characterized by control theory. (Author)

Some practical regularity conditions for nonlinear programs

Abstract: : Some practical sufficient conditions are given for a program with linear constraints and nonlinear objective to have well-behaved duality properties.

Minimum Weight Design of Enclosed Antennas

Abstract: The minimization of the total weight of the tiltable component of an enclosed antenna structure with a given topology and a given initial design subject to relative deformation constraints (rms limits) in both face-up and face-side attitudes as well as to certain structural constraints is formulated, and the solution to this problem is sought by a modified version of the gradient projection method of nonlinear programming. The given initial design should satisfy all the structural constraints, but it may violate one or both deformation constraints. The solution is obtained in two steps by first finding a feasible solution, that is a solution which satisfies all the constraints, and then minimizing the total weight subject to structural and deformation constraints. The structural constraints, for the sake of simplicity, have been assumed to be linear. The method is tested by optimizing a plane truss of a large radio telescope antenna.

Optimum design of spatial structures

Abstract: : A systematic approach is presented to the optimal design of spatial frames for minimum weight subject to constraints on stress and geometry. The optimization procedures discussed are general and may be applied to structures which can be analyzed by matrix displacement or finite element methods. Two methods of mathematical programming are applied to obtain a minimum weight design. Both of these techniques require derivatives of the objective and constraint functions to improve estimates of the optimum design. In order to take full advantage of existing analysis capability, the programming techniques in this research have been assuming that such derivatives are not available.

Comparative study of various minimization techniques used in mathematical programming

Abstract: Nine different techniques of unconstrained minimization are applied to the same problem and their relative efficiency is compared. The techniques used are of gradient, variable metric type, used in conjunction with the sequential unconstrained minimization techniques (SUMT) program. The problem tested has nonlinear inequality constraints up to the ninth order.

A branch-and-bound algorithm for allocation problems in which constraint coefficients depend on decision variables.

Abstract: : A branch-and-bound algorithm is developed for a class of allocation problems in which some constraint coefficients depend on the values of certain of the decision variables Were it not for these dependencies, the problems could be solved by linear programming The algorithm is developed in terms of a strategic-deployment problem in which it is desired to find a least-cost transportation fleet, subject to constraints on men and/or materiel requirements in the event of certain hypothesized contingencies Among the transportation vehicles available for selection are aircraft with the characteristic that the amount of goods deliverable by an aircraft on a particular route in a given time period (called aircraft productivity and measured in kilotons per aircraft per month) depends on the ratio of type 1 to type 2 aircraft used on that particular route A model is formulated in which these relations are first approximated by piecewise linear functions A branch-and-bound algorithm for solving the resultant nonlinear problem is then presented; the algorithm solves a sequence of linear programming problems The algorithm is illustrated by a sample problem, and comments concerning its practicality are made (Author)

Statistical control for non-linear optimization.

Abstract: : The paper delineates an algorithm which provides a statistical point estimate as well as a statistical upper confidence point for the global optimum, say g*, in a non-convex mathematical programming problem. Further, the procedure will supply a value of the input vector, x dot, so that g(x dot), the objective function evaluated at x dot, is close to g*. Two examples involving the optimization of second order response surface designs with known global optima are given for the purpose of comparing the point estimate and the upper confidence point estimate with the known values. (Author)

Optimal Scheduling for Cyclic Steam Injection Projects

Abstract: A well steam-injection scheduling model which maximizes the profit per day for oil production leases is discussed. The model can be used to produce operating resteaming schedules and to evaluate the profitability for any number of steam generators. There are no limitations on the size of the problem which can be handled by this approach exclusive of computer storage capability and run time. The model consists of nonlinear regression, nonlinear optimization, and a scheduling algorithm. Weekly well-production data from a production-reporting system are utilized to extend the system to a reporting, operating control, and plannning system that not only provides management with production history, but analysis tools for better decisions as well.

A relaxed interior approach to nonlinear programming

Abstract: This paper contains the mathematical validation of a new approach to mathematical programming problems based on a penalty function method. The given problem is replaced by a second „auxiliary“ problem which, in many cases may be solved by standard methods since it involves the maximization of a concave function of a single variable over an interval. The auxiliary problem is defined implicitly in therms of the constituents of the original problem. Examples are presented in order to illustrate the theoretical results.

A simple proof of a general duality theorem of convex programming

Abstract: The first part of this note presents concisely and partially proves in logical terms the relations betweenUzawa's andKuhn andTucker's equivalence theorems of nonlinear programming. In the second part we give simple mathematical proofs of two lemmata linking theKuhn-Tucker conditions and dual solutions and use them to establish a nonlinear duality theorem of considerable generality.