Showing papers on "Discrete optimization published in 1971"

Discrete splines via mathematical programming.

Abstract: Existence, uniqueness and characterizing properties are given for a class of constrained minimization problems in real Euclidean space. These problems are the discrete analogues of minimization problems in Banach space whose solutions are generalized splines. Solutions of these discrete problems, which are called discrete splines, can be obtained by algorithms of mathematical programming.

A basic optimization problem in linear systems.

Abstract: This problem is extensively considered in the literature and includes the familiar optimal regulator and optimal tracking problems for linear systems with a quadratic performance criterion (see [1]). The solution may be obtained by the following rather direct method (see [2], Section 4.4). Observe that H1 × H3, equipped with the usual inner product, is a Hilbert space whose norm is computed by [(u, x)] 2 = I] u []2 + I I x 112 for (u, x) ~ Hx × H3. Note also that the infimum of J over H1 is the distance of (0, N~:) ~ H1 x H3 from the graph of T. Since the graph of T is a closed linear subspace, this infimum is attained uniquely by the orthogonal projection of(0, N~:) on this subspace. It is then easily established that the optimal control is given by

Automated Optimum Design from Discrete Components

Abstract: Optimization in structural design using discrete variable can exploit the discrete characteristics of the design problem and avoid artificial and often self-defeating discretation from problem solutions in continuous variables. A method for solving member-sizing problems using wide flange tables, a current design code, and linear elastic behavior due to arbitrary loading conditions is shown to be within reasonable economic bounds. The method consists of a combinatorial algorithm classified as a branch-and-bound type. Starting with a lower-bound infeasible solution the algorithm detects a feasible solution for an upper bound and explores the region for a local optimum, bouncing along the constraints if the value of the objective function can be improved. Reanalysis is performed at each step with an exact method which performs only part of normal analysis steps. Included as part of the STRUDL information system, the programmed algorithm allows an engineer to assist in improving the optimization process and perform optimization on a structure on a piece-wise basis.

On the Maximum Principle for a Class of Discrete Dynamical Systems with Lags

Abstract: A maximum principle for a class of discrete systems with lags is established constructively within the framework of discrete optimal control theory. An application of the maximum principle to a problem in advertising concludes the paper.PDF Version: Sethi, S.P. and Burdet, C.-A., "On the Maximum Principle for Discrete Dynamical Systems with Lags," Management Sciences Research Report No. 262, GSIA, Carnegie-Mellon University, November 1971.

Optimization under aeroelastic constraints

Abstract: Optimization of aeroelastic constraints for aircraft design using differential equation idealization and finite element approximation

Discrete optimization on a multivariable boolean lattice

Abstract: Consider the problem of finding the minimum value of a scalar objective function whose arguments are theN components of 2N vector elements partially ordered as a Boolean lattice. If the function is strictly decreasing along any shortest path from the minimum point to its logical complement, then the minimum can be located precisely after sequential measurement of the objective function atN + 1 points. This result suggests a new line of research on discrete optimization problems.

On computational methods for dynamic and static optimization

Abstract: In the 1960s the joint impact of fast digital computers and the space age stimulated the ingenuity of a number of researchers in optimization and prompted them to invent new and/or better (faster, more elegant) methods for solving, numerically, optimization problems of different types. Two areas that received much attention were trajectory optimization and parameter optimization. That is, the computational problems of determining optimal trajectories (generally, functions of time) and optimal parameter values by minimizing (maximizing) appropriate performance criteria.

Some problems in discrete optimization

Abstract: The present paper concentrates on several problems of network flows and discrete optimization. Progress has been made on some of the problems while little is known about others. Some of the problems discussed are shortest paths, multi-commodity flows, traveling salesman problems, m-center problem, telepak problems and binary trees.

The analysis of discrete time systems with deterministic inputs by the serial/matrix method†

Abstract: A matrix method is presented for the determination of the response sequences of discrete time-invariant linear systems for a large range of deterministic input sequences. Using the z-transform formulation the transfer function of the discrete system is defined and, assuming that the input sequence is a linear combination of a certain set of basic sequences, it is demonstrated that an explicit formulation of the output sequence may be obtained by purely matrix operations. This approach not only simplifies the application of the z-transform technique but can also be used as the basis of a digital computer programme. Examples are given to illustrate the use of the method including its application to the solution of general nth-order difference equations

A Survey of Current Optimization Methods.

Abstract: : The report surveys a number of optimization methods which can be applied to non-differentiable functions. The methods include both deterministic and non-deterministic approaches to the solution of problems in optimization. The methods are discussed and compared, both in theory and by means of sample problems using computer programs supplied with the report, on the basis of such factors as efficiency, handling of constraints, and termination mechanisms. Some guidelines are offered for selection of techniques and programs most suitable for several types of problems. (Author)