Showing papers on "Piecewise linear function published in 1979"

The Knapsack Sharing Problem

Abstract: The knapsack sharing problem has a utility or tradeoff function for each variable and seeks to maximize the value of the smallest tradeoff function (a maximin objective function). A single constraint places an upper bound on the sum of the non-negative variables. We develop efficient algorithms for piecewise linear, nonlinear, and piecewise nonlinear tradeoff functions and for any knapsack sharing problem with integer variables. These algorithms for the knapsack sharing problem extend the sharing problem algorithm in a companion paper to any piecewise linear, nonlinear, or piecewise nonlinear tradeoff functions.

On a class of transformations which have unique absolutely continuous invariant measures

Abstract: A class of piecewise C2 transformations from an interval into itself with slopes greater than 1 in absolute value, and having the property that it takes partition points into partition points is shown to have unique absolutely continuous invariant measures. For this class of functions, a central limit theorem holds for all real measurable functions. For the subclass of piecewise linear transformations having a fixed point, it is shown that the unique absolutely continuous invariant measures are piecewise constant.

Alternatives to the exponential spline in tension

Abstract: A general setting is given for smooth interpolating splines depending on a parameter such that as this parameter approaches infinity the spline converges to the piecewise linear interpolant. The theory includes the standard exponential spline in tension, a rational spline, and several cubic splines. An algorithm is given for one of the cubics; the parameter for this example controls the spacing of new knots which are introduced.

Stochastic optimization of a water supply system

Abstract: Stochastic dynamic programming (DP) is used to find the operating policy with the least expected steady state cost for a water supply system consisting of a reservoir and an alternative source. The set of possible decisions consists of a number of release rules, each expressing release as a function of storage, rather than a number of discrete releases, as in the conventional DP approach. A flexible procedure is developed which permits inflow to be described by piecewise linear probability density functions, and removes the constraint that inflow and release must be multiples of the discrete unit of storage. The techniques are applied to a reservoir-river system, and through simulation, the results are compared with the solution found by conventional DP.

Piecewise linear elastic–plastic analysis

Abstract: The purpose of this paper is to show that a mathematical programming method, due to Maier and his co-workers, of nonlinear structural analysis can be rerformulated so that greater computational efficiences are achived. The methods are designed for a class of elastic–plastic structures under ‘piecewise linear’ assumptions and solve among others the problem of determining the stresses and strains in the structure. The reformulation gives rise to a class of mathematical programming problems calles a ‘n by dn linear complementarity problem’, for which the author has developed an efficient algorithm. It will be explained why and by how much the proposed method (the reformulation and its solution by the author's algorithm) solves the structural problems more efficiently than the existing one. Results of a systematic computer experiment supporting the efficiency of the proposed method are also presented.

An Approximation Procedure for Nonsymmetric, Nonlinear Hyperbolic Systems with Integral Boundary Conditions

Abstract: Optimal order error estimates for two linearizations of a collocation process using tensor products of continuous, piecewise linear functions in space and time are derived for a class of nonsymmetric, nonlinear hyperbolic systems with integral boundary conditions. This procedure is a variant of the so-called box scheme.Error estimates are also derived for a generalization of these procedures to collocation based on continuous, piecewise polynomials of degree r in space tensored with continuous, piecewise linear functions in time.

A Complementary Pivoting Approach to Parametric Nonlinear Programming

Abstract: A family of nonlinear programming problems: minimize θt, u subject to gt, u ≤ 0 and u ≥ 0 is considered, where t is a scalar parameter changing from 0 to +∞, u an n-dimensional variable vector, θ a continuous real valued mapping defined on the 1 + n-dimensional space and g a continuous mapping from the 1 + n-dimensional space into the m-dimensional space. It is assumed that for each fixed t, θt, · and gjt, · j = 1,..., m have partial derivatives ∂θt, u/∂ui and ∂gjt, u/∂uii = 1,..., n, j = 1,..., m which are continuous with respect to both t and u. Under a moderate constraint qualification, this paper applies the complementary pivoting theory to the computation of a piecewise linear path in the product space of the parameter t and the variable vector u from which we can approximately know how a Kuhn-Tucker's stationary solution to the problem moves as the parameter t changes from 0 to any given nonnegative number.

A String Model Etching Algorithm

Abstract: : The basic algorithm and software implementation of a string model for simulation of surface etching are presented. The algorithm models the time evolution of a line edge profile by advancing nodes or points on a piecewise linear curve representing the profile. The specific formulas for the direction and rate of advance, insertion and deletion of points and deletion of loops are shown. Complete software documentation in the form of parameter definitions and a listing of the FORTRAN code for a CDC 6400 machine are included.

A Study of ${\text{PC}}^1 $ Homeomorphisms on Subdivided Polyhedrons

Abstract: In this paper we consider the problem of establishing conditions when a given piecewise continuously differentiable mapping is a homeomorphism of a closed convex polyhedral set. These conditions are a generalization of the ones used by Gale–Nikaido and are similar in spirit to those of Mas-Colell. For the special case when the mapping is piecewise linear, we give an apparently new sufficiency condition for the mapping to be a homeomorphism of $R^n $. The results are further extended to include the case when the Jacobians may be singular.

On the mixed finite element approximation for the buckling of plates

Abstract: We consider the mixed finite element method for the buckling problem of the thin plate by using piecewise linear polynomials. We give error estimates for the approximate eigenvalues and the eigenfunctions.

An algorithm for a piecewise linear model of trade and production with negative prices and bankruptcy

Abstract: The general equilibrium model is approximated as a piecewise linear convex model and solved from the point of view of welfare economics using linear programming and fixed point methods.

The fixed point approach to nonlinear programming

Abstract: In this paper we consider the application of the recent algorithms that compute fixed points in unbounded regions to the nonlinear programming problem. It is shown that these algorithms solve the inequality constrained problem with functions that are not necessarily differentiable. The application to convex and piecewise linear problems is also discussed.

An implementation of estimation techniques to a hydrological model for prediction of run-off to a hydroelectric power station

Abstract: Parameter and state estimation algorithms have been applied to a hydrological model of a catchment area in southern Norway to yield improved control of the household of water resources and better economy and efficiency in the running of the power station, as experience proves since the system was installed on-line in the summer of 1978. A nonlinear conceptual state-space model of a Nordic hydrological system is presented in this paper. The model is transformed analytically to a piecewise linear time-discrete form in order to obtain straightforward updating of the gain matrix in the estimator through solution of the Riccati equation. Estimator gains can then be expressed approximately as a function of the state. Discharge coefficients (time constants) are estimated by the maximum likelihood method. Results are presented which show estimated and observed runoffs, and estimates of the state variables.

Wild embeddings of piecewise linear manifolds in codimension two

Abstract: Publisher Summary This chapter discusses wild embeddings of piecewise linear manifolds in codimension two. It presents the construction of many topological embeddings of non-simply connected piecewise linear (=PL) 2n -manifolds into the 2 n +2-space, which cannot be approximated by PL embeddings. The main idea is a generalization of Giffen and Eaton, Pixley and Venema's, and to apply Giffen's shift spinning construction to PL spineless manifolds. The existence of PL spineless manifolds was proved by Cappell and Shaneson. The chapter presents a theorem that states that if 2 n ≥6, then the equivalence classes of wild embeddings f (N ) (N=1,2,,…) are mutually distinct. Any wild embedding f (N ) cannot be approximated by a locally flat topological embedding.

Scalar labelings for homotopy paths

Abstract: Two scalar labelings are introduced for obtaining approximate solutions to systems of nonlinear equations by simplicial approximation. Under reasonable assumptions the new scalar-labeling algorithms are shown to follow, in a limiting sense, homotopy paths which can also be tracked by piecewise linear vector labeling algorithms. Though the new algorithms eliminate the need to pivot on a system of linear equations, the question of relative computational efficiency is unresolved.

Error analysis of finite element methods for mildly nonlinear variational inequalities

Abstract: The finite element method is applied through the use of a variational inequality to an obstacle problem involving nonhomogeneous boundary data. For piecewise linear conforming trial functions energy norm error bounds are derived.

Optimisation non Differentiable: Methodes de Faisceaux

Abstract: We define a class of descent methods to minimize a nondifferentiable function. These methods are based on a representation of the objective which combines a quadratic approximation and the usual approximation by a piecewise linear function. Hence, they realize a synthesis between quasi-Newton methods and cutting plane methods. In addition, they have the particularity of requiring no sophisticated line search. They are also presented in ref. [7] (in English).

Piecewise linear paths to minimize convex functions may not be monotonic

Abstract: The exact homotopy path when seeking the minimum of a convex function is monotonic in the homotopy parameter. This monotonicity is not inherited by the piecewise linear approximations to such paths produced by fixed-point algorithms.

An ¹ remainder theorem for an integrodifferential equation with asymptotically periodic solution

Abstract: For a certain integrodifferential equation of Volterra type on (0, oo), with piecewise linear convolution kernel, it is shown that the solution is u(t) = a cos fit + p(t), with p E L'(0, oo) and a and /3 constant; u' is represented similarly.

Optimum plastic design of a variable-thickness orthotropic plate under in-plane loading

Abstract: Using Hill's criterion for yielding of an orthotropic material, the plastic design of a plate having minimum volume is investigated. A stress-function approach is proposed for finding an approximate solution to the non-linear optimization problem. The possibility of using piecewise linear representations of the yield surface in order to reduce the problem to a linear one is also explored. Numerical results are presented for a circular plate under axisymmetric loading, and it is demonstrated that solutions to the linearized problems provide bounds upon the volume of material required by the Hill condition.

On some properties of a piecewise linear basis set

Abstract: A set of piecewise linear basis functions called the integral Walsh basis set is defined, and some of its properties are discussed.

Perturbations in fixed point algorithms

Abstract: If the Jacobian of a differentiable function is singular at a zero of the function, any piecewise linear approximation to it may not have a zero, even when the function has one. In this paper we present a technique for perturbing such functions so that the recent fixed point algorithms that trace zeros of piecewise linear homotopies will succeed in finding a zero of such functions. We also show how to unperturb in case the Jacobian is nonsingular at the solution, and thus not impede the super linear convergence attained by these algorithms.

Bounds to internal forces for elastic-plastic solids subjected to variable loads

Abstract: Considering an elastic-plastic workhardening solid with piecewise linear yield surfaces and a piecewise linear workhardening law, we give a method for constructing bounds to the internal forces and to the (hardened) yield stresses produced by the action of variable loads at any point of the body and at any time. The loading history is supposed to be unknown, but the loads range within a given domain.