Algorithms for network piecewise-linear programs: A comparative study

Piecewise-Linear Programming (PLP) is an important area of Mathematical Programming and concerns the minimisation of a convex separable piecewise-linear objective function, subject to linear constraints. In this paper a subarea of PLP called Network Piecewise-Linear Programming (NPLP) is explored. The paper presents four specialised algorithms for NPLP: (Strongly Feasible) Primal Simplex, Dual Method, Out-of-Kilter and (Strongly Polynomial) Cost-Scaling and their relative efficiency is studied. A statistically designed experiment is used to perform a computational comparison of the algorithms. The response variable observed in the experiment is the CPU time to solve randomly generated network piecewise-linear problems classified according to problem class (Transportation, Transshipment and Circulation), problem size, extent of capacitation, and number of breakpoints per arc. Results and conclusions on performance of the algorithms are reported.

A Branch and Bound Algorithm for a Transportation Type Problem with Piecewise Linear Convex Objective Function

A branch and bound type algorithm is presented in this paper to the problem of finding a transportation schedule which minimises the total transportation cost, where the transportation cost over each route is assumed to be a piecewice linear continuous convex function with increasing slopes. The algorithm is an extension of the work done by Balachandran and Perry, in which the transportation cost over each route is assumed to beapiecewise linear discontinuous function with decreasing slopes. A numerical example is solved illustrating the algorithm.

Extrinsic Analog Synthesis using Piecewise Linear Current-Mode Circuits

A methodology is presented for the synthesis of analog circuits using piecewise linear (PWL) approximations. The function to be synthesized is divided into PWL segments such that each segment can be realized using elementary MOS current-mode programmable-gain circuits. A number of these elementary current-mode circuits when connected in parallel, it is possible to realize piecewise linear approximation of any arbitrary analog function with in the allowed approximation error bounds. Simulation results show a close agreement between the desired function and the synthesized output. The number of PWL segments used for approximation and hence the circuit area is determined by the required accuracy and the smoothness of the resulting function.

On the design of the piecewise linear vibration absorber

This paper explores the potential of the piecewise linear vibration absorber in a system subject to narrow band harmonic loading. Such a spring is chosen because the design of linear springs is common knowledge among engineers. The two-degrees-of-freedom system is solved by using the Incremental Harmonic Balance method, and response aspects such as stiffness crossing frequency and jump behaviour are discussed. The effects of mass, stiffness, natural frequency ratios, and stiffness crossing positions on the suppression zone are probed. It is shown that a hardening absorber can deliver a wider bandwidth than a linear one over a range of frequencies. The absorber parameters needed to produce good designs have been determined and the quality of the realized suppression zone is discussed. Design guidelines are formulated to aid the parameter selection process.

Spectral density of first order Piecewise linear systems excited by white noise

The Fokker-Planck (FP) equation is used to develop a general method for finding the spectral density for a class of randomly excited first order systems. This class consists of systems satisfying stochastic differential equations of form ẋ + f(x) = m/Ʃ/j = 1 h_j(x)n_j(t) where f and the h_j are piecewise linear functions (not necessarily continuous), and the n_j are stationary Gaussian white noise. For such systems, it is shown how the Laplace-transformed FP equation can be solved for the transformed transition probability density. By manipulation of the FP equation and its adjoint, a formula is derived for the transformed autocorrelation function in terms of the transformed transition density. From this, the spectral density is readily obtained. The method generalizes that of Caughey and Dienes, J. Appl. Phys., 32.11.

This method is applied to 4 subclasses: (1) m = 1, h₁ = const. (forcing function excitation); (2) m = 1, h₁ = f (parametric excitation); (3) m = 2, h₁ = const., h₂ = f, n₁ and n₂ correlated; (4) the same, uncorrelated. Many special cases, especially in subclass (1), are worked through to obtain explicit formulas for the spectral density, most of which have not been obtained before. Some results are graphed.

Dealing with parametrically excited first order systems leads to two complications. There is some controversy concerning the form of the FP equation involved (see Gray and Caughey, J. Math. Phys., 44.3); and the conditions which apply at irregular points, where the second order coefficient of the FP equation vanishes, are not obvious but require use of the mathematical theory of diffusion processes developed by Feller and others. These points are discussed in the first chapter, relevant results from various sources being summarized and applied. Also discussed is the steady-state density (the limit of the transition density as t → ∞).