Linear Bounding Functions Research Articles

Piecewise linear bounding functions in univariate global optimization

Piecewise linear bounding functions in univariate global optimization

A Framework for Globally Convergent Algorithms Using Gradient Bounding Functions

This paper presents a novel framework for developing globally convergent algorithms without evaluating the value of a given function. One way to implement such a framework for a twice continuously differentiable function is to apply linear bounding functions (LBFs) to its gradient function. The algorithm thus obtained can get a better point in each iteration without using a line search. Under certain conditions, it can achieve at least superlinear convergent rate 1.618 without calculating explicitly the Hessian. Furthermore, the strategy of switching from the negative gradient direction to the Newton-alike direction is derived in a natural manner and is computationally effective. Numerical examples are used to show how the algorithm works.

A Multivariate Global Optimization Using Linear Bounding Functions

Recently linear bounding functions (LBFs) were proposed and used to find e-global minima. This paper presents an LBF-based algorithm for multivariate global optimization problems. The algorithm consists of three phases. In the global phase, big subregions not containing a solution are quickly eliminated and those which possibly contain the solution are detected. An efficient scheme for the local phase is developed using our previous local minimization algorithm, which is globally convergent with superlinear/quadratic rate and does not require evaluation of gradients and Hessian matrices. To ensure that the found minimizers are indeed the global solutions or save computation effort, a third phase called the verification phase is often needed. Under adequate conditions the algorithm finds the e-global solution(s) within finite steps. Numerical testing results illustrate how the algorithm works, and demonstrate its potential and feasibility.

Conversion estimates for gas-solid reactions: Kinetics of arbitrary power-law type

The special case of power-law type kinetics which is first order in the gas concentration and arbitrary order in the solid concentration has been widely analysed by many research workers using the cumulative gas concentration method. That work, with the pseudo-steady-state assumption applied, provides bounds on this cumulative gas concentration and these are used to derive bounds on the solid concentration. This paper shows that these bounds for the cumulative gas concentration are valid for arbitrary power law kinetics. By providing suitable upper and lower linear bounding functions on the gas concentration in the kinetic term and utilising the cumulative gas concentration method, bounds are provided for the solid concentration when the reaction rate takes the power law form of arbitrary order in both the gas and solid concentrations.