Piecewise Polynomial Interpolation Function Approach for Solving Nonlinear Programming Problems with Disjoint Feasible Regions: Mathematical Proofs

TL;DR

This paper introduces the Piecewise Polynomial Interpolation (PPI) function approach to effectively solve nonlinear programming problems with disjoint feasible regions caused by prohibited operating zones, supported by mathematical proofs.

Contribution

It defines the PPI function and proves its properties, enabling the transformation of problems with disjoint feasible regions into forms suitable for gradient-based optimization methods.

Findings

PPI function accurately models disjoint feasible regions.

Mathematical proofs confirm the properties of PPI functions.

The approach facilitates solving complex nonlinear programming problems.

Abstract

The Piecewise Polynomial Interpolation (PPI) function approach is aimed at solving nonlinear programming problems with disjoint feasible regions. In such problems, disjointedness is generally associated with prohibited operating zones, which correspond to bands of values that a variable is not allowed to assume. An analytical implication of such prohibited operating zones is to make the objective function, as well as its domain, discontinuous. The PPI function approach consists in replacing the constraints associated with prohibited operating zones by an equivalent set of equality and inequality constraints, thereby allowing the application of any efficient gradient-based optimization method for solving the equivalent problem. In this paper, we present the definition of the PPI function and provide the mathematical proofs for its properties.