Theory of functional connections applied to quadratic and nonlinear programming under equality constraints

TL;DR

This paper presents a novel method using the Theory of Functional Connections to efficiently solve quadratic and nonlinear programming problems with linear equality constraints, avoiding traditional Lagrange multipliers.

Contribution

It introduces a new approach that provides closed-form solutions for quadratic problems and combines Newton's method with elimination for nonlinear problems, enhancing computational efficiency.

Findings

Closed-form solutions for quadratic programming problems.

Effective combination of Newton's method with elimination scheme.

Numerical example demonstrating convergence and efficiency.

Abstract

This paper introduces an efficient approach to solve quadratic and nonlinear programming problems subject to linear equality constraints via the Theory of Functional Connections. This is done without using the traditional Lagrange multiplier technique. More specifically, two distinct expressions (fully satisfying the equality constraints) are provided, to first solve the constrained quadratic programming problem as an unconstrained one for closed-form solution. Such expressions are derived via using an optimization variable vector, which is called the free vector $\boldsymbol{g}$ by the Theory of Functional Connections. In the spirit of this Theory, for the equality constrained nonlinear programming problem, its solution is obtained by the Newton's method combining with elimination scheme in optimization. Convergence analysis is supported by a numerical example for the proposed approach.