  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.
