  Nonlinear programming is explicitly analyzed via a novel perspective/method
and from a bottom-up manner. The philosophy is based on the recent findings on
convex quadratic equation (CQE), which help clarify a geometric interpretation
that relates CQE to convex quadratic function (CQF). More specifically,
regarding the solvability of CQE, its necessary and sufficient condition as
well as a unified parameterization of all the solutions has recently been
analytically formulated. Moving forward, the understanding of CQE is utilized
to describe the geometric structure of CQF, and the CQE-CQF relation. All these
results are shown closely related to a basis in the optimization literature,
namely quadratic programming (QP). For the first time from this viewpoint, the
QPs subject to equality, inequality, equality-and-inequality, and extended
constraints can be algebraically solved in derivative-free closed formulae,
respectively. All the results are derived without knowing a feasible point, a
priori and any time during the process.
