# Exact Optimization: Part I

## TL;DR

This paper introduces a novel geometric approach to nonlinear programming, providing explicit algebraic solutions to quadratic programming problems without requiring initial feasible points.

## Contribution

It offers a new perspective based on convex quadratic equations, enabling derivative-free, explicit solutions to various quadratic programming problems.

## Key findings

- Closed-form solutions for QPs with various constraints
- Unified parameterization of solutions to CQE
- Enhanced understanding of CQF and CQE relations

## Abstract

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.