Exact Optimization Part I

• Published in: Taiwanese journal of mathematics Vol. 27; no. 1; pp. 169 - 205
• Main Authors: Lin, Li-Gang, Liang, Yew-Wen
• Format: Journal Article
• Language: English
• Published: Mathematical Society of the Republic of China 01.02.2023
• ISSN: 1027-5487; 2224-6851
• DOI: 10.11650/tjm/220907

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.