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.
Funding Statement
This work was supported in part by the Max Planck Society, Germany; Ministry of Science and Technology, Taiwan; and Delta Electronics, Inc.
Acknowledgments
The authors acknowledge Julian Mellor and Scott Read for their assistance in the native use of this language.
Citation
Li-Gang Lin. Yew-Wen Liang. "Exact Optimization: Part I." Taiwanese J. Math. 27 (1) 169 - 205, February, 2023. https://doi.org/10.11650/tjm/220907
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