Exact Optimization: Part I

Li Gang Lin, Yew Wen Liang

Research output: Contribution to journalArticlepeer-review


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.

Original languageEnglish
Pages (from-to)169-205
Number of pages37
JournalTaiwanese Journal of Mathematics
Issue number1
StatePublished - Feb 2023


  • and phrases. nonlinear/quadratic programming
  • convex quadratic function/equation
  • matrix algebra
  • parametric optimization


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