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Considering that linear outer approximations sacrifice accuracy when approximating highly nonlinear functions and recognizing the recent advancements in the efficiency and robustness of available methods to solve optimization problems with quadratic objectives and constraints, we contemplate here the construction of quadratic outer approximations of twice-differentiable convex functions for use in deterministic global optimization. To this end, we present a novel cutting-plane algorithm that determines the tightest scaling parameter,\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$\\alpha $$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mi>\u03b1<\/mml:mi>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    , in the second-order Taylor series approximation quadratic underestimator proposed by\u00a0Su et al. [25]. We use a representative set of convex functions extracted from optimization benchmark libraries to showcase\u2013qualitatively and quantitatively\u2013the tightness of the constructed quadratic underestimators and to demonstrate the overall computational efficiency of our algorithm. Furthermore, we extend our construction procedure to generate even tighter quadratic underestimators by allowing overestimation in infeasible polyhedral regions of optimization problems, as informed by the latter\u2019s linear constraints.\n                  <\/jats:p>","DOI":"10.1007\/s10898-025-01565-7","type":"journal-article","created":{"date-parts":[[2025,12,8]],"date-time":"2025-12-08T06:01:42Z","timestamp":1765173702000},"page":"923-952","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Constructing Tight Quadratic Relaxations for Global Optimization: I. 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