Optimality Conditions at Infinity in Semialgebraic Vector Optimization

References

• [1] Bao TQ, Mordukhovich BS (2010) Relative Pareto minimizers for multiobjective problems: Existence and optimality conditions. Math. Programming 122(2):301–347.Crossref, Google Scholar
• [2] Bochnak J, Coste M, Roy MF (1998) Real Algebraic Geometry, Ergebnisse der Mathematik und ihrer Grenzgebiete/A Series of Modern Surveys in Mathematics, vol. 36 (Springer, Berlin, Heidelberg).Crossref, Google Scholar
• [3] Drusvyatskiy D, Ioffe AD (2015) Quadratic growth and critical point stability of semi-algebraic functions. Math. Programming 153(2):635–653.Crossref, Google Scholar
• [4] Ehrgott M (2005) Multicriteria Optimization, 2nd ed. (Springer, Berlin, Heidelberg).Google Scholar
• [5] Fernando JF, Gamboa JM, Ueno C (2016) The open quadrant problem: A topological proof. A Mathematical Tribute to Professor José María Montesinos Amilibia (Dep. Geom. Topol. Fac. Cien. Mat. Universidad Complutense de Madrid, Madrid), 337–350.Google Scholar
• [6] Flores-Bazán F (2002) Ideal, weakly efficient solutions for vector optimization problems. Math. Programming 93(3):453–475.Crossref, Google Scholar
• [7] Gutiérrez C, Jiménez B, Novo V (2006) A unified approach and optimality conditions for approximate solutions of vector optimization problems. SIAM J. Optim. 17(3):688–710.Crossref, Google Scholar
• [8] Ha TXD (2006) Variants of the Ekeland variational principle for a set-valued map involving the Clarke normal cone. J. Math. Anal. Appl. 316(1):346–356.Crossref, Google Scholar
• [9] Hà HV, Phạm TS (2008) Global optimization of polynomials using the truncated tangency variety and sums of squares. SIAM J. Optim. 19(2):941–951.Crossref, Google Scholar
• [10] Hà HV, Phạm TS (2009) Solving polynomial optimization problems via the truncated tangency variety and sums of squares. J. Pure Appl. Algebra 213(11):2167–2176.Crossref, Google Scholar
• [11] Hà HV, Phạm TS (2017) Genericity in Polynomial Optimization, Series in Optimization and Applications, vol. 3 (World Scientific, Singapore).Crossref, Google Scholar
• [12] Jahn J (2004) Vector Optimization: Theory, Applications, and Extensions (Springer, Berlin, Heidelberg).Crossref, Google Scholar
• [13] Jiao LG, Lee JH, Phạm TS (2024) Fermat’s rule at infinity in non-degenerate semi-algebraic optimization. J. Nonlinear Convex Anal. 25(12):3105–3117.Google Scholar
• [14] Khan AA, Tammer C, Zălinescu C (2015) Set-Valued Optimization: An Introduction with Applications (Springer, Berlin, Heidelberg).Crossref, Google Scholar
• [15] Kim DS, Nguyen MT, Phạm TS (2025) Subdifferentials at infinity and applications in optimization. Math. Programming, ePub ahead of print Janaury 4, https://doi.org/10.1007/s10107-024-02187-9.Crossref, Google Scholar
• [16] Kim DS, Phạm TS, Tuyen NV (2019) On the existence of Pareto solutions for polynomial vector optimization problems. Math. Programming 177(1–2):321–341.Crossref, Google Scholar
• [17] Kim DS, Mordukhovich B, Phạm TS, Tuyen NV (2021) Existence of efficient and properly efficient solutions to problems of constrained vector optimization. Math. Programming 190(1–2):259–283.Crossref, Google Scholar
• [18] Kouchnirenko AG (1976) Polyhèdres de Newton et nombre de Milnor. Inventiones Math. 32(1):1–31.Crossref, Google Scholar
• [19] Lasserre JB (2015) An Introduction to Polynomial and Semi-Algebraic Optimization (Cambridge University Press, Cambridge, UK).Crossref, Google Scholar
• [20] Löhne A (2011) Vector Optimization with Infimum and Supremum (Springer, Berlin, Heidelberg).Crossref, Google Scholar
• [21] Luc DT (1989) Theory of Vector Optimization, Lecture Notes in Economics and Mathematical Systems, vol. 319 (Springer, Berlin, Heidelberg).Crossref, Google Scholar
• [22] Mordukhovich BS (2006) Variational Analysis and Generalized Differentiation, I: Basic Theory; II: Applications (Springer, Berlin, Heidelberg).Google Scholar
• [23] Mordukhovich BS (2018) Variational Analysis and Applications (Springer, New York).Crossref, Google Scholar
• [24] Némethi A, Zaharia A (1990) On the bifurcation set of a polynomial function and Newton boundary. Publ. Res. Inst. Math. Sci. 26(4):681–689.Crossref, Google Scholar
• [25] Nguyen MT, Phạm TS (2024) Clarke’s tangent cones, subgradients, optimality conditions and the Lipschitzness at infinity. SIAM J. Optim. 34(2):1732–1754.Crossref, Google Scholar
• [26] Nie J, Yang Z (2024) The multi-objective polynomial optimization. Math. Oper. Res. 49(4):2723–2748.Link, Google Scholar
• [27] Nie J, Demmel J, Sturmfels B (2006) Minimizing polynomials via sum of squares over the gradient ideal. Math. Programming 106(3):587–606.Crossref, Google Scholar
• [28] Phạm TS (2019) Optimality conditions for minimizers at infinity in polynomial programming. Math. Oper. Res. 44(4):1381–1395.Link, Google Scholar
• [29] Phạm TS (2020) Local minimizers of semi-algebraic functions from the viewpoint of tangencies. SIAM J. Optim. 30(3):1777–1794.Crossref, Google Scholar
• [30] Phạm TS (2023) Tangencies and polynomial optimization. Math. Programming 199(1–2):1239–1272.Crossref, Google Scholar
• [31] Rockafellar RT, Wets R (1998) Variational Analysis, Grundlehren Der Mathematischen Wissenschaften, vol. 317 (Springer, Berlin, Heidelberg).Crossref, Google Scholar