Exploiting Sign Symmetries in Minimizing Sums of Rational Functions
References
- [1] (2005) A numerical evaluation of several stochastic algorithms on selected continuous global optimization test problems. J. Global Optim. 31:635–672.Crossref, Google Scholar
- [2] (2000) The Mosek interior point optimizer for linear programming: An implementation of the homogeneous algorithm. Frenk H, Roos K, Terlaky T, Zhang S, eds. High Performance Optimization, Applied Optimization, vol. 33 (Springer, Boston), 197–232.Crossref, Google Scholar
- [3] (2022) Effective representations in real algebraic geometry and polynomial optimization. PhD thesis, Inria d’Université Côte d’Azur, Valbonne, France.Google Scholar
- [4] (2023) On the effective Putinar’s Positivstellensatz and moment approximation. Math. Programming 200(1):71–103.Crossref, Google Scholar
- [5] (1998) Real Algebraic Geometry, A Series of Modern Surveys in Mathematics, vol. 36 (Springer, Berlin, Heidelberg).Crossref, Google Scholar
- [6] (2016) Minimizing the sum of many rational functions. Math. Programming Comput. 8(1):83–111.Crossref, Google Scholar
- [7] (2005) Sparse Fisher discriminant analysis for computer aided detection. Kargupta H, Kamath C, Srivastava J, Goodman A, eds. Proc. 2005 SIAM Internat. Conf. Data Mining (SDM) (Society for Industrial and Applied Mathematics, Philadelphia), 476–480.Google Scholar
- [8] (2007) On sparse Fisher discriminant method for microarray data analysis. Bioinformation 2(5):230–234.Crossref, Google Scholar
- [9] (2014) Minimizing rational functions by exact Jacobian SDP relaxation applicable to finite singularities. J. Global Optim. 58(2):261–284.Crossref, Google Scholar
- [10] (2013) Verifying global minima for L2 minimization problems in multiple view geometry. Internat. J. Comput. Vision 101:288–304.Crossref, Google Scholar
- [11] (2006) Global optimization of rational functions: A semidefinite programming approach. Math. Programming 106(1):93–109.Crossref, Google Scholar
- [12] (2008) Practical global optimization for multiview geometry. Internat. J. Comput. Vision 79:271–284.Crossref, Google Scholar
- [13] (2018) Convergence rates of moment-sum-of-squares hierarchies for volume approximation of semialgebraic sets. Optim. Lett. 12(3):435–442.Crossref, Google Scholar
- [14] (2017) Convergence rates of moment-sum-of-squares hierarchies for optimal control problems. Systems Control Lett. 100:1–5.Crossref, Google Scholar
- [15] (2021) Minimizing rational functions: A hierarchy of approximations via pushforward measures. SIAM J. Optim. 31(3):2285–2306.Crossref, Google Scholar
- [16] (2009) Pre- and post-processing sum-of-squares programs in practice. IEEE Trans. Automatic Control 54(5):1007–1011.Crossref, Google Scholar
- [17] (2021) TSSOS: A Julia library to exploit sparsity for large-scale polynomial optimization. Effective Methods Algebraic Geometry (Tromso, Norway).Google Scholar
- [18] (1967) The arithmetic-geometric inequality. Shisha O, ed. Inequalities (Academic Press, New York), 205–224.Google Scholar
- [19] (2016) Maximizing the sum of a generalized Rayleigh quotient and another Rayleigh quotient on the unit sphere via semidefinite programming. J. Global Optim. 64(2):399–416.Crossref, Google Scholar
- [20] (2013) An exact Jacobian SDP relaxation for polynomial optimization. Math. Programming 137:225–255.Crossref, Google Scholar
- [21] (2007) On the complexity of Putinar’s Positivstellensatz. J. Complexity 23(1):135–150.Crossref, Google Scholar
- [22] (2008) Global minimization of rational functions and the nearest GCDs. J. Global Optim. 40(4):697–718.Crossref, Google Scholar
- [23] (1998) The Symmetric Eigenvalue Problem (Society for Industrial and Applied Mathematics, Philadelphia, PA).Crossref, Google Scholar
- [24] (2006) Towards a joint optimization of scheduling and beamforning for MIMO downlink. 2006 IEEE Ninth Internat. Sympos. Spread Spectrum Tech. Appl. (IEEE, Pisctaway, NJ), 493–497.Google Scholar
- [25] (1993) Positive polynomials on compact semi-algebraic sets. Indiana Univ. Math. J. 42(3):969–984.Crossref, Google Scholar
- [26] (2000) Some concrete aspects of Hilbert’s 17th problem. Contemporary Mathematics, vol. 253 (American Mathematical Society, Providence, RI), 251–272.Google Scholar
- [27] (2003) Fractional programming: The sum-of-ratios case. Optim. Methods Software 18(2):219–229.Crossref, Google Scholar
- [28] (2026) Convergence rates for the moment-SOS hierarchy. Numerical Algebra, Control Optim. 16:105–156.Google Scholar
- [29] (2009) Semi-infinite programming, duality, discretization and optimality conditions. Optimization 58(2):133–161.Crossref, Google Scholar
- [30] (2000) Duality, optimality conditions and perturbation analysis. Wolkowicz H, Saigal R, Vandenberghe L, eds. Handbook of Semidefinite Programming: Theory, Algorithms, and Applications (Kluwer Academic Publisher, Boston), 67–110.Crossref, Google Scholar
- [31] (2022) Asymptotic analysis of semidefinite bounds for polynomial optimization and independent sets in geometric hypergraphs. PhD thesis, Tilburg University, Tilburg, Netherlands.Google Scholar
- [32] (1963) Theory of Approximation of Functions of a Real Variable, International Series of Monographs on Pure and Applied Mathematics (Pergamon Press, Oxford, UK).Google Scholar
- [33] (2005) Strong duality conditions in semidefinite programming. J. Electr. Engrg. 56(1):1–5.Google Scholar
- [34] (2021) TSSOS: A moment-SOS hierarchy that exploits term sparsity. SIAM J. Optim. 31(1):30–58.Crossref, Google Scholar
- [35] (2018) An efficient global optimization algorithm for maximizing the sum of two generalized Rayleigh quotients. Comput. Appl. Math. 37(4):4412–4422.Crossref, Google Scholar
- [36] (2022) CS-TSSOS: Correlative and term sparsity for large-scale polynomial optimization. ACM Trans. Math. Software 48(4):1–26.Crossref, Google Scholar
- [37] (2008) Solving the sum-of-ratios problem by a stochastic search algorithm. J. Global Optim. 42(1):91–109.Crossref, Google Scholar
- [38] (2009) Sparse linear discriminant analysis for simultaneous testing for the significance of a gene set/pathway and gene selection. Bioinformatics 25(9):1145–1151.Crossref, Google Scholar
- [39] (2013) On optimizing the sum of the Rayleigh quotient and the generalized Rayleigh quotient on the unit sphere. Comput. Optim. Appl. 54(1):111–139.Crossref, Google Scholar

