Skip to main content Accessibility help
×
Hostname: page-component-76fb5796d-skm99 Total loading time: 0 Render date: 2024-04-28T09:33:54.297Z Has data issue: false hasContentIssue false

References

Published online by Cambridge University Press:  05 February 2015

Jean Bernard Lasserre
Affiliation:
Centre National de la Recherche Scientifique (CNRS), Toulouse
Get access
Type
Chapter
Information
Publisher: Cambridge University Press
Print publication year: 2015

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Acquistapace, F., Andradas, C., and Broglia, F. 2000. The strict Positivstellensatz for global analytical functions and the moment problem for semianalytic sets. Math. Ann., 316, 609–616.CrossRefGoogle Scholar
Acquistapace, F., Andradas, C., and Broglia, F. 2002. The Positivstellensatz for definable functions on O-minimal structures. Illinois J. Math., 46, 685–693.Google Scholar
Adams, W. W. and Loustaunau, P. 1994. An Introduction to Gröbner Bases. Providence, RI: American Mathematical Society.CrossRefGoogle Scholar
Ahmadi, A. A. and Parrilo, P. A. 2010. On the equivalence of algebraic conditions for convexity and quasi convexity of polynomials. In: 49th IEEE Conference on Decision and Control, pp. 3343–3348. New York: IEEE.Google Scholar
Ahmadi, A. A., Olshevsky, A., Parrilo, P. A., and Tsitsiklis, J. N. 2013. NP-hardness of deciding convexity of quartic polynomials and related problems. Math. Prog., 137, 453–476.CrossRefGoogle Scholar
Ahmed, S. and Guan, Yongpei. 2005. The inverse optimal value problem. Math. Prog. Ser. A, 102, 91–110.CrossRefGoogle Scholar
Ahuja, R. ~K. and Orlin, J. B. 2001. Inverse optimization. Oper. Res., 49, 771–783.CrossRefGoogle Scholar
Akhiezer, N. I. 1965. The Classical Moment Problem. New York: Hafner.Google Scholar
Ali, M. M., Khompatraporn, C., and Zabinsky, Z. B. 2005. A numerical evaluation of several stochastic algorithms on selected continuous global optimization test problems. J. Global Optim., 31, 635–672.CrossRefGoogle Scholar
Anastassiou, G. A. 1993. Moments in Probability and Approximation Theory. New York: Longman Scientific and Technical.Google Scholar
Andronov, V. G., Belousov, E. G., and Shironin, V. M. 1982. On solvability of the problem of polynomial programming. Izv. Akad. Nauk SSSR, Teckh. Kibern., 4, 194–197.Google Scholar
Androulakis, I. P., Maranas, C. D., and Floudas, C. A. 1995. alphaBB: a global optimization method for general constrained nonconvex problems. J. Global Optim., 7, 337–363.CrossRefGoogle Scholar
Anjos, M. 2001. New Convex Relaxations for the Maximum Cut and VLSI Layout Problems. Ph.D. thesis, University of Waterloo, Ontario, Canada. orion.math. uwaterloo.ca/∼hwolkowi.Google Scholar
Anjos, M. and Lasserre, J. B. (editors) 2012. Handbook on Semidefinite, Conic and Polynomial Optimization. International Series in Operations Research and Management Science, vol. 166. New York: Springer.CrossRefGoogle Scholar
Ash, R. B. 1972. Real Analysis and Probability. San Diego, CA: Academic Press.Google Scholar
Balas, E., Ceria, S., and Cornuéjols, G. 1993. A lift-and-project cutting plane algorithm for mixed 0/1 programs. Math. Prog., 58, 295–324.CrossRefGoogle Scholar
Baldoni, V., Berline, N., De Loera, J. A., Köppe, M., and Vergne, M. 2011. How to integrate a polynomial over a simplex. Math. Comp., 80, 297–325.Google Scholar
Ball, K. 1992. Ellipsoids of maximal volume in convex bodies. Geom. Dedicata, 41, 241–250.CrossRefGoogle Scholar
Ball, K. 2001. Convex geometry and functional analysis. In: Johnson, W. B. and Lindenstrauss, J. (editors), Handbook of the Geometry of Banach Spaces I, pp. 161–194. Amsterdam: North Holland.Google Scholar
Barvinok, A. 2002. A Course in Convexity. Providence, RI: American Mathematical Society.CrossRefGoogle Scholar
Basu, S., Pollack, R., and Roy, M.-F. 2003. Algorithms in Real Algebraic Geometry. Algorithms and Computations in Mathematics, vol. 10. Berlin: Springer.Google Scholar
Bayer, C. and Teichmann, J. 2006. The proof of Tchakaloff's theorem. Proc. Amer. Math. Soc., 134, 3035–3040.CrossRefGoogle Scholar
Becker, E. and Schwartz, N. 1983. Zum Darstellungssatz von Kadison–Dubois. Arch. Math. (Basel), 40, 421–428.CrossRefGoogle Scholar
Belousov, E. G. 1977. Introduction to Convex Analysis and Integer Programming. Moscow: Moscow University Publications.Google Scholar
Belousov, E. G. and Klatte, D. 2002. A Frank–Wolfe type theorem for convex polynomial programs. Comp. Optim. Appl., 22, 37–48.CrossRefGoogle Scholar
Benabbas, S. and Magen, A. 2010. Extending SDP integrality gaps to Sherali–Adams with applications to quadratic programming and MaxCutGain. In: Eisenbrand, F. and Shepherd, F. B. (editors), Integer Programming and Combinatorial Optimization, pp. 299–312. Lecture Notes in Computer Science, vol. 6080. Berlin: Springer.CrossRefGoogle Scholar
Benabbas, S., Georgiou, K., Magen, A., and Tulsiani, M. 2012. SDP gaps from pairwise independence. Theory Comp., 8, 269–289.Google Scholar
Ben-Tal, A. and Nemirovski, A. 2001. Lectures on Modern Convex Optimization. Philadelphia, PA: SIAM.CrossRefGoogle Scholar
Ben-Tal, A., El Ghaoui, L., and Nemirovski, A. 2000. Robustness. In: Wolkowicz, H., Saigal, R., and Vandenberghe, L. (editors), Handbook of Semidefinite Programming: Theory, Algorithms, and Applications. Boston, MA: Kluwer Academic.Google Scholar
Ben-Tal, A., Boyd, S., and Nemirovski, A. 2006. Extending scope of robust optimization: comprehensive robust counterparts of uncertain problems. Math. Prog. Ser. B, 107, 63–89.CrossRefGoogle Scholar
Berg, C. 1987. The multidimensional moment problem and semi-groups. In: Landau, H. J. (editor), Moments in Mathematics, pp. 110–124. Proceedings of Symposia in Applied Mathematics, vol. 37. Providence, RI: American Mathematical Society.CrossRefGoogle Scholar
Bernstein, S. 1921. Sur la représentation des polynômes positifs. Math. Z., 9, 74–109.Google Scholar
Bertsimas, D. 1995. Nonlinear Programming, second edition. Boston, MA: Athena Scientific.Google Scholar
Bertsimas, D. and Sim, M. 2004. The price of robustness. Oper. Res., 52, 35–53.CrossRefGoogle Scholar
Bertsimas, D. and Sim, M. 2006. Tractable approximations to robust conic optimization problems. Math. Prog. Ser. B, 107, 5–36.CrossRefGoogle Scholar
Blekherman, G. 2006. There are significantly more nonnegative polynomials than sums of squares. Isr. J. Math., 153, 355–380.CrossRefGoogle Scholar
Blekherman, G. 2012. Nonnegative polynomials and sums of squares. J. Amer. Math. Soc., 25, 617–635.CrossRefGoogle Scholar
Blekherman, G. 2014. Positive Gorenstein ideals. Proc. Amer. Math. Soc. To appear.CrossRefGoogle Scholar
Blekherman, G. and Lasserre, J. B. 2012. The truncated K-moment problem for closure of open sets. J. Func. Anal., 263, 3604–3616.CrossRefGoogle Scholar
Bochnak, J., Coste, M., and Roy, M.-F. 1998. Real Algebraic Geometry. New York: Springer.CrossRefGoogle Scholar
Bomze, I. M. 2012. Copositive optimization – recent developments and applications. Eur. J. Oper. Res, 216, 509–520.CrossRefGoogle Scholar
Bomze, I. M. and de Klerk, E. 2002. Solving standard quadratic optimization problems via linear, semidefinite and copositive programming. J. Global Optim., 24, 163–185.CrossRefGoogle Scholar
Bomze, I. M., Dürr, M., de Klerk, E., Roos, C., Quist, A. J., and Terlaky, T. 2000. On copositive programming and standard quadratic optimization problems. J. Global Optim., 18, 301–320.CrossRefGoogle Scholar
Bomze, I. M., Schachinger, W., and Uchida, G. 2012. Think co(mpletely) positive! – matrix properties, examples and a clustered bibliography on copositive optimization. J. Global Optim., 52, 423–445.CrossRefGoogle Scholar
Bonnans, J. F. and Shapiro, A. 2000. Perturbation Analysis of Optimization Problems. New York: Springer.CrossRefGoogle Scholar
Buelens, P. F. and Hellinckx, L. J. 1974. Optimal control of linear multivariable systems with quadratic performance index, and the inverse optimal control problem. Int. J. Control, 20, 113–127.CrossRefGoogle Scholar
Bugarin, F., Henrion, D., and Lasserre, J. B. 2011. Minimizing the Sum of Many Rational Functions. Technical Report. LAAS-CNRS, Toulouse, France. arXiv: 1102. 4954.Google Scholar
Burer, S. 2009. On the copositive representation of binary and continuous nonconvex quadratic programs. Math. Prog. Ser. A, 120, 479–495.CrossRefGoogle Scholar
Burer, S. 2012. Copositive programming. In Anjos, M. and Lasserre, J. B. (editors), Handbook on Semidefinite, Conic and Polynomial Optimization, pp. 201–218.Google Scholar
Burer, S.International Series in Operations Research and Management Science, vol. 166. New York: Springer.
Burgdorf, S., Scheiderer, C., and Schweighofer, M. 2012. Pure states, nonnegative polynomials and sums of squares. Comment. Math. Helv., 87, 113–140.Google Scholar
Burton, D. and Toint, P. 1992. On an instance of the inverse shortest path problem. Math. Prog., 63, 45–61.Google Scholar
Cafieri, S., Lee, J., and Liberti, L. 2010. On convex relaxations of quadrilinear terms. J. Global Optim., 47, 661–685.CrossRefGoogle Scholar
Carleman, T. 1926. Les Fonctions Quasi-analytiques. Paris: Gauthier-Villars.Google Scholar
Cassier, G. 1984. Problème des moments sur un compact de ℝn et représentation de polynômes à plusieurs variables. J. Func. Anal., 58, 254–266.CrossRefGoogle Scholar
Castaing, C., de Fitte, P. R., and Valadier, M. 2004. Young Measures on Topological Spaces. With Applications in Control Theory and Probability Theory. Mathematics and its Applications, vol. 571. Dordrecht: Kluwer Academic.Google Scholar
Chlamtac, E. 2007. Approximation algorithms using hierarchies of semidefinite programming relaxations. In: 48th Annual IEEE Symposium on Foundations of Computer Science (FOCS'07), pp. 691–701. New York: IEEE.Google Scholar
Chlamtac, E. and Singh, G. 2008. Improved approximation guarantees through higher levels of SDP hierarchies. In: Approximation, Randomization and Combinatorial Optimization Problems, pp. 49–62. Lecture Notes in Computer Science, vol. 5171. Berlin: Springer.Google Scholar
Chlamtac, E. and Tulsiani, M. 2012. Convex relaxations and integrality gaps. In: Anjos, M. and Lasserre, J. B. (editors), Handbook on Semidefinite, Conic and Polynomial Optimization, pp. 139–170. International Series in Operations Research and Management Science, vol. 166. New York: Springer.Google Scholar
Curto, R. and Fialkow, L. 1991. Recursiveness, positivity, and truncated moment problems. Houston Math. J., 17, 603–635.Google Scholar
Curto, R. and Fialkow, L. 1996. Solution of the Truncated Complex Moment Problem for Flat Data. Memoirs of the American Mathematical Society, vol. 119. Providence, RI: American Mathematical Society.Google Scholar
Curto, R. and Fialkow, L. 1998. Flat Extensions of Positive Moment Matrices: Recursively Generated Relations. Memoirs of the American Mathematical Society, vol. 136. Providence, RI: American Mathematical Society.Google Scholar
Curto, R. and Fialkow, L. 2000. The truncated complex K-moment problem. Trans. Amer. Math. Soc., 352, 2825–2855.CrossRefGoogle Scholar
Curto, R. E. and Fialkow, L. A. 2008. An analogue of the Riesz–Haviland theorem for the truncated moment problem. J. Funct. Anal., 255, 2709–2731.CrossRefGoogle Scholar
d'Aspremont, A. 2008. Smooth optimization with approximate gradient. SIAM J. Optim., 19, 1171–1183.CrossRefGoogle Scholar
de Klerk, E. and Laurent, M. 2011. On the Lasserre hierarchy of semidefinite programming relaxations of convex polynomial optimization problems. SIAM J. Optim., 21, 824–832.CrossRefGoogle Scholar
de Klerk, E. and Pasechnik, D. V. 2002. Approximation of the stability number of a graph via copositive programming. SIAM J. Optim., 12, 875–892.CrossRefGoogle Scholar
de Klerk, E. and Pasechnik, D. V. 2007. A linear programming formulation of the standard quadratic optimization problem. J. Global Optim., 75–84.Google Scholar
de Klerk, E., Laurent, M., and Parrilo, P. A. 2006. A PTAS for the minimization of polynomials of fixed degree over the simplex. Theor. Comp. Sci., 361, 210–225.CrossRefGoogle Scholar
de Klerk, E., Pasechnik, D. V., and Schrijver, A. 2007. Reduction of symmetric semidefinite programs using the regular ⋆-representation. Math. Prog., 109, 613–624.CrossRefGoogle Scholar
Diaconis, P. and Freedman, D. 2006. The Markov moment problem and de Finetti's Theorem Part I.Math. Z., 247, 183–199.Google Scholar
Dunkl, C. F. and Xu, Y. 2001. Orthogonal Polynomials of Several Variables. Encyclopedia of Mathematics and its Applications, vol. 81. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
Dürr, M. 2010. Copositive programming – a survey. In: Diehl, M., Glineur, F., Jarlebring, E., and Michiels, W. (editors), Recent Advances in Optimization and its Applications in Engineering, pp. 3–20. New York: Springer.Google Scholar
Faraut, J. and Korányi, A. 1994. Analysis on Symmetric Cones. Oxford: Clarendon Press.Google Scholar
Fekete, M. 1935. Proof of three propositions of Paley. Bull. Amer. Math. Soc., 41, 138–144.CrossRefGoogle Scholar
Feller, W. 1966. An Introduction to Probability Theory and Its Applications, second edition. New York: John Wiley & Sons.Google Scholar
Fidalgo, C. and Kovacec, A. 2011. Positive semidefinite diagonal minus tail forms are sums of squares. Math. Z., 269, 629–645.CrossRefGoogle Scholar
Floudas, C. A. 2000. Deterministic Global Optimization Theory, Methods and Applications. Dordrecht: Kluwer Academic.CrossRefGoogle Scholar
Floudas, C. A. and Pardalos, P. (editors) 2001. Encyclopedia of Optimization. Dordrecht: Kluwer Academic.CrossRef
Floudas, C. A., Pardalos, P. M., Adjiman, C. S., Esposito, W. R., Gümüs, Z. H., Harding, S. T., Klepeis, J. L., Meyer, C. A., and Schweiger, C. A. 1999. Handbook of Test Problems in Local and Global Optimization. Boston, MA: Kluwer. titan.princeton.edu/TestProblems.CrossRefGoogle Scholar
Freeman, R. A. and Kokotovic, P. V. 1996. Inverse optimality in robust stabilization. SIAM J. Control Optim., 34, 1365–1391.CrossRefGoogle Scholar
Freitag, E. and Busam, R. 2009. Complex Analysis, second edition. Berlin: Springer.Google Scholar
Fujii, K. 2011. Beyond the Gaussian. SIGMA, 7, arXiv: 0912,2135.Google Scholar
Gaddum, J. W. 1958. Linear inequalities and quadratic forms. Pacific J. Math., 8, 411–414.CrossRefGoogle Scholar
Gaterman, K. and Parrilo, P. A. 2004. Symmetry group, semidefinite programs and sums of squares. J. Pure Appl. Alg., 192, 95–128.CrossRefGoogle Scholar
Gelfand, I. M. and Graev, M. I. 1999. GG-Functions and their relations to general hypergeometric functions. Lett. Math. Phys., 50, 1–28.CrossRefGoogle Scholar
Ghasemi, M. and Marshall, M. 2010. Lower bounds for polynomials using geometric programming. Arch. Math. (Basel), 95, 343–353.Google Scholar
Ghasemi, M. and Marshall, M. 2012. Lower bounds for a polynomial in terms of its coefficients. SIAM J. Optim., 22, 460–473.Google Scholar
Goemans, M. X. and Williamson, D. P. 1995. Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming. J. ACM, 42, 1115–1145.CrossRefGoogle Scholar
Golub, G. and Loan, C. F. Van. 1996. Matrix Computations, third edition. New York: John Hopkins University Press.Google Scholar
Gounaris, C. E. and Floudas, C. A. 2008. Tight convex underestimators for C2-continuous problems: II. Multivariate functions. J. Global Optim., 42, 69–89.Google Scholar
Gouveia, J. and Netzer, T. 2011. Positive polynomials and projections of spectrahedra. SIAM J. Optim., 21, 960–976.CrossRefGoogle Scholar
Gouveia, J. and Thomas, R. 2012. Convex hulls of algebraic sets. In: Anjos, M. and Lasserre, J. B. (editors), Handbook on Semidefinite, Conic and Polynomial Optimization, pp. 113–138. International Series in Operations Research and Management Science, vol. 166. New York: Springer.Google Scholar
Gouveia, J., Parrilo, P. A., and Thomas, R. 2010. Theta bodies for polynomial ideals. SIAM J. Optim., 20, 2097–2118.CrossRefGoogle Scholar
Gouveia, J., Laurent, M., Parrilo, P. A., and Thomas, R. 2012. A new hierarchy of semidefinite programming relaxations for cycles in binary matroids and cuts in graphs. Math. Prog. Ser. A. To appear.Google Scholar
Güler, O. 2010. Foundations of Optimization. New York: Springer.CrossRefGoogle Scholar
Guzman, Y. A., Hasan, M. M. F., and Floudas, C. A. 2014. Computational comparison of convex underestimators for use in a branch-and-bound global optimization framework. In: Rassias, T. M., Floudas, C. A., and Butenko, S. (editors), Optimization in Science and Engineering: In Honor of the 60th Birthday of Panos Pardalos. New York: Springer.Google Scholar
Handelman, D. 1988. Representing polynomials by positive linear functions on compact convex polyhedra. Pacific J. Math., 132, 35–62.CrossRefGoogle Scholar
Hanzon, B. and Jibetean, D. 2003. Global minimization of a multivariate polynomial using matrix methods. J. Global Optim., 27, 1–23.CrossRefGoogle Scholar
Hausdorff, F. 1915. Summationsmethoden und Momentfolgen I. Soobshch. Kharkov matem. ob-va Ser. 2, 14, 227–228.Google Scholar
Haviland, E. K. 1935. On the momentum problem for distributions in more than one dimension, I. Amer. J. Math., 57, 562–568.CrossRefGoogle Scholar
Haviland, E. K. 1936. On the momentum problem for distributions in more than one dimension, II. Amer.J.Math., 58, 164–168.CrossRefGoogle Scholar
Helton, J. W. and Nie, J. 2009. Sufficient and necessary conditions for semidefinite representability of convex hulls and sets. SIAM J. Optim., 20, 759–791.CrossRefGoogle Scholar
Helton, J. W. and Nie, J. 2010. Semidefinite representation of convex sets. Math. Prog. Ser. A, 122, 21–64.CrossRefGoogle Scholar
Helton, J. W. and Nie, J. 2012. Semidefinite representation of convex sets and convex hulls. In: Anjos, M. and Lasserre, J. B. (editors), Handbook on Semidefinite, Conic and Polynomial Optimization, pp. 77–112. International Series in Operations Research and Management Science, vol. 166. New York: Springer.Google Scholar
Helton, J. W. and Putinar, M. 2007. Positive polynomials in scalar and matrix variables, the spectral theorem and optimization. In: Bakonyi, M., Gheondea, A., and Putinar, M. (editors), Operator Theory, Structured Matrices and Dilations, pp. 229–306. Bucharest: Theta.Google Scholar
Helton, J. W., Lasserre, J. B., and Putinar, M. 2008. Measures with zeros in the inverse of their moment matrix. Ann. Prob., 36, 1453–1471.CrossRefGoogle Scholar
Henk, M. 2012. Löwner–John ellipsoids. Doc. Math., 95–106. extra volume: Optimization stories, 95–106.Google Scholar
Henrion, D. and Lasserre, J. B. 2003. GloptiPoly: global optimization over polynomials with Matlab and SeDuMi. ACM Trans. Math. Software, 29, 165–194.CrossRefGoogle Scholar
Henrion, D. and Lasserre, J. B. 2005. Detecting global optimality and extracting solution in GloptiPoly. In: Henrion, D. and Garulli, A. (editors), Positive Polynomials in Control, pp. 293–310. Lecture Notes in Control and Information Science, vol. 312. Berlin: Springer.CrossRefGoogle Scholar
Henrion, D. and Lasserre, J. B. 2006. Convergent relaxations of polynomial matrix inequalities and static output feedback. IEEE Trans. Autom. Control, 51, 192–202.CrossRefGoogle Scholar
Henrion, D. and Lasserre, J. B. 2012. Inner approximations for polynomial matrix inequalities and robust stability regions. IEEE Trans. Autom. Control, 57, 1456–1467.CrossRefGoogle Scholar
Henrion, D., Lasserre, J. B., and Savorgnan, C. 2009a. Approximate volume and integration for basic semi-algebraic sets. SIAM Review, 51, 722–743.CrossRefGoogle Scholar
Henrion, D., Lasserre, J. B., and Löfberg, J. 2009b. GloptiPoly 3: moments, optimization and semidefinite programming. Optim. Methods Software, 24, 761–779.CrossRefGoogle Scholar
Hernández-Lerma, O. and Lasserre, J. B. 1996. Discrete-Time Markov Control Processes: Basic Optimality Criteria. New York: Springer.CrossRefGoogle Scholar
Heuberger, C. 2004. Inverse combinatorial optimization: a survey on problems, methods and results. J. Comb. Optim., 8, 329–361.CrossRefGoogle Scholar
Hol, C. W. J. and Scherer, C. W. 2005. A sum of squares approach to fixed-order H∞synthesis. In: Henrion, D. and Garulli, A. (editors), Positive Polynomials in Control, pp. 45–71. Lecture Notes in Control and Information Science, vol. 312. Berlin: Springer.CrossRefGoogle Scholar
Hong, H. and Safey El Din, M. 2012. Variant quantifier elimination. J. Symb. Comput., 47, 883–901.CrossRefGoogle Scholar
Huang, S. and Liu, Z. 1999. On the inverse problem of linear programming and its application to minimum weight perfect k-matching. Eur. J. Oper. Res., 112, 421–426.CrossRefGoogle Scholar
Jacobi, T. and Prestel, A. 2001. Distinguished representations of strictly positive polynomials. J. Reine. Angew. Math., 532, 223–235.Google Scholar
Jibetean, D. and de Klerk, E. 2006. Global optimization of rational functions: a semidefinite programming approach. Math. Prog., 106, 93–109.CrossRefGoogle Scholar
Jibetean, D. and Laurent, M. 2005. Semidefinite approximations for global unconstrained polynomial optimization. SIAM J. Optim., 16, 490–514.CrossRefGoogle Scholar
John, F. 1948. Extremum problems with inequalities as subsidiary conditions. In: Friedrichs, K. O., Neugebauer, O. E., and Stoker, J. J. (editors), Studies and Essays Presented to R. Courant on his 60th Birthday, pp. 187–204. New York: Interscience.Google Scholar
Jordan, M. I. 2004. Graphical models. Stat. Sci., 19, 140–155. Special Issue on Bayesian Statistics.CrossRefGoogle Scholar
Karlin, A. R., Mathieu, C., and Nguyen, C. Thach. 2011. Integrality gaps of linear and semi-definite programming relaxations for knapsack. In: Günlük, O. and Woeginger, G. J. (editors), Integer Programming and Combinatorial Optimization, pp. 301–314. Lecture Notes in Computer Science, vol. 6655. Berlin: Springer.Google Scholar
Karush, W. 1939. Minima of Functions of Several Variables with Inequality as Side Constraints. Ph.D. thesis, Department of Mathematics, University of Chicago, Chicago, IL.Google Scholar
Kemperman, J. H. B. 1968. The general moment problem, a geometric approach. Ann. Math. Stat., 39, 93–122.CrossRefGoogle Scholar
Kemperman, J. H. B. 1987. Geometry of the moment problem. In: Landau, H. J. (editors), Moments in Mathematics, pp. 110–124. Proceedings of Symposia in Applied Mathematics, vol. 37. Providence, RI: American Mathematical Society.CrossRefGoogle Scholar
Kim, S., Kojima, M., and Waki, H. 2009. Exploiting sparsity in SDP relaxation for sensor network localization. SIAM J. Optim., 20, 192–215.CrossRefGoogle Scholar
Kojima, M. and Maramatsu, M. 2007. An extension of sums of squares relaxations to polynomial optimization problems over symmetric cones. Math. Prog., 110, 315–336.CrossRefGoogle Scholar
Kojima, M. and Maramatsu, M. 2009. A note on sparse SOS and SDP relaxations for polynomial optimization problems over symmetric cones. Comp. Optim. Appl., 42, 31–41.CrossRefGoogle Scholar
Kojima, M., Kim, S., and Maramatsu, M. 2005. Sparsity in sums of squares of squares of polynomials. Math. Prog., 103, 45–62.CrossRefGoogle Scholar
Krivine, J. L. 1964a. Anneaux préordonnés. J. Anal. Math., 12, 307–326.CrossRefGoogle Scholar
Krivine, J. L. 1964b. Quelques propriétés des préordres dans les anneaux commutatifs unitaires. C. R. Acad. Sci. Paris, 258, 3417–3418.Google Scholar
Kuhlmann, S. and Putinar, M. 2007. Positive polynomials on fibre products. C. R. Acad. Sci. Paris, Ser. 1, 1344, 681–684.Google Scholar
Kuhlmann, S. and Putinar, M. 2009. Positive polynomials on projective limits of real algebraic varieties. Bull. Sci. Math., 133, 92–111.CrossRefGoogle Scholar
Kuhlmann, S., Marshall, M., and Schwartz, N. 2005. Positivity, sums of squares and the multi-dimensional moment problem II. Adv. Geom., 5, 583–606.CrossRefGoogle Scholar
Landau, H. (editor) 1987. Moments in Mathematics. Proceedings of Symposia in Applied Mathematics, vol. 37. Providence, RI: American Mathematical Society.CrossRef
Lasserre, J. B. 2000. Optimisation globale et théorie des moments. C. R. Acad. Sci. Paris, Ser. 1, 331, 929–934.Google Scholar
Lasserre, J. B. 2001. Global optimization with polynomials and the problem of moments. SIAM J. Optim., 11, 796–817.CrossRefGoogle Scholar
Lasserre, J. B. 2002a. An explicit equivalent positive semidefinite program for nonlinear 0-1 programs. SIAM J. Optim., 12, 756–769.CrossRefGoogle Scholar
Lasserre, J. B. 2002b. Polynomials nonnegative on a grid and discrete optimization. Trans. Amer. Math. Soc., 354, 631–649.CrossRefGoogle Scholar
Lasserre, J. B. 2002c. Semidefinite programming vs. LP relaxations for polynomial programming. Math. Oper. Res., 27, 347–360.CrossRefGoogle Scholar
Lasserre, J. B. 2004. Polynomial programming: LP-relaxations also converge. SIAM J. Optim., 15, 383–393.Google Scholar
Lasserre, J. B. 2005. SOS approximations of polynomials nonnegative on a real algebraic set. SIAM J. Optim., 16, 610–628.CrossRefGoogle Scholar
Lasserre, J. B. 2006a. Convergent SDP-relaxations in polynomial optimization with sparsity. SIAM J. Optim., 17, 822–843.CrossRefGoogle Scholar
Lasserre, J. B. 2006b. Robust global optimization with polynomials. Math. Prog. Ser. B, 107, 275–293.CrossRefGoogle Scholar
Lasserre, J. B. 2006c. A sum of squares approximation of nonnegative polynomials. SIAM J. Optim, 16, 751–765.CrossRefGoogle Scholar
Lasserre, J. B. 2007. Sufficient conditions for a real polynomial to a sum of squares. Arch. Math. (Basel), 89, 390–398.CrossRefGoogle Scholar
Lasserre, J. B. 2008. Representation of nonnegative convex polynomials. Arch. Math. (Basel), 91, 126–130.CrossRefGoogle Scholar
Lasserre, J. B. 2009a. Convex sets with semidefinite representation. Math. Prog. Ser. A, 120, 457–477.CrossRefGoogle Scholar
Lasserre, J. B. 2009b. Convexity in semi-algebraic geometry and polynomial optimization. SIAM J. Optim., 19, 1995–2014.CrossRefGoogle Scholar
Lasserre, J. B. 2009c. Moments, Positive Polynomials and Their Applications. London: Imperial College Press.CrossRefGoogle Scholar
Lasserre, J. B. 2010a. Certificates of convexity for basic semi-algebraic sets. Appl. Math. Lett., 23, 912–916.CrossRefGoogle Scholar
Lasserre, J. B. 2010b. A “joint+marginal” approach to parametric polynomial optimization. SIAM J. Optim., 20, 1995–2022.CrossRefGoogle Scholar
Lasserre, J. B. 2011. A new look at non negativity on closed sets and polynomial optimization. SIAM J. Optim., 21, 864–885.CrossRefGoogle Scholar
Lasserre, J. B. 2013a. Borel measures with a density on a compact semi-algebraic set. Arch. Math. (Basel), 101, 361–371.CrossRefGoogle Scholar
Lasserre, J. B. 2013b. A generalization of Löwner–John's ellipsoid problem. Math. Prog. Ser. A. To appear.Google Scholar
Lasserre, J. B. 2013c. Inverse polynomial optimization. Math. Oper. Res., 38, 418–436.CrossRefGoogle Scholar
Lasserre, J. B. 2013d. The K-moment problem with continuous linear functionals. Trans. Amer. Math. Soc., 365, 2489–2504.Google Scholar
Lasserre, J. B. 2013e. A Lagrangian relaxation view of linear and semidefinite hierarchies. SIAM J. Optim., 23, 1742–1756.CrossRefGoogle Scholar
Lasserre, J. B. 2013f. Tractable approximations of sets defined with quantifiers. Math. Prog. Ser. B. To appear.Google Scholar
Lasserre, J. B. 2014a. Level sets and non Gaussian integrals of positively homogeneous functions. Int. Game Theory Rev. To appear.Google Scholar
Lasserre, J. B. 2014b. New approximations for the cone of copositive matrices and its dual. Math. Prog., 144, 265–276.CrossRefGoogle Scholar
Lasserre, J. B. and Netzer, T. 2007. SOS approximations of nonnegative polynomials via simple high degree perturbations. Math. Z., 256, 99–112.CrossRefGoogle Scholar
Lasserre, J. B. and Putinar, M. 2010. Positivity and optimization for semi-algebraic functions. SIAM J. Optim., 20, 3364–3383.CrossRefGoogle Scholar
Lasserre, J. B. and Putinar, M. 2012. Positivity and optimization: beyond polynomials. In: Anjos, M., and Lasserre, J. B. (editors), Handbook on Semidefinite, Conic and Polynomial Optimization, pp. 407–436. International Series in Operations Research and Management Science, vol. 166. New York: Springer.Google Scholar
Lasserre, J. B. and Thanh, T. P. 2011. Convex underestimators of polynomials. In: Proceedings of the 50th IEEE CDC Conference on Decision and Control, December 2011, pp. 7194–7199. New York: IEEE.Google Scholar
Lasserre, J. B. and Thanh, T. P. 2012. A “joint+marginal” heuristic for 0/1 programs. J. Global Optim. 54, 729–744.CrossRefGoogle Scholar
Lasserre, J. B. and Thanh, T. P. 2013. Convex underestimators of polynomials. J. Global Optim., 56, 1–25.CrossRefGoogle Scholar
Laurent, M. 2003. A comparison of the Sherali–Adams, Lovász–Schrijver and Lasserre relaxations for 0-1 programming. Math. Oper. Res., 28, 470–496.CrossRefGoogle Scholar
Laurent, M. 2005. Revisiting two theorems of Curto and Fialkow on moment matrices. Proc. Amer. Math. Soc., 133, 2965–2976.CrossRefGoogle Scholar
Laurent, M. 2007a. Semidefinite representations for finite varieties. Math. Prog. Ser. A, 109, 1–26.CrossRefGoogle Scholar
Laurent, M. 2007b. Strengthened semidefinite programming bounds for codes. Math. Prog. Ser. B, 109, 239–261.CrossRefGoogle Scholar
Laurent, M. 2008. Sums of squares, moment matrices and optimization over polynomials. In: Putinar, M. and Sullivant, S. (editors), Emerging Applications of Algebraic Geometry, pp. 157–270. IMA Volumes in Mathematics and its Applications, vol. 149. New York: Springer.Google Scholar
Löfberg, J. 2004. YALMIP: a toolbox for modeling and optimization in MATLAB. In: Proceedings of the IEEE Symposium on Computer-Aided Control System Design (CACSD), Taipei, Taiwan. New York: IEEE.Google Scholar
Lombardi, H., Perrucci, D., and Roy, M.-F. 2014. An Elementary Recursive Bound for Effective Positivstellensatz and Hilbert 17th Problem. Technical Report IRMAR, Rennes, France. arXiv:1404.2338v1.Google Scholar
Lovász, L. 2003. Semidefinite programs and combinatorial optimization. In: Reed, B. A. and Sales, C. L. (editors), Recent Advances in Algorithms and Combinatorics, pp. 137–194. CMS Books Math./Ouvrages Math. SMC, 11. New York: Springer.Google Scholar
Lovász, L. and Schrijver, A. 1991. Cones of matrices and set-functions and 0-1 optimization. SIAM J. Optim., 1, 166–190.CrossRefGoogle Scholar
Marshall, M. 2006. Representation of non-negative polynomials with finitely many zeros. Ann. Fac. Sci. Toulouse, 15, 599–609.CrossRefGoogle Scholar
Marshall, M. 2008. Positive Polynomials and Sums of Squares. AMS Mathematical Surveys and Monographs, vol. 146. Providence, RI: American Mathematical Society.CrossRefGoogle Scholar
Marshall, M. 2009. Representation of non-negative polynomials, degree bounds and applications to optimization. Canad. J. Math., 61, 205–221.CrossRefGoogle Scholar
Marshall, M. 2010. Polynomials non-negative on a strip. Proc. Amer. Math. Soc., 138, 1559–1567.Google Scholar
Marshall, M. and Netzer, T. 2012. Positivstellensätze for real function algebras. Math. Z., 270, 889–901.CrossRefGoogle Scholar
Morozov, A. and Shakirov, S. 2009a. Introduction to integral discriminants. J. High Energy Phys., 12(12).Google Scholar
Morozov, A. and Shakirov, S. 2009b. New and Old Results in Resultant Theory. Technical Report ITEP, Moscow. Theor. Math. Phys. To appear.Google Scholar
Moylan, P. J. and Anderson, B. D. O. 1973. Nonlinear regulator theory and an inverse optimal control problem. IEEE Trans. Autom Control, 18, 460–465.CrossRefGoogle Scholar
Mulholland, H. P. and Rogers, C. A. 1958. Representation theorems for distribution functions. Proc. London Math. Soc., 8, 177–223.Google Scholar
Nesterov, Y. 2000. Squared functional systems and optimization problems. In: Frenk, H., Roos, K., Terlaky, T., and Zhang, S. (editors), High Performance Optimization, pp. 405–440. New York: Springer.Google Scholar
Netzer, T., Plaumann, D., and Schweighofer, M. 2010. Exposed faces of semidefinitely representable sets. SIAM J. Optim., 20, 1944–1955.CrossRefGoogle Scholar
Nie, J. 2014. Optimality conditions and finite convergence of Lasserre's hierarchy. Math. Prog. Ser. A, 146, 97–121.CrossRefGoogle Scholar
Nie, J. and Schweighofer, M. 2007. On the complexity of Putinars’ Positivstellensatz. J. Complexity, 23, 135–150.CrossRefGoogle Scholar
Nie, J., Demmel, J., and Sturmfels, B. 2006. Minimizing polynomials via sum of squares over the gradient ideal. Math. Prog. Ser. A, 106, 587–606.CrossRefGoogle Scholar
Nussbaum, A. E. 1966. Quasi-analytic vectors. Ark. Mat., 5, 179–191.Google Scholar
Park, J. G. and Lee, K. Y. 1975. An inverse optimal control problem and its application to the choice of performance index for economic stabilization policy. IEEE Trans. Syst. Man Cybern., 5, 64–76.Google Scholar
Parrilo, P. A. 2000. Structured Semidefinite Programs and Semialgebraic Geometry Methods in Robustness and Optimization. Ph.D. thesis, California Institute of Technology, Pasadena, CA.Google Scholar
Parrilo, P. A. 2002. An Explicit Construction of Distinguished Representations of Polynomials Nonnegative over Finite Sets. Technical Report Auto-02, IfA, ETH, Zurich, Switzerland.Google Scholar
Parrilo, P. A. 2003. Semidefinite programming relaxations for semialgebraic problems. Math. Prog., 96, 293–320.CrossRefGoogle Scholar
Pedregal, P. 1999. Optimization, relaxation and Young measures. Bull. Amer. Math. Soc., 36, 27–58.CrossRefGoogle Scholar
Pena, J., Vera, J., and Zuluaga, L. 2007. Computing the stability number of a graph via linear and semidefinite programming. SIAM J. Optim., 87–105.Google Scholar
Pólya, G. 1974. Collected Papers, vol. II. Cambridge, MA: MIT Press.Google Scholar
Pólya, G. and Szegö, G. 1976. Problems and Theorems in Analysis II. Berlin: Springer.CrossRefGoogle Scholar
Powers, V. and Reznick, B. 2000. Polynomials that are positive on an interval. Trans. Amer. Math. Soc., 352, 4677–4692.CrossRefGoogle Scholar
Prajna, S., Papachristodoulou, A., and Parrilo, P. A. 2002. Introducing SOSTOOLS: a general purpose sum of squares programming solver. In: Proceedings of the 41st IEEE Conference on Decision and Control. New York: IEEE.Google Scholar
Prestel, A. and Delzell, C. N. 2001. Positive Polynomials. Berlin: Springer.CrossRefGoogle Scholar
Putinar, M. 1993. Positive polynomials on compact semi-algebraic sets. Indiana Univ. Math. J., 42, 969–984.CrossRefGoogle Scholar
Putinar, M. 2000. A note on Tchakaloff's theorem. Proc. Amer. Math. Soc., 125, 2409–2414.Google Scholar
Reznick, B. 1995. Uniform denominators in Hilbert's seventeenth problem. Math. Z., 220, 75–98.CrossRefGoogle Scholar
Reznick, B. 2000. Some concrete aspects of Hilbert's 17th problem. In: Delzell, C. N. and Madden, J. J. (editors), Real Algebraic Geometry and Ordered Structures. Contemporary Mathematics, vol. 253. Providence, RI: American Mathematical Society.Google Scholar
Richter, H. 1957. Parameterfreie Abschätzung und Realisierung von Erwartungswerten. Bl. Dtsch. Ges. Versicherungsmath., 3, 147–161.Google Scholar
Riener, C., Theobald, T., Jansson, L., and Lasserre, J. B. 2013. Exploiting symmetries in SDP-relaxations for polynomial optimization. Math. Oper. Res., 38, 122–141.CrossRefGoogle Scholar
Rockafellar, R. T. 1970. Convex Analysis. Princeton, NJ: Princeton University Press.CrossRefGoogle Scholar
Rogosinsky, W. W. 1958. Moments of non-negative mass. Proc. R. Soc. London Ser. A, 245, 1–27.Google Scholar
Royden, H. L. 1988. Real Analysis, third edition. New York: Macmillan.Google Scholar
Rugh, W. J. 1971. On an inverse optimal control problem. IEEE Trans. Autom. Control, 16, 87–88.CrossRefGoogle Scholar
Schaefer, A. 2004. Inverse integer programming. Optim. Lett., 3, 483–489.Google Scholar
Scheiderer, C. 2008. Positivity and sums of squares: a guide to some recent results. In: Putinar, M. and Sullivant, S. (editors), Emerging Applications of Algebraic Geometry, pp. 271–324. IMA Volumes in Mathematics and its Applications, vol 149. New York. Springer.Google Scholar
Scherer, C. W. and Hol, C. W. J. 2004. Sum of squares relaxation for polynomial semidefinite programming. Proceedings of the 16th International Symposium on Mathematical Theory of Networks and Systems, Leuven, pp. 1–10.Google Scholar
Schichl, H. and Neumaier, A. 2006. Transposition theorems and qualification-free optimality conditions. SIAM J. Optim., 17, 1035–1055.Google Scholar
Schmid, J. 1998. On the Degree Complexity of Hilbert's 17th Problem and the Real Nullstellensatz. Ph.D. thesis, University of Dortmund. Habilitationsschrift zur Erlangug der Lehrbefignis für das Fach Mathematik an der Universität Dortmund.Google Scholar
Schmüdgen, K. 1991. The K-moment problem for compact semi-algebraic sets. Math. Ann., 289, 203–206.CrossRefGoogle Scholar
Schneider, R. 1994. Convex Bodies: The Brunn–Minkowski Theory. Cambridge: Cambridge University Press.Google Scholar
Schoenebeck, G. 2008. Linear level Lasserre lower bounds for certain k-CSPs. In: 49th Annual IEEE Symposium on Foundations of Computer Science (FOCS'08), pp. 593–602. New York: IEEE.Google Scholar
Schrijver, A. 2005. New codes upper bounds from the Terwilliger algebra and semidefinite pogramming. IEEE Trans. Inf. Theory, 51, 2859–2866.CrossRefGoogle Scholar
Schweighofer, M. 2005. On the complexity of Schmüdgen's Positivstellensatz. J. Complexity, 20, 529–543.Google Scholar
Schweighofer, M. 2006. Global optimization of polynomials using gradient tentacles and sums of squares. SIAM J. Optim., 17, 920–942.CrossRefGoogle Scholar
Shakirov, S. R. 2010. Nonperturbative approach to finite-dimensional non-Gaussian integrals. Theor. Math. Phys., 163, 804–812.CrossRefGoogle Scholar
Sherali, H. D. and Adams, W. P. 1990. A hierarchy of relaxations between the continuous and convex hull representations for zero-one programming problems. SIAM J. Discr. Math., 3, 411–430.CrossRefGoogle Scholar
Sherali, H. D. and Adams, W. P. 1999. A Reformulation-Linearization Technique for Solving Discrete and Continuous Nonconvex Problems. Dordrecht: Kluwer.CrossRefGoogle Scholar
Sherali, H. D., Adams, W. P., and Tuncbilek, C. H. 1992. A global optimization algorithm for polynomial programming problems using a reformulation-linearization technique. J. Global Optim., 2, 101–112.Google Scholar
Shor, N. Z. 1987. Quadratic optimization problems. Tekh. Kibern., 1, 128–139.Google Scholar
Shor, N. Z. 1998. Nondifferentiable Optimization and Polynomial Problems. Dordrecht: Kluwer.CrossRefGoogle Scholar
Simon, B. 1998. The classical moment problem as a self-adjoint finite difference operator. Adv. Math., 137, 82–203.CrossRefGoogle Scholar
Stengle, G. 1974. A Nullstellensatz and a Positivstellensatz in semialgebraic geometry. Math. Ann., 207, 87–97.CrossRefGoogle Scholar
Sturm, J. F. 1999. Using SeDuMi 1.02, a MATLAB toolbox for optimizing over symmetric cones. Optim. Methods Software, 11-12, 625–653.Google Scholar
Tawarmalani, M. and Sahinidis, N. V. 2002. Convex extensions and envelopes of lower semi-continuous functions. Math. Prog., 93, 247–263.CrossRefGoogle Scholar
Tawarmalani, M. and Sahinidis, N. V. 2005. A polyhedral branch-and-cut approach to global optimization. Math. Prog., 103, 225–249.CrossRefGoogle Scholar
Tchakaloff, V. 1957. Formules de cubature mécanique à coefficients non négatifs. Bull. Sci. Math., 81, 123–134.Google Scholar
Tuncel, L. 2000. Potential reduction and primal-dual methods. In: Wolkowicz, H., Saigal, R., and Vandenberghe, L. (editors), Handbook of Semidefinite Programming: Theory, Algorithms, and Applications, pp. 235–265. Boston, MA: Kluwer Academic.Google Scholar
Vallentin, F. 2007. Symmetry in semidefinite programs. Linear Alg. Appl., 430, 360–369.Google Scholar
Vandenberghe, L. and Boyd, S. 1996. Semidefinite programming. SIAM Rev., 38, 49–95.CrossRefGoogle Scholar
Vasilescu, F.-H. 2003. Spectral measures and moment problems. Spectral Theory Appl., 173–215.Google Scholar
Vui, H. H. and Pham, T. S. 2008. Global optimization of polynomials using the truncated tangency variety and sums of squares. SIAM J. Optim., 19, 941–951.CrossRefGoogle Scholar
Waki, S., Kim, S., Kojima, M., and Maramatsu, M. 2006. Sums of squares and semidefinite programming relaxations for polynomial optimization problems with structured sparsity. SIAM J. Optim., 17, 218–242.CrossRefGoogle Scholar
Waki, H., Kim, S., Kojima, M., Muramatsu, M., and Sugimoto, H. 2008. SparsePOP : a sparse semidefinite programming relaxation of polynomial optimization problems. ACM Trans. Math. Software, 35, article 15.Google Scholar
Zhang, J. and Liu, Z. 1996. Calculating some inverse linear programming problems. J. Comp. Appl. Math., 72, 261–273.Google Scholar
Zhang, J. and Liu, Z. 1999. A further study on inverse linear programming problems. J. Comp. Appl. Math., 106, 345–359.CrossRefGoogle Scholar
Zhang, J., Ma, Z., and Yang, C. 1995. A column generation method for inverse shortest path problems. Math. Methods Oper. Res., 41, 347–358.CrossRefGoogle Scholar

Save book to Kindle

To save this book to your Kindle, first ensure coreplatform@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

  • References
  • Jean Bernard Lasserre
  • Book: An Introduction to Polynomial and Semi-Algebraic Optimization
  • Online publication: 05 February 2015
  • Chapter DOI: https://doi.org/10.1017/CBO9781107447226.022
Available formats
×

Save book to Dropbox

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.

  • References
  • Jean Bernard Lasserre
  • Book: An Introduction to Polynomial and Semi-Algebraic Optimization
  • Online publication: 05 February 2015
  • Chapter DOI: https://doi.org/10.1017/CBO9781107447226.022
Available formats
×

Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

  • References
  • Jean Bernard Lasserre
  • Book: An Introduction to Polynomial and Semi-Algebraic Optimization
  • Online publication: 05 February 2015
  • Chapter DOI: https://doi.org/10.1017/CBO9781107447226.022
Available formats
×