Pozdyayev, V. V. Atomic optimization. I: Search space transformation and one-dimensional problems. (English. Russian original) Zbl 1284.49038 Autom. Remote Control 74, No. 12, 2069-2092 (2013); translation from Avtom. Telemekh. 2013, No. 12, 39-80 (2013). Summary: The paper studies optimization problems with polynomial objective functions and constraints in the form of inequalities. We propose a transformation of the solution method based on the theory of moments. This transformation allows to construct equivalent solution algorithms in the augmented original search space instead of the space of moments. One-dimensional optimization problems are explored in detail. Cited in 1 ReviewCited in 3 Documents MSC: 49M30 Other numerical methods in calculus of variations (MSC2010) 44A60 Moment problems Keywords:optimization problems; polynomial objective functions; inequality constraints; search space transformation; theory of moments; solution algorithms Software:GloptiPoly; PENBMI; PENNON PDFBibTeX XMLCite \textit{V. V. Pozdyayev}, Autom. Remote Control 74, No. 12, 2069--2092 (2013; Zbl 1284.49038); translation from Avtom. Telemekh. 2013, No. 12, 39--80 (2013) Full Text: DOI References: [1] Apkarian, P. and Tuan, H.D., Parameterized LMIs in Control Theory, SIAM J. Control Optimiz., 2000, vol. 38, no. 4, pp. 1241–1264. · Zbl 0960.93012 [2] Blondel, V. and Tsitsiklis, J., NP-hardness of Some Linear Control Design Problems, SIAM J. Control Optimiz., 1997, vol. 35, no. 6, pp. 2118–2127. · Zbl 0892.93050 [3] Boyd, S., Ghaoui, L.E., Feron, E., and Balakrishnan, V., Linear Matrix Inequalities in System and Control Theory, Philadelphia: SIAM, 1994. · Zbl 0816.93004 [4] Boyd, S. and Vandenberghe, L., Convex Optimization, Cambridge: Cambridge Univ. Press, 2004. · Zbl 1058.90049 [5] Curto, R.E. and Fialkow, L.A., The Truncated Complex K-moment Problem, Trans. AMS, 2000, vol. 352, no. 6, pp. 2825–2856. · Zbl 0955.47011 [6] Delibaşı, A. and Henrion, D., Hermite Matrix in Lagrange Basis for Scaling Static Output Feedback Polynomial Matrix Inequalities, Int. J. Control, 2010, vol. 83, no. 12, pp. 2494–2505. · Zbl 1208.90174 [7] Henrion, D. and Lasserre, J.-B., Detecting Global Optimality and Extracting Solutions in GloptiPoly, in Positive Polynomials in Control, Berlin: Springer-Verlag, 2005, pp. 1–18. [8] Henrion, D. and Lasserre, J.-B., Convergent Relaxations of Polynomial Matrix Inequalities and Static Output Feedback, IEEE Trans. Autom. Control, 2006, vol. 51, no. 2, pp. 192–202. · Zbl 1366.93180 [9] Henrion, D., Lasserre, J.-B., and Löfberg, J., GloptiPoly 3: Moments, Optimization and Semidefinite Programming, Optimiz. Methods Software, 2009, vol. 24, no. 4–5, pp. 761–779. · Zbl 1178.90277 [10] Henrion, D., Löfberg, J., Kočvara, M., and Stingl, M., Solving Polynomial Static Output Feedback Problems with PENBMI, Proc. 44th IEEE Conf. on Decision and Control, 2005, pp. 7581–7586 [11] Iwasaki, T. and Skelton, R., Parametrization of All Stabilizing Controllers via Quadratic Lyapunov Functions, J. Optimiz. Theory Appl., 1995, vol. 85, pp. 291–307. · Zbl 0826.93058 [12] Kočvara, M. and Stingl, M., PENBMI, http://www.penopt.com (cited November 1, 2011). [13] Kočvara, M. and Stingl, M., PENNON: A Code for Convex Nonlinear and Semidefinite Programming, Optimiz. Methods Software, 2003, vol. 18, no. 3, pp. 317–333. · Zbl 1037.90003 [14] Lasserre, J.-B., Global Optimization with Polynomials and the Problem of Moments, SIAM J. Optimiz., 2001, vol. 11, no. 3, pp. 796–817. · Zbl 1010.90061 [15] Petersen, K.B. and Pedersen, M.S., The Matrix Cookbook, http://citeseer.ist.psu.edu/viewdoc/summary?doi=10.1.1.113.6244 (cited November 1, 2011). This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.