Sequential randomized algorithms for sampled convex optimization

Motivated by the complexity of solving convex scenario problems in one-shot, two new algorithms for the sequential solution of sampled convex optimization problems are presented, for full constraint satisfaction and partial constraint satisfaction, respectively. A rigorous analysis of the theoretical properties of the algorithms is provided, and the related sample complexity is derived. Extensive numerical simulations for a non-trivial example testify the goodness of the proposed solution.

[1]  Gaston H. Gonnet,et al.  On the LambertW function , 1996, Adv. Comput. Math..

[2]  Giuseppe Carlo Calafiore,et al.  RACT: Randomized Algorithms Control Toolbox for MATLAB , 2008 .

[3]  Giuseppe Carlo Calafiore,et al.  Uncertain convex programs: randomized solutions and confidence levels , 2005, Math. Program..

[4]  F. Tuteur,et al.  The use of a quadratic performance index to design multivariable control systems , 1966 .

[5]  Roberto Tempo,et al.  Randomized control design through probabilistic validation , 2012, 2012 American Control Conference (ACC).

[6]  Giuseppe Carlo Calafiore,et al.  Research on probabilistic methods for control system design , 2011, Autom..

[7]  Mario Sznaier,et al.  Randomized Algorithms for Analysis and Control of Uncertain Systems with Applications, Second Edition, Roberto Tempo, Giuseppe Calafiore, Fabrizio Dabbene (Eds.). Springer-Verlag, London (2013), 357, ISBN: 978-1-4471-4609-4 , 2014, Autom..

[8]  Fabrizio Dabbene,et al.  A Randomized Cutting Plane Method with Probabilistic Geometric Convergence , 2010, SIAM J. Optim..

[9]  Minyue Fu,et al.  Worst-case properties of the uniform distribution and randomized algorithms for robustness analysis , 1998, Math. Control. Signals Syst..

[10]  Roberto Tempo,et al.  On the Sample Complexity of Probabilistic Analysis and Design Methods , 2010 .

[11]  Yasuaki Oishi,et al.  Polynomial-time algorithms for probabilistic solutions of parameter-dependent linear matrix inequalities , 2007, Autom..

[12]  Marco C. Campi,et al.  The Exact Feasibility of Randomized Solutions of Uncertain Convex Programs , 2008, SIAM J. Optim..

[13]  B. Anderson,et al.  Optimal control: linear quadratic methods , 1990 .

[14]  Eduardo F. Camacho,et al.  Randomized Strategies for Probabilistic Solutions of Uncertain Feasibility and Optimization Problems , 2009, IEEE Transactions on Automatic Control.

[15]  Giuseppe Carlo Calafiore,et al.  Random Convex Programs , 2010, SIAM J. Optim..

[16]  Giuseppe Carlo Calafiore,et al.  The scenario approach to robust control design , 2006, IEEE Transactions on Automatic Control.

[17]  E. Yaz.  Linear Matrix Inequalities In System And Control Theory , 1998, Proceedings of the IEEE.

[18]  Giuseppe Carlo Calafiore,et al.  A probabilistic analytic center cutting plane method for feasibility of uncertain LMIs , 2007, Autom..