Escalarização de Tchebychev ao longo de raios para construção de frentes de pareto de problemas de otimização com dois objetivos

This dissertation is about Multiobjective Optimization. We present the basic concepts, the optimality conditions and three scalarization techniques for constructing the solutions of optimization problems with two objectives. Two of these techniques, namely the Weighting Sum and the Classic Tchebychev Scalarization are well known by the specialists. The third, the Tchebychev Scalarization Along Rays, which is a modification of the Classic Tchebychev Scalarization, is broadly descussed. We present the main features of those scalarizations, their weaknesses and compare them through numerical problems.Este trabalho é sobre Otimização Multi-objetivo. Apresentamos os conceitos básicos, as condições de otimalidade e três técnicas de escalarização para a construção de soluções de problemas de otimização com dois objetivos. Duas dessas técnicas, o Método das Somas Ponderadas e a Escalarização Clássica de Tchebychev são de amplo conhecimento dos estudiosos da área. A terceira, a Escalarização de Tchebychev ao Longo de Raios, que se trata de uma modificação da Escalarização Clássica de Tchebychev, será discutida em detalhes. Apresentamos as principais características dessas escalarizações, suas limitações e fazemos comparações com a utilização de exemplos numéricosCAPES - Coordenação de Aperfeiçoamento de Pessoal de Nível Superio

An inexact modified subgradient algorithm for nonconvex optimization

Nonconvex optimization, Nonsmooth optimization, Sharp augmented Lagrangian, Modified subgradient method, Inexact minimization, Bang-bang control,

On asymptotic Lagrangian duality for nonsmooth optimization

Zero duality gap for nonconvex optimization problems requires the use of a generalized Lagrangian function in the definition of the dual problem. We analyze the situation in which the original problem is associated with a sequence of Lagrangian functions, which in turn defines a sequence of dual problems. Under a set of basic assumptions, we prove that the generated sequence of optimal dual values converges to the optimal primal value, and call the latter situation strong duality for the sequence of Lagrangian functions. As an application of our theory, we construct two sequences of augmented Lagrangians for general equality constrained optimization problems in finite dimensions which exhibit strong duality in this new sense. References R. S. Burachik, W. P. Freire and C. Y. Kaya. Interior epigraph directions method for nonsmooth and nonconvex optimization via generalized augmented Lagrangian duality. Journal of Global Optimization, 60(3):501–529, 2014. doi:10.1007/s10898-013-0108-4 R. S. Burachik, R. N. Gasimov, N. A. Ismayilova and C. Y. Kaya. On a modified subgradient algorithm for dual problems via sharp augmented Lagrangian. Journal of Global Optimization, 34(1):55–78, 2006. doi:10.1007/s10898-005-3270-5 R. S. Burachik, A. N. Iusem and J. G. Melo. The exact penalty map for nonsmooth and nonconvex optimization. Optimization, 64(4):717–738, 2015. doi:10.1080/02331934.2013.830117 R. S. Burachik, A. N. Iusem and J. G. Melo. An inexact modified subgradient algorithm for primal-dual problems via augmented Lagrangians. Journal of Optimization Theory and Applications, 157(1):108–131, 2013. doi:10.1007/s10957-012-0158-7 R. S. Burachik, A. N. Iusem and J. G. Melo. A primal dual modified subgradient algorithm with sharp Lagrangian. Journal of Global Optimization, 46(3): 347–361, 2010. doi:10.1007/s10898-009-9429-8 R. S. Burachik, A. N. Iusem and J. G. Melo. Duality and exact penalization for general augmented Lagrangians, Journal of Optimization Theory and Applications, 147(1):125–140, 2010. doi:10.1007/s10957-010-9711-4 R. S. Burachik and C. Y. Kaya. An augmented penalty function method with penalty parameter updates for nonconvex optimization, Nonlinear Analysis: Theory, Methods and Applications, 75(3):1158–1167, 2012. doi:10.1016/j.na.2011.03.013 R. S. Burachik and C. Y. Kaya. An update rule and a convergence result for a penalty function method. Journal of Industrial Management and Optimization, 3(2): 381–398, 2007. doi:10.3934/jimo.2007.3.381 R. S. Burachik, C. Y. Kaya and M. Mammadov. An inexact modified subgradient algorithm for nonconvex optimization. Computational Optimization and Applications, 45(1):1–24, 2010. doi:10.1007/s10589-008-9168-7 R. S. Burachik and X. Q. Yang, Asymptotic strong duality, Numerical Algebra, Control and Optimization (NACO), 1(3):539–548, 2011. doi:10.3934/naco.2011.1.539 R. S. Burachik and A. M. Rubinov. Abstract convexity and augmented Lagrangians. SIAM Journal on Optimization, 18(2):413–436, 2007. doi:10.1137/050647621 R. S. Burachik and A. M. Rubinov, On the absence of duality gap for Lagrange-type functions, Journal of Industrial and Management Optimization, 1(1):33–38, 2005. doi:10.3934/jimo.2005.1.33 R. N. Gasimov. Augmented Lagrangian duality and nondifferentiable optimization methods in nonconvex programming. Journal of Global Optimization, 24(2):187–203, 2002. doi:10.1023/A:1020261001771 R. N. Gasimov and A. M. Rubinov. On augmented Lagrangians for optimization problems with a single constraint. Journal of Global Optimization, 28(2):153–173, 2004. doi:10.1023/B:JOGO.0000015309.88480.2b O. Guler. Foundations of Optimization, Graduate texts in Mathematics 258, Springer, Berlin, 2010. doi:10.1007/978-0-387-68407-9 G. Li. Global error bounds for piecewise convex polynomials. Mathematical Programming, 137(1):37–64, 2013. doi:10.1007/s10107-011-0481-z X. X. Huang, K. L. Teo and X. Q. Yang, Approximate augmented Lagrangian functions and nonlinear semidefinite programs, Acta Mathematica Sinica, English Series, 22:1283–1296, 2006. doi:10.1007/s10114-005-0702-6 R. T. Rockafellar and R. J.-B. Wets, Variational Analysis, Springer, Berlin, 1998. doi:10.1007/978-3-642-02431-3 A. M. Rubinov, X. X. Huang and X. Q. Yang, The zero duality gap property and lower semicontinuity of the perturbation function, Mathematics of Operations Research, 27:775–791, 2002. doi:10.1287/moor.27.4.775.295 C. Y. Wang, X. Q. Yang and X. M. Yang, A unified nonlinear Lagrangian approach to duality and optimal paths, Journal of Optimization Theory and Applications 135:85–100, 2007. doi:10.1007/s10957-007-9225-