A one-step worst-case optimal algorithm for bi-objective univariate optimization

A bi-objective optimization problem with univariate non-convex objectives is considered. The objective functions are assumed Lipschitz continuous. An algorithm is developed implementing the concept of one-step worst case optimality. The possibility of extension of the proposed algorithm to the multivariate case is discussed. Some illustrative numerical examples are included where the proposed algorithm is compared with an average case optimal algorithm.

[1]  Lothar Thiele,et al.  Multiobjective Optimization Using Evolutionary Algorithms - A Comparative Case Study , 1998, PPSN.

[2]  Daniel Scholz.  Deterministic Global Optimization: Geometric Branch-and-bound Methods and their Applications , 2011 .

[3]  János D. Pintér,et al.  Global optimization in action , 1995 .

[4]  Yaroslav D. Sergeyev,et al.  Lipschitz gradients for global optimization in a one-point-based partitioning scheme , 2012, J. Comput. Appl. Math..

[5]  Y. D. Sergeyev,et al.  Global Optimization with Non-Convex Constraints - Sequential and Parallel Algorithms (Nonconvex Optimization and its Applications Volume 45) (Nonconvex Optimization and Its Applications) , 2000 .

[6]  Sanjeev Arora,et al.  Computational Complexity: A Modern Approach , 2009 .

[7]  Mikhail Posypkin,et al.  Nonuniform covering method as applied to multicriteria optimization problems with guaranteed accuracy , 2013 .

[8]  Yaroslav D. Sergeyev,et al.  Global Search Based on Efficient Diagonal Partitions and a Set of Lipschitz Constants , 2006, SIAM J. Optim..

[9]  Kaisa Miettinen,et al.  Nonlinear multiobjective optimization , 1998, International series in operations research and management science.

[10]  Antanas Zilinskas,et al.  A statistical model-based algorithm for ‘black-box’ multi-objective optimisation , 2014, Int. J. Syst. Sci..

[11]  A. G. Sukharev.  Optimal strategies of the search for an extremum , 1971 .

[12]  Panos M. Pardalos,et al.  Introduction to Global Optimization , 2000, Introduction to Global Optimization.

[13]  Lothar Thiele,et al.  Multiobjective evolutionary algorithms: a comparative case study and the strength Pareto approach , 1999, IEEE Trans. Evol. Comput..

[14]  Rudolf Mathar,et al.  A class of test functions for global optimization , 1994, J. Glob. Optim..

[15]  Aimo A. Törn,et al.  Global Optimization , 1999, Science.

[16]  Mikhail Posypkin,et al.  A deterministic approach to global box-constrained optimization , 2012, Optimization Letters.

[17]  Roman G. Strongin,et al.  Introduction to Global Optimization Exploiting Space-Filling Curves , 2013 .

[18]  Antanas Zilinskas.  On the worst-case optimal multi-objective global optimization , 2013, Optim. Lett..

[19]  A. G. Sukharev.  A sequentially optimal algorithm for numerical integration , 1979 .

[20]  Julius Zilinskas,et al.  Investigation of selection strategies in branch and bound algorithm with simplicial partitions and combination of Lipschitz bounds , 2010, Optim. Lett..

[21]  Lothar Thiele,et al.  Comparison of Multiobjective Evolutionary Algorithms: Empirical Results , 2000, Evolutionary Computation.

[22]  Kalyanmoy Deb,et al.  Multi-objective optimization using evolutionary algorithms , 2001, Wiley-Interscience series in systems and optimization.

[23]  Peter J. Fleming,et al.  On the Performance Assessment and Comparison of Stochastic Multiobjective Optimizers , 1996, PPSN.

[24]  Marco Laumanns,et al.  Performance assessment of multiobjective optimizers: an analysis and review , 2003, IEEE Trans. Evol. Comput..

[25]  A. G. Sukharev.  Best sequential search strategies for finding an extremum , 1972 .

[26]  Deterministic Global Optimization , 2012 .

[27]  S. A. Piyavskii.  An algorithm for finding the absolute extremum of a function , 1972 .

[28]  Yaroslav D. Sergeyev,et al.  Acceleration of Univariate Global Optimization Algorithms Working with Lipschitz Functions and Lipschitz First Derivatives , 2013, SIAM J. Optim..

[29]  B. Shubert.  A Sequential Method Seeking the Global Maximum of a Function , 1972 .

[30]  Yaroslav D. Sergeyev,et al.  Lipschitz Global Optimization , 2011 .

[31]  Y. Sergeyev,et al.  Univariate geometric Lipschitz global optimization algorithms , 2012 .

[32]  John N. Hooker,et al.  Testing heuristics: We have it all wrong , 1995, J. Heuristics.