When to be Discrete: Analyzing Algorithm Performance on Discretized Continuous Problems

The domain of an optimization problem is seen as one of its most important characteristics. In particular, the distinction between continuous and discrete optimization is rather impactful. Based on this, the optimizing algorithm, analyzing method, and more are specified. However, in practice, no problem is ever truly continuous. Whether this is caused by computing limits or more tangible properties of the problem, most variables have a finite resolution. In this work, we use the notion of the resolution of continuous variables to discretize problems from the continuous domain. We explore how the resolution impacts the performance of continuous optimization algorithms. Through a mapping to integer space, we are able to compare these continuous optimizers to discrete algorithms on the exact same problems. We show that the standard $(\mu_W, \lambda)$-CMA-ES fails when discretization is added to the problem.

[1]  N. Hansen,et al.  Anytime Performance Assessment in Blackbox Optimization Benchmarking , 2022, IEEE Transactions on Evolutionary Computation.

[2]  Anna V. Kononova,et al.  One-shot optimization for vehicle dynamics control systems: towards benchmarking and exploratory landscape analysis , 2022, GECCO Companion.

[3]  Shota Saito,et al.  Benchmarking CMA-ES with margin on the bbob-mixint testbed , 2022, GECCO Companion.

[4]  Thomas Bäck,et al.  Learning the characteristics of engineering optimization problems with applications in automotive crash , 2022, GECCO.

[5]  Shota Saito,et al.  CMA-ES with margin: lower-bounding marginal probability for mixed-integer black-box optimization , 2022, GECCO.

[6]  Thomas Bäck,et al.  IOHanalyzer: Detailed Performance Analyses for Iterative Optimization Heuristics , 2020, ACM Trans. Evol. Learn. Optim..

[7]  Thomas Bäck,et al.  IOHexperimenter: Benchmarking Platform for Iterative Optimization Heuristics , 2021, Evolutionary computation.

[8]  Thomas Bartz-Beielstein,et al.  Benchmarking in Optimization: Best Practice and Open Issues , 2020, ArXiv.

[9]  Dimo Brockhoff,et al.  Mixed-integer benchmark problems for single- and bi-objective optimization , 2019, GECCO.

[10]  Lorena A. Barba,et al.  Reproducible and Replicable Computational Fluid Dynamics: It’s Harder Than You Think , 2017, Computing in Science & Engineering.

[11]  Charles Audet,et al.  Derivative-Free and Blackbox Optimization , 2017 .

[12]  Nikolaus Hansen,et al.  The CMA Evolution Strategy: A Tutorial , 2016, ArXiv.

[13]  Xuefeng Yan,et al.  Self-adaptive differential evolution algorithm with discrete mutation control parameters , 2015, Expert Syst. Appl..

[14]  Andrew Lewis,et al.  S-shaped versus V-shaped transfer functions for binary Particle Swarm Optimization , 2013, Swarm Evol. Comput..

[15]  Andrew M. Sutton,et al.  Genetic Algorithms - A Survey of Models and Methods , 2012, Handbook of Natural Computing.

[16]  Günter Rudolph,et al.  Evolutionary Strategies , 2012, Handbook of Natural Computing.

[17]  N. Hansen.  A CMA-ES for Mixed-Integer Nonlinear Optimization , 2011 .

[18]  Bernd Bischl,et al.  Exploratory landscape analysis , 2011, GECCO '11.

[19]  Kusum Deep,et al.  A real coded genetic algorithm for solving integer and mixed integer optimization problems , 2009, Appl. Math. Comput..

[20]  Anne Auger,et al.  Real-Parameter Black-Box Optimization Benchmarking 2009: Noiseless Functions Definitions , 2009 .

[21]  Mehmet Fatih Tasgetiren,et al.  A discrete differential evolution algorithm for the permutation flowshop scheduling problem , 2007, GECCO '07.

[22]  Hans-Paul Schwefel,et al.  Evolution strategies – A comprehensive introduction , 2002, Natural Computing.

[23]  Nikolaus Hansen,et al.  Adapting arbitrary normal mutation distributions in evolution strategies: the covariance matrix adaptation , 1996, Proceedings of IEEE International Conference on Evolutionary Computation.

[24]  C. Floudas.  Nonlinear and Mixed-Integer Optimization: Fundamentals and Applications , 1995 .

[25]  Günter Rudolph,et al.  An Evolutionary Algorithm for Integer Programming , 1994, PPSN.

[26]  Thomas Bäck,et al.  An Overview of Evolutionary Algorithms for Parameter Optimization , 1993, Evolutionary Computation.

[27]  John H. Holland,et al.  Adaptation in Natural and Artificial Systems: An Introductory Analysis with Applications to Biology, Control, and Artificial Intelligence , 1992 .