On the Design of Constraint Covariance Matrix Self-Adaptation Evolution Strategies Including a Cardinality Constraint

This paper describes the algorithm's engineering of a covariance matrix self-adaptation evolution strategy (CMSA-ES) for solving a mixed linear/nonlinear constrained optimization problem arising in portfolio optimization. While the feasible solution space is defined by the (probabilistic) simplex, the nonlinearity comes in by a cardinality constraint bounding the number of linear inequalities violated. This gives rise to a nonconvex optimization problem. The design is based on the CMSA-ES and relies on three specific techniques to fulfill the different constraints. The resulting algorithm is then thoroughly tested on a data set derived from time series data of the Dow Jones Index.

[1]  Petros Koumoutsakos,et al.  Reducing the Time Complexity of the Derandomized Evolution Strategy with Covariance Matrix Adaptation (CMA-ES) , 2003, Evolutionary Computation.

[2]  Andreas Zell,et al.  Main vector adaptation: a CMA variant with linear time and space complexity , 2001 .

[3]  Nikolaus Hansen,et al.  Completely Derandomized Self-Adaptation in Evolution Strategies , 2001, Evolutionary Computation.

[4]  S. Uryasev,et al.  Chapter 8 ALGORITHMS FOR OPTIMIZATION OF VALUE- AT-RISK , 2001 .

[5]  L. Darrell Whitley,et al.  Searching for Balance: Understanding Self-adaptation on Ridge Functions , 2006, PPSN.

[6]  Kalyanmoy Deb,et al.  Simulated Binary Crossover for Continuous Search Space , 1995, Complex Syst..

[7]  Bernhard Sendhoff,et al.  Covariance Matrix Adaptation Revisited - The CMSA Evolution Strategy - , 2008, PPSN.

[8]  Petros Koumoutsakos,et al.  A Method for Handling Uncertainty in Evolutionary Optimization With an Application to Feedback Control of Combustion , 2009, IEEE Transactions on Evolutionary Computation.

[9]  Oliver Kramer,et al.  A Review of Constraint-Handling Techniques for Evolution Strategies , 2010, Appl. Comput. Intell. Soft Comput..

[10]  Christian Igel,et al.  Evolutionary Optimization of Sequence Kernels for Detection of Bacterial Gene Starts , 2006, ICANN.

[11]  Dirk V. Arnold,et al.  On the Behaviour of the (1+1)-ES for a Simple Constrained Problem , 2008, PPSN.

[12]  Silja Meyer-Nieberg,et al.  Self-adaptation in evolution strategies , 2008 .

[13]  Nikolaus Hansen,et al.  Benchmarking a BI-population CMA-ES on the BBOB-2009 function testbed , 2009, GECCO '09.

[14]  P. Suganthan,et al.  Problem Definitions and Evaluation Criteria for the CEC 2010 Competition on Constrained Real- Parameter Optimization , 2010 .

[15]  Zbigniew Michalewicz,et al.  Evolutionary Algorithms for Constrained Parameter Optimization Problems , 1996, Evolutionary Computation.

[16]  Isao Ono,et al.  Theoretical analysis of evolutionary computation on continuously differentiable functions , 2010, GECCO '10.

[17]  Yoram Singer,et al.  Efficient projections onto the l1-ball for learning in high dimensions , 2008, ICML '08.

[18]  Helmut Mausser,et al.  ALGORITHMS FOR OPTIMIZATION OF VALUE­ AT-RISK* , 2002 .

[19]  Jing J. Liang,et al.  Problem Definitions and Evaluation Criteria for the CEC 2005 Special Session on Real-Parameter Optimization , 2005 .

[20]  Tetsuyuki Takahama,et al.  Constrained optimization by the ε constrained differential evolution with an archive and gradient-based mutation , 2010, IEEE Congress on Evolutionary Computation.

[21]  Hans-Georg Beyer,et al.  Self-adaptation of evolution strategies under noisy fitness evaluations , 2006, Genetic Programming and Evolvable Machines.

[22]  Stefan Roth,et al.  Covariance Matrix Adaptation for Multi-objective Optimization , 2007, Evolutionary Computation.

[23]  Romeo Rizzi,et al.  A mixed integer linear programming formulation of the optimal mean/Value-at-Risk portfolio problem , 2007, Eur. J. Oper. Res..

[24]  P. N. Suganthan,et al.  Differential Evolution: A Survey of the State-of-the-Art , 2011, IEEE Transactions on Evolutionary Computation.