A New Scalarization Technique to Approximate Pareto Fronts of Problems with Disconnected Feasible Sets

We introduce and analyze a novel scalarization technique and an associated algorithm for generating an approximation of the Pareto front (i.e., the efficient set) of nonlinear multiobjective optimization problems. Our approach is applicable to nonconvex problems, in particular to those with disconnected Pareto fronts and disconnected domains (i.e., disconnected feasible sets). We establish the theoretical properties of our new scalarization technique and present an algorithm for its implementation. By means of test problems, we illustrate the strengths and advantages of our approach over existing scalarization techniques such as those derived from the Pascoletti–Serafini method, as well as the popular weighted-sum method.

[1]  C. Goh,et al.  A method for convex curve approximation , 1997 .

[2]  Lotfi A. Zadeh,et al.  Optimality and non-scalar-valued performance criteria , 1963 .

[3]  Ulf Schlichtmann,et al.  A Successive Approach to Compute the Bounded Pareto Front of Practical Multiobjective Optimization Problems , 2009, SIAM J. Optim..

[4]  R. Lyndon While,et al.  A review of multiobjective test problems and a scalable test problem toolkit , 2006, IEEE Transactions on Evolutionary Computation.

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

[6]  H. P. Benson,et al.  The Vector Maximization Problem: Proper Efficiency and Stability , 1977 .

[7]  Joydeep Dutta,et al.  A new scalarization and numerical method for constructing the weak Pareto front of multi-objective optimization problems , 2011 .

[8]  Kim Fung Man,et al.  Multiobjective Optimization , 2011, IEEE Microwave Magazine.

[9]  Matthias Ehrgott,et al.  Multicriteria Optimization , 2005 .

[10]  H. P. Benson,et al.  Optimization over the efficient set , 1984 .

[11]  T. L. Saaty,et al.  The computational algorithm for the parametric objective function , 1955 .

[12]  José Mario Martínez,et al.  Augmented Lagrangian methods under the constant positive linear dependence constraint qualification , 2007, Math. Program..

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

[14]  P. Serafini,et al.  Scalarizing vector optimization problems , 1984 .

[15]  A. M. Geoffrion.  Proper efficiency and the theory of vector maximization , 1968 .

[16]  Gabriele Eichfelder,et al.  Adaptive Scalarization Methods in Multiobjective Optimization , 2008, Vector Optimization.

[17]  Kok Lay Teo,et al.  Approximation methods for non-convex curves , 1999, Eur. J. Oper. Res..

[18]  Yaochu Jin,et al.  Dynamic Weighted Aggregation for evolutionary multi-objective optimization: why does it work and how? , 2001 .

[19]  M. Ehrgott,et al.  Improved ε-Constraint Method for Multiobjective Programming , 2008 .

[20]  Patrick Siarry,et al.  An attempt to classify multiobjective optimization methods , 2004 .

[21]  Margaret M. Wiecek,et al.  Generating epsilon-efficient solutions in multiobjective programming , 2007, Eur. J. Oper. Res..

[22]  José Mario Martínez,et al.  On Augmented Lagrangian Methods with General Lower-Level Constraints , 2007, SIAM J. Optim..

[23]  Jerald P. Dauer,et al.  Optimization over the efficient set , 1995, J. Glob. Optim..

[24]  Gabriele Eichfelder,et al.  Scalarizations for adaptively solving multi-objective optimization problems , 2009, Comput. Optim. Appl..

[25]  Brian W. Kernighan,et al.  AMPL: A Modeling Language for Mathematical Programming , 1993 .

[26]  Marc Despontin,et al.  Multiple Criteria Optimization: Theory, Computation, and Application, Ralph E. Steuer (Ed.). Wiley, Palo Alto, CA (1986) , 1987 .

[27]  Johan Philip,et al.  Algorithms for the vector maximization problem , 1972, Math. Program..

[28]  C. Yalçin Kaya,et al.  Optimization Over the Efficient Set of Multi-objective Convex Optimal Control Problems , 2010, J. Optim. Theory Appl..

[29]  Yacov Y. Haimes,et al.  Multiobjective Decision Making: Theory and Methodology , 1983 .

[30]  Kalyanmoy Deb,et al.  A fast and elitist multiobjective genetic algorithm: NSGA-II , 2002, IEEE Trans. Evol. Comput..

[31]  Masahiro Tanaka,et al.  GA-based decision support system for multicriteria optimization , 1995, 1995 IEEE International Conference on Systems, Man and Cybernetics. Intelligent Systems for the 21st Century.

[32]  Jasbir S. Arora,et al.  Survey of multi-objective optimization methods for engineering , 2004 .

[33]  Gabriele Eichfelder,et al.  An Adaptive Scalarization Method in Multiobjective Optimization , 2008, SIAM J. Optim..

[34]  John E. Dennis,et al.  Normal-Boundary Intersection: A New Method for Generating the Pareto Surface in Nonlinear Multicriteria Optimization Problems , 1998, SIAM J. Optim..