Abstract
The global resolution of constrained non-linear bi-objective optimization problems (NLBOO) aims at covering their Pareto-optimal front which is in general a one-manifold in \(\mathbb {R}^2\). Continuation methods can help in this context as they can follow a continuous component of this front once an initial point on it is provided. They constitute somehow a generalization of the classical scalarization framework which transforms the bi-objective problem into a parametric single-objective problem. Recent works have shown that they can play a key role in global algorithms dedicated to bi-objective problems, e.g. population based algorithms, where they allow discovering large portions of locally Pareto optimal vectors, which turns out to strongly support diversification. The contribution of this paper is twofold: we first provide a survey on continuation techniques in global optimization methods for NLBOO, identifying relations between several work and usual limitations, among which the ability to handle inequality constraints. We then propose a rigorous active set management strategy on top of a continuation method based on interval analysis, certified with respect to feasibility, local optimality and connectivity. This allows overcoming the latter limitation as illustrated on a representative bi-objective problem.
Similar content being viewed by others
Notes
More precisely, the number of parameters can be reduced to one.
A turning point is somehow a U-turn, where the direction of continuation with respect to the fixed parameter locally changes, hence is singular for this parametrization.
The activation (or disactivation) of \(k\) constraints at the same time occurs when the one dimensional curve Pareto frontier crosses the intersection of the \(k\) constraints boundaries, which is stable only if \(k=1\).
Parallelotopes allow a cheap and certified sampling of the Pareto frontier they include.
References
Allgower, E.L., Georg, K.: Introduction to numerical continuation methods. In: Classics in Applied Mathematics, vol. 45. SIAM, Philadelphia (2003)
Askar, S., Tiwari, A.: Multi-objective optimisation problems: a symbolic algorithm for performance measurement of evolutionary computing techniques. In: Ehrgott, M., et al. (eds.) Evolutionary Multi-Criterion Optimization. Lecture Notes in Computer Science, vol. 5467, pp. 169–182. Springer, Berlin (2009)
Beltrán, C., Leykin, A.: Certified numerical homotopy tracking. Exp. Math. 21(1), 69–83 (2012)
Beyn, W.-J., Effenberger, C., Kressner, D.: Continuation of eigenvalues and invariant pairs for parameterized nonlinear eigenvalue problems. Numer. Math. 119, 489–516 (2011)
Boissonnat, J.-D., Ghosh, A.: Triangulating smooth submanifolds with light scaffolding. Math. Comput. Sci. 4, 431–461 (2010)
Das, I., Dennis, J.: Normal-boundary intersection: a new method for generating the pareto surface in nonlinear multicriteria optimization problems. SIAM J. Optim. 8(3), 631–657 (1998)
Ehrgott, M.: Multicriteria Optimization, 2nd edn. Springer, Berlin (2005)
Eichfelder, G.: Scalarizations for adaptively solving multi-objective optimization problems. Comput. Optim. Appl. 44(2), 249–273 (2009)
Erfani, T., Utyuzhnikov, S.: Directed search domain: a method for even generation of the pareto frontier in multiobjective optimization. Eng. Optim. 43(5), 467–484 (2011)
Faudot, D., Michelucci, D.: A new robust algorithm to trace curves. Reliab. Comput. 13(4), 309–324 (2007)
Goldsztejn, A., Granvilliers, L.: A new framework for sharp and efficient resolution of NCSP with manifolds of solutions. Constraints 15(2), 190–212 (2010)
Goualard F.: GAOL 3.1.1: Not Just Another Interval Arithmetic Library. LINA, 4.0 edition (2006)
Granvilliers, L., Benhamou, F.: Algorithm 852: realPaver: an interval solver using constraint satisfaction techniques. ACM Trans. Math. Softw. 32(1), 138–156 (2006)
Guddat, J., Th. Jongen, H., Nowack, D.: Parametric optimization: pathfollowing with jumps. In: Gmez-Fernandez, J.A., et al. (eds.) Approximation and Optimization. Lecture Notes in Mathematics, vol. 1354, pp. 43–53. Springer, Berlin (1988)
Guddat, J., Vazquez, F.G., Nowack, D., Ruckmann, J.: A modified standard embedding with jumps in nonlinear optimization. Eur. J. Oper. Res. 169(3), 1185–1206 (2006)
Harada, K., Sakuma, J., Kobayashi, S.: Local search for multiobjective function optimization: pareto descent method. In: GECCO, pp. 659–666. ACM (2006)
Harada, K., Sakuma, J., Kobayashi, S., Ono, I.: Uniform sampling of local pareto-optimal solution curves by pareto path following and its applications in multi-objective GA. In: GECCO, pp. 813–820. ACM (2007)
Harada, K., Sakuma, J., Ono, I., Kobayashi, S.: Constraint-handling method for multi-objective function optimization: pareto descent repair operator. In: Obayashi, S., et al. (eds.) Evolutionary Multi-Criterion Optimization. volume 4403 of LNCS, pp. 156–170. Springer, Berlin (2007)
Hartikainen, M., Miettinen, K., Wiecek, M.M.: Constructing a pareto front approximation for decision making. Math. Meth. Oper. Res. 73(2), 209–234 (2011)
Hartikainen, M., Miettinen, K., Wiecek, M.M.: Paint: pareto front interpolation for nonlinear multiobjective optimization. Comput. Optim. Appl. 52(3), 845–867 (2012)
Hillermeier, C.: Generalized homotopy approach to multiobjective optimization. J. Optim. Theor. Appl. 110(3), 557–583 (2001)
Hillermeier, C.: Nonlinear Multiobjective Optimization: A Generalized Homotopy Approach, vol. 135. Birkäuser, Basel (2001)
Kearfott, B., Xing, Z.: An interval step control for continuation methods. SIAM J. Numer. Anal. 31(3), 892–914 (1994)
Lara, A., Sanchez, G., Coello, C., Schütze, O.: HCS: a new local search strategy for memetic multiobjective evolutionary algorithms. IEEE Trans. Evol. Comput. 14(1), 112–132 (2010)
Leyffer, S.: A complementarity constraint formulation of convex multiobjective optimization problems. INFORMS J. Comput. 21(2), 257–267 (April 2009)
Lovison, A.: Singular continuation: generating piecewise linear approximations to pareto sets via global analysis. SIAM J. Optim. 21(2), 463–490 (2011)
Lovison, A.: Global search perspectives for multiobjective optimization. J. Glob. Optim. 57(2), 385–398 (2013)
Lundberg, B., Poore, A.: Numerical continuation and singularity detection methods for parametric nonlinear programming. SIAM J. Optim. 3(1), 134–154 (1993)
Martin, B., Goldsztejn, A., Granvilliers, L., Jermann, C.: Certified parallelotope continuation for one-manifolds. SIAM J. Numer. Anal. 51(6), 3373–3401 (2013)
Messac, A., Ismail-Yahaya, A., Mattson, C.: The normalized normal constraint method for generating the pareto frontier. Struct. Multidiscip. Optim. 25(2), 86–98 (2003)
Messac, A., Mattson, C.: Generating well-distributed sets of pareto points for engineering design using physical programming. Optim. Eng. 3(4), 431–450 (2002)
Messac, A., Mattson, C.: Normal constraint method with guarantee of even representation of complete pareto frontier. AIAA J. 42, 2101–2111 (2004)
Miettinen, K.: Nonlinear Multiobjective Optimization, volume 12 of International Series in Operations Research and Management Science. Kluwer, Dordrecht (1999)
Neumaier, A.: Interval Methods for Systems of Equations. Cambridge University Press, Cambridge (1991)
Pereyra, V.: Fast computation of equispaced pareto manifolds and pareto fronts for multiobjective optimization problems. Math. Comput. Simul. 79(6), 1935–1947 (2009)
Pereyra, V., Saunders, M., Castillo, J.: Equispaced pareto front construction for constrained bi-objective optimization. Math. Comput. Model. 57(9–10), 2122–2131 (2013)
Porta, J.-M., Jaillet, L., Bohigas, O.: Randomized path planning on manifolds based on higher-dimensional continuation. Int. J. Robot. Res. 31(2), 201–215 (2012)
Potschka, A., Logist, F., Van Impe, J., Bock, H.: Tracing the Pareto frontier in bi-objective optimization problems by ODE techniques. Numer. Algorithms 57(2), 217–233 (2011)
Rakowska, J., Haftka, R., Watson, L.: An active set algorithm for tracing parametrized optima. Struct. Optim. 3(1), 29–44 (1991)
Rakowska, J., Haftka, R., Watson, L.: Multi-objective control-structure optimization via homotopy methods. SIAM J. Optim. 3(3), 654–667 (1993)
Rao, J., Papalambros, P.: A non-linear programming continuation strategy for one parameter design optimization problems. In: ASME Design Automation Conference, pp. 77–89 (1989)
Rigoni, E., Poles, S.: NBI and MOGA-II, two complementary algorithms for multi-objective optimizations. In: Practical Approaches to Multi-Objective Optimization, number 04461 in Dagstuhl, Seminar (2005)
Ringkamp, M., Ober-Blbaum, S., Dellnitz, M., Schütze, O.: Handling high-dimensional problems with multi-objective continuation methods via successive approximation of the tangent space. Eng. Optim. 44(9), 1117–1146 (2012)
Ruzika, S., Wiecek, M.M.: Approximation methods in multiobjective programming. J. Optim. Theor. Appl. 126(3), 473–501 (2005)
Schütze, O., Coello, C.: Coello, S. Mostaghim, E. Talbi, and M. Dellnitz. Hybridizing evolutionary strategies with continuation methods for solving multi-objective problems. Eng. Optim. 40, 383–402 (2008)
Schütze, O., Dell’Aere, A., Dellnitz, M.: On continuation methods for the numerical treatment of multi-objective optimization problems. In: Branke, J. et al. (ed.) Practical Approaches to Multi-Objective Optimization, number 04461 in Dagstuhl Seminar (2005)
Schütze, O., Lara, A., Coello Coello, C.: Evolutionary continuation methods for optimization problems. In: GECCO, pp. 651–658. ACM (2009)
Smale, S.: Newton’s method estimates from data at one point. In: Ewing, R.E., et al. (eds.) The Merging of Disciplines in Pure, Applied and Computational Mathematics, pp. 185–196. Springer, New York (1986)
Utyuzhnikov, S., Fantini, P., Guenov, M.: A method for generating a well-distributed pareto set in nonlinear multiobjective optimization. J. Comput. Appl. Math. 223(2), 820–841 (2009)
Zhang, Z.: Immune optimization algorithm for constrained nonlinear multiobjective optimization problems. Appl. Soft Comput. 7(3), 840–857 (2007)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Martin, B., Goldsztejn, A., Granvilliers, L. et al. On continuation methods for non-linear bi-objective optimization: towards a certified interval-based approach. J Glob Optim 64, 3–16 (2016). https://doi.org/10.1007/s10898-014-0201-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10898-014-0201-3