New LP bound in multivariate Lipschitz optimization: Theory and applications

Most numerically promising methods for solving multivariate unconstrained Lipschitz optimization problems of dimension greater than two use rectangular or simplicial branch-and-bound techniques with computationally cheap but rather crude lower bounds.Generalizations to constrained problems, however, require additional devices to detect sufficiently many infeasible partition sets. In this article, a new lower bounding procedure is proposed for simplicial methods yielding considerably better bounds at the expense of two linear programs in each iteration. Moreover, the resulting approach can solve easily linearly constrained problems, since in this case infeasible partition sets are automatically detected by the lower bounding procedure.Finally, it is shown that the lower bounds can be further improved when the method is applied to solve systems of inequalities. Implementation issues, numerical experiments, and comparisons are discussed in some detail.

[1]  Glenn W. Graves,et al.  AN ALGORITHM FOR NONCONVEX PROGRAMMING , 1969 .

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

[3]  M. Todd.  The Computation of Fixed Points and Applications , 1976 .

[4]  Reiner Horst,et al.  An algorithm for nonconvex programming problems , 1976, Math. Program..

[5]  D. Mayne,et al.  Outer approximation algorithm for nondifferentiable optimization problems , 1984 .

[6]  Efim A. Galperin,et al.  The cubic algorithm , 1985 .

[7]  János D. Pintér,et al.  Globally convergent methods for n-dimensional multiextremal optimization , 1986 .

[8]  J. Pintér.  Global optimization on convex sets , 1986 .

[9]  Regina Hunter Mladineo.  An algorithm for finding the global maximum of a multimodal, multivariate function , 1986, Math. Program..

[10]  R. Horst,et al.  On the convergence of global methods in multiextremal optimization , 1987 .

[11]  E. A. Galperin,et al.  Precision, complexity, and computational schemes of the cubic algorithm , 1988 .

[12]  C. C. Meewella,et al.  An algorithm for global optimization of Lipschitz continuous functions , 1988 .

[13]  R. Hoprst,et al.  Deterministic global optimization with partition sets whose feasibility is not known: Application to concave minimization, reverse convex constraints, DC-programming, and Lipschitzian optimization , 1988 .

[14]  J. Pintér.  Branch- and bound algorithms for solving global optimization problems with Lipschitzian structure , 1988 .

[15]  R. Horst,et al.  Branch-and-bound methods for solving systems of Lipschitzian equations and inequalities , 1988 .

[16]  Reiner Horst,et al.  Convergence and restart in branch-and-bound algorithms for global optimization. Application to concave minimization and D.C. Optimization problems , 1988, Math. Program..

[17]  János D. Pintér,et al.  An application of Lipschitzian global optimization to product design , 1991, J. Glob. Optim..

[18]  G. R. Wood,et al.  Multidimensional bisection applied to global optimisation , 1991 .

[19]  Graham R. Wood,et al.  The bisection method in higher dimensions , 1992, Math. Program..

[20]  Roman G. Strongin,et al.  Algorithms for multi-extremal mathematical programming problems employing the set of joint space-filling curves , 1992, J. Glob. Optim..

[21]  Regina Hunter Mladineo.  Convergence rates of a global optimization algorithm , 1992, Math. Program..

[22]  R. Horst,et al.  Global Optimization: Deterministic Approaches , 1992 .

[23]  Graham R. Wood,et al.  Multidimensional bisection: The performance and the context , 1993, J. Glob. Optim..

[24]  Abdellah Salhi,et al.  Global Optimization: Deterministic Approaches (2nd Edition) , 1994 .

[25]  Dimitri P. Bertsekas,et al.  Nonlinear Programming , 1997 .

[26]  Brigitte Jaumard,et al.  An On-Line Cone Intersection Algorithm for Global Optimization of Multivariate Lipschitz Functions , 1995 .