10,652 research outputs found
Multidimensional Constrained Global Optimization in Domains with Computable Boundaries
Multidimensional constrained global optimization problem with objective function under Lipschitz condition and constraints generating a feasible domain with computable boundaries is considered. For solving this problem the dimensionality reduction approach on the base of the nested optimization scheme is used. This scheme reduces initial multidimensional problem to a family of one-dimensional subproblems and allows applying univariate methods for the execution of multidimensional optimization. Sequential and parallel modifications of well-known information-statistical methods of Lipschitz optimization are proposed for solving the univariate subproblems arising inside the nested scheme in the case of domains with computable boundaries. A comparison with classical penalty function method being traditional means of taking into account the constraints is carried out. The results of experiments demonstrate a significant advantage of the methods proposed over the penalty function method
Lipschitz gradients for global optimization in a one-point-based partitioning scheme
A global optimization problem is studied where the objective function
is a multidimensional black-box function and its gradient satisfies the
Lipschitz condition over a hyperinterval with an unknown Lipschitz constant
. Different methods for solving this problem by using an a priori given
estimate of , its adaptive estimates, and adaptive estimates of local
Lipschitz constants are known in the literature. Recently, the authors have
proposed a one-dimensional algorithm working with multiple estimates of the
Lipschitz constant for (the existence of such an algorithm was a
challenge for 15 years). In this paper, a new multidimensional geometric method
evolving the ideas of this one-dimensional scheme and using an efficient
one-point-based partitioning strategy is proposed. Numerical experiments
executed on 800 multidimensional test functions demonstrate quite a promising
performance in comparison with popular DIRECT-based methods.Comment: 25 pages, 4 figures, 5 tables. arXiv admin note: text overlap with
arXiv:1103.205
Index Information Algorithm with Local Tuning for Solving Multidimensional Global Optimization Problems with Multiextremal Constraints
Multidimensional optimization problems where the objective function and the
constraints are multiextremal non-differentiable Lipschitz functions (with
unknown Lipschitz constants) and the feasible region is a finite collection of
robust nonconvex subregions are considered. Both the objective function and the
constraints may be partially defined. To solve such problems an algorithm is
proposed, that uses Peano space-filling curves and the index scheme to reduce
the original problem to a H\"{o}lder one-dimensional one. Local tuning on the
behaviour of the objective function and constraints is used during the work of
the global optimization procedure in order to accelerate the search. The method
neither uses penalty coefficients nor additional variables. Convergence
conditions are established. Numerical experiments confirm the good performance
of the technique.Comment: 29 pages, 5 figure
Deterministic global optimization using space-filling curves and multiple estimates of Lipschitz and Holder constants
In this paper, the global optimization problem with
being a hyperinterval in and satisfying the Lipschitz condition
with an unknown Lipschitz constant is considered. It is supposed that the
function can be multiextremal, non-differentiable, and given as a
`black-box'. To attack the problem, a new global optimization algorithm based
on the following two ideas is proposed and studied both theoretically and
numerically. First, the new algorithm uses numerical approximations to
space-filling curves to reduce the original Lipschitz multi-dimensional problem
to a univariate one satisfying the H\"{o}lder condition. Second, the algorithm
at each iteration applies a new geometric technique working with a number of
possible H\"{o}lder constants chosen from a set of values varying from zero to
infinity showing so that ideas introduced in a popular DIRECT method can be
used in the H\"{o}lder global optimization. Convergence conditions of the
resulting deterministic global optimization method are established. Numerical
experiments carried out on several hundreds of test functions show quite a
promising performance of the new algorithm in comparison with its direct
competitors.Comment: 26 pages, 10 figures, 4 table
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