Numerical Methods for Multiextremal Nonlinear Programming Problems with Nonconvex Constraints

On strong homogeneity of a class of global optimization algorithms working with infinite and infinitesimal scales

“…Due to the presence of multiple local minima and non-differentiability of the objective function, classical local optimization techniques cannot be used for solving these problems and global optimization methods should be developed (see, e.g., [8,14,18,19,30,35,36,37,39,41]). …”

“…One of the desirable properties of global optimization methods (see [7,35,41]) is their strong homogeneity meaning that a method produces the same sequences of trial points (i.e., points where the objective function f (x) is evaluated) independently of both shifting f (x) vertically and its multiplication by a scaling constant. In other words, it can be useful to optimize a scaled function g(x) = g(x; α, β) = αf (x) + β, α > 0,…”

“…The concept of strong homogeneity has been introduced in [41] where it has been shown that both the P-algorithm (see [40]) and the one-step Bayesian algorithm (see [17]) are strongly homogeneous. The case α = 1, β = 0 was considered in [7,35] where a number of methods enjoying this property and called homogeneous were studied. It should be mentioned that there exist global optimization methods that are homogeneous or strongly homogeneous and algorithms (see, for instance, the DIRECT algorithm from [14] and a huge number of its modifications) that do not possess this property.…”

“…It should be mentioned that there exist global optimization methods that are homogeneous or strongly homogeneous and algorithms (see, for instance, the DIRECT algorithm from [14] and a huge number of its modifications) that do not possess this property. All the methods mentioned above have been developed to work with Lipschitz global optimization problems that can be met very frequently in practical applications (see, e.g., [18,19,30,35,36,39]). These methods belong to the class of "Divide-the-best" algorithms introduced in [23].…”

Index information algorithm with local tuning for solving multidimensional global optimization problems with multiextremal constraints

“…It uses two powerful ideas for solving problem (i)-(iv). The first one is the index scheme (see [35,38,39,41,42]), allowing to solve Lipschitz problems where both the objective function ϕ(y) and constraints G i (y), 1 ≤ i ≤ m, may be multiextremal and partially defined. Its importance increases in this case because it is not clear how to solve such problems by using, for example, the penalty approach.…”