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We consider the extremal problem of interpolation of convex scattered data in ℝ3 by smooth edge convex curve networks with minimal Lp-norm of the second derivative for 1 < p ≤ ∞. The problem for p = 2 was set and solved by Andersson et al. (1995). Vlachkova (2019) extended the results in (Andersson et al., 1995) and solved the problem for 1 < p < ∞. The minimum edge convex Lp-norm network for 1 < p < ∞ is obtained from the solution to a system of nonlinear equations with coeffificients determined by the d ata. The solution in the case 1 < p < ∞ is unique forstrictly convex data. The approach used in (Vlachkova, 2019) can not be applied to the corressponding extremal problem for p = ∞. In this case the solution is not unique. Here we establish the existence of a solution to the extremal interpolation problem for p = ∞. This solution is a quadratic spline function with at most one knot on each edge of the underlying triangulation. We also propose suffificient conditions for solving the problem for p = ∞. [ABSTRACT FROM AUTHOR]

We consider the extremal problem of interpolation of scattered data in ℝ3 by smooth curve networks with minimal Lp-norm of the second derivative for 1 < p ≤ ∞. The problem for p = 2 was set and solved by Nielson [1]. Andersson et al. [2] gave a new proof of Nielson's result by using a different approach. Vlachkova [3] extended the results in [2] and solved the problem for 1 < p < ∞. The minimum Lp-norm network for 1 < p < ∞ is obtained from the solution to a system of nonlinear equations with coefficients determined by the data. The solution in the case 1 < p < ∞ is unique. We denote the corresponding minimum Lp-norm network by Fp. In the case where p = ∞ we establish the existence of a solution of the same type as in the case where 1 < p < ∞. This solution on each edge of the underlying triangulation is a quadratic spline function with at most one knot. We denote this solution by F∞ and prove that the minimum Lp-norm networks Fp converge to the minimum L∞-norm network F∞ as p → ∞. [ABSTRACT FROM AUTHOR]

In this work we re-derive and improve upon quasi-Newton methods commonly used to find the zeros or extrema of functions, using the maximum entropy method. Unlike Newton's method, in which the Jacobian or Hessian matrix is calculated at each iteration, quasi-Newton methods find an approximation to the matrix by updating it from the previous iteration. The updates generally follow the under-determined secant equation. The methodology used here differs from previous maximum entropy quasi-Newton derivations found in the literature, in that it updates the average values of the Jacobian or Hessian rather than updating a covariance matrix while keeping the mean fixed at zero. By approaching the derivation differently to previous studies, new insights were obtained into how quasi-Newton methods behave and how they can be improved. Numerical experiments demonstrate several improvements on existing quasi-Newton methods. [ABSTRACT FROM AUTHOR]

Lipschitz global optimization appears in many practical problems: decision making, optimal control, stability problems, finding the minimal root problems, etc. In many engineering applications the objective function is a "black-box", multiextremal, non-differentiable and hard to evaluate. Another common property of the function to be optimized very often is the Lipschitz condition. In this talk, the Lipschitz global optimization problem is considered and several nature-inspired and Lipschitz global optimization algorithms are briefly described and compared with respect to the number of evaluations of the objective function. [ABSTRACT FROM AUTHOR]

Multidimensional unconstrained global optimization problem with objective function under Lipschitz condition 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 being Lipschitzian as well and thus allows applying univariate methods for the execution of multidimensional optimization. For two well-known one-dimensional methods of Lipschitz optimization the modifications providing the acceleration of the search process in the situation when the objective function is continuously differentiable in a vicinity of the global minimum are considered and compared. Results of computational experiments on conventional test class of multiextremal functions confirm efficiency of the modified methods. [ABSTRACT FROM AUTHOR]

In this paper the global optimization problem where the objective function is multiextremal and satisfying the Lipschitz condition over a hyperinterval is considered. An algorithm that uses Peano-type space-filling curves to reduce the original Lipschitz multi-dimensional problem to a univariate one satisfying the Hölder condition is proposed. The algorithm at each iteration applies a new geometric technique working with a number of possible Hölder 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 Hölder global optimization, as well. Convergence condition are given. Numerical experiments show quite a promising performance of the new technique. [ABSTRACT FROM AUTHOR]

The paper contains the results of investigation of a parallel global optimization algorithm combined with a dimension reduction scheme. This allows solving multidimensional problems by means of reducing to data-independent subproblems with smaller dimension solved in parallel. The new element implemented in the research consists in using several graphic accelerators at different computing nodes. The paper also includes results of solving problems of wellknown multiextremal test class GKLS on Lobachevsky supercomputer using tens of thousands of GPU cores. [ABSTRACT FROM AUTHOR]

We define GSt(α,δ) be the class of function f,f(0) = f′(0)-1 = 0 and satisfying Re{eiαzf′(z)g(z)}>δ,0≤δ<1 in D = {z:|z|<1} for |α| < π, cosα > δ and g(z) = z1-z2. For this class of function, we find certain extremal properties and the bound of GSt(α,δ). [ABSTRACT FROM AUTHOR]

When modeling extreme events there are a few primordial parameters, among which we refer the extreme value index and the extremal index. The extreme value index measures the right tail-weight of the underlying distribution and the extremal index characterizes the degree of local dependence in the extremes of a stationary sequence. Most of the semi-parametric estimators of these parameters show the same type of behaviour: nice asymptotic properties, but a high variance for small values of k, the number of upper order statistics to be used in the estimation, and a high bias for large values of k. This shows a real need for the choice of k. Choosing some well-known estimators of those parameters we revisit the application of a heuristic algorithm for the adaptive choice of k. The procedure is applied to some simulated samples as well as to some real data sets. [ABSTRACT FROM AUTHOR]

A dynamic Walrasian economic equilibrium problem introduced in [3] is considered and, by applying the duality theory in infinite dimensional spaces, a characterization of equilibrium in terms of Lagrange variables is obtained. Thanks to this characterization we are able to give a description of the behavior of the economic market during the evolution in time. [ABSTRACT FROM AUTHOR]

The method of lower and upper solutions provides, as well as results of existence, other important properties such as location of solution, extremal solutions,..., which have been under used and, moreover, its potential has not been optimized, either in theory either in applications. This work will present some cases to emphasize both items: two fourth order problems with functional boundary conditions (including an application to a continuous model for the deformation of the human spine under the action of some forces) and a third order periodic problem where unbounded nonlinearities are allowed, provided that an one-sided Nagumo-type condition is verified. [ABSTRACT FROM AUTHOR]

Pontryagin's Maximum Principle [8] is considered as an outstanding achievement of optimal control theory. This Principle does not give sufficient conditions to compute an optimal trajectory; it only provides necessary conditions. Thus only candidates to be optimal trajectories, called extremals, are found. The Maximum Principle gives rise to different kinds of them and, particularly, the so-called abnormal extremals have been studied because they can be optimal, as Liu and Sussmann, and Montgomery proved in subRiemannian geometry [5, 7]. We build up a presymplectic algorithm, similar to those defined in [2, 3, 4, 6], to determine where the different kinds of extremals of an optimal control problem can be. After describing such an algorithm, we apply it to the study of extremals, specially the abnormal ones, in optimal control problems for affine connection control systems [1]. These systems model the motion of different types of mechanical systems such as rigid bodies, nonholonomic systems and robotic arms [1]. [ABSTRACT FROM AUTHOR]

This lecture brings a survey of topics and results concerning differential equations which arise as equations for extremals of higher-order variational functionals on fibered manifolds. © 2004 American Institute of Physics [ABSTRACT FROM AUTHOR]

In an extremal problem, instead of f (x) one has a sequence fn(x) of functions approximating in some sense f (x) and on the basis of which one has to find an extremum of f (x), such problems are usually called limit extremal problems. In this study, a minimization algorithm for limit extremal problems is proposed under some certain constraints. The algorithm is based on the linearization method of Pshenichnyi [1, 2]. [ABSTRACT FROM AUTHOR]

Necessary Conditions of Optimality (NCO) play an important role in the characterization of and search for solutions of Optimization Problems. They enable us to identify a small set of candidates to local minimizers among the overall set of admissible solutions. However, in constrained optimization problems it may happen that necessary conditions of optimality are merely state a relation between the constraints and do not use the objective function to select candidates to minimizers. To avoid this phenomenon, it is necessary to strengthened the NCO. Here, we overview and describe strengthened forms NCO for calculus of variations with inequality constraints. [ABSTRACT FROM AUTHOR]