Lyudmila N. Polyakova

Quasidifferentiability in nonsmooth, nonconvex mechanics

Nonconvex and nonsmooth optimization problems arise in advanced engineering analysis and structural analysis applications. In fact the set of inequality and complementarity relations that describe the structural analysis problem are generated as optimality conditions by the quasidifferential potential energy optimization problem. Thus new kind of variational expressions arise for these problems, which generalize the classical variational equations of smooth mechanics, the variational inequalities of convex, nonsmooth mechanics and give a solid, computationally efficient explication of hemivariational inequalities of nonconvex, nonsmooth mechanics. Moreover quasidifferential calculus and optimization software make this approach applicable for a large number of problems. The connection of quasidifferential optimization and nonsmooth, nonconvex mechanics is discussed in this paper. A number of representative examples from elastostatic analysis applications are treated in details. Numerical examples illustrate the theory.

Difference convex optimization techniques in nonsmooth computational mechanics

The impact and the usefulness of difference convex optimization techniques for the numerical solution of problems arising in nonsmooth and nonconvex computational mechanics are investigated in this paper. Algorithms for the numerical solution of the problem are proposed and studied. The relation to the more general quasi- and co-differentiable optimization techniques is also discussed. The link to classical, smooth and nonsmooth computational mechanics’thms is also presented

Exact penalty methods for nonsmooth optimization

Methods of penalty functions are widely used in nonlinear programming. The idea of penalty function methods consists in reducing the problem of conditional optimization to a problem of the unconstrained optimization. Among the different approaches existing for such reduction we shall consider the method of exact penalty nondifferentiable functions. The implementation of exact penalty function methods first of all depends on the properties of an objective function and a function defining a constraint. Therefore various conditions are imposed on these functions to make it possible to solve our problem.

On global unconstrained minimization of the difference of polyhedral functions

The problem of finding a global minimizer of the difference of polyhedral functions is considered. By means of conjugate functions, necessary and sufficient conditions for the unboundedness and the boundedness of such functions in Rn are derived. Using hypodifferentials of polyhedral functions, necessary and sufficient conditions for a global unconstrained minimum on Rn are proved.

On Global Properties of D.C.Functions

Necessary and sufficient optimality conditions of the difference of convex functions are derived for unconstrained optimization problems. Connection between extremal properties of the difference of convex functions and the extremal properties of the difference of their conjugate functions is established. Duality theorems are proved. A smooth approximation of a d.c. function is investigated.

About constructing a dual polyhedral cone in R 3

The problem of constructing a dual cone of a convex polyhedral cone arises in many mathematical and applied researches. In the paper the method of constructing the dual cone to the acute convex polyhedral cone in R3 is proposed. The Householder transformation, creating a convex hull on the plane and projecting the point lying on the z-axis onto a corresponding face are the basic operations which we used.

A stochastic model of rumour spreading

In this paper, the new model for dynamics of spreading of rumour in a continuous time is suggested. Model depends on a parameter representing the probability of the outcome of interaction between the two spreaders. The process of spreading rumors is described by a system of linear differential equations.

On minimizing the maximum of two quadratic functions

The problem of minimizing the maximum of two strongly convex quadratic functions on Rn is considered. It is shown that in some cases this problem is equivalent to finding the positive root of a polynomial of the degree 2n or less.

About probabilistic model of the terminal operation

Suppose that similar terminals of several companies are in relative proximity to each other. Each company carries out transportation and storage of cargoes. If the terminal is crowded then the company can rent a portion of a terminal of another company. On contrary, if the terminal of some company is not sufficiently loaded, then the portion of the terminal can be suggested in the lease to another company.

On a continuous method for minimizing of nonsmooth functions

A method for minimizing of functions from one class of nonsmooth functions (namely, continuously hypodifferentiable functions) is considered. In it a direction of descent is found by projecting the zero element on a continuous hypodifferential. Step multipliers are calculated either from the Armijo condition or from a one-dimensional optimization. Theorems of convergence are proved.