Yu. G. Evtushenko

Parallel global optimization of functions of several variables

On the basis of the method of nonuniform coverings, a parallel method for the global optimization of Lipschitzian functions is developed. This method is implemented in C-MPI for the global minimization of functions whose gradient satisfies the Lipschitz condition. The performance of the algorithm is demonstrated using the calculation of the structure of a protein molecule as an example.

Nonuniform covering method as applied to multicriteria optimization problems with guaranteed accuracy

The nonuniform covering method is applied to multicriteria optimization problems. The ɛ-Pareto set is defined, and its properties are examined. An algorithm for constructing an ɛ-Pareto set with guaranteed accuracy ɛ is described. The efficiency of implementing this approach is discussed, and numerical results are presented.

A deterministic algorithm for global multi-objective optimization

The paper describes a method for solving multi-objective optimization problems with box constraints. Unlike existing approaches, the proposed method not only constructs a finite approximation of Pareto frontier, but also proves its ϵ-optimality. The paper gives a detailed explanation of basic theoretical concepts behind the method and describes the algorithmic implementation. A practically important application of the proposed method to finding the working space of a robotic manipulator is presented.

Augmented Lagrangian method for large-scale linear programming problems

The augmented Lagrangian and Newton methods are used to simultaneously solve the primal and dual linear programming problems. The proposed approach is applied to the primal linear programming problem with a very large number (≈106) of nonnegative variables and a moderate (≈103) number of equality-type constraints. Computation results such as the solution of a linear programme with 10 million primal variables are presented.

An application of the nonuniform covering method to global optimization of mixed integer nonlinear problems

The nonuniform covering method for global optimization of functions of several variables is extended to nonlinear programs. It is shown that this method can be used for solving problems that, in addition to conventional constraints, involve partial integrality conditions. Estimates for the accuracy of the solution and for the number of steps required for finding a minimum with a prescribed tolerance are derived. New minorants based on an estimate for the spectrum of the Hessian matrix of the objective function and the constraints are given. New formulas for covering sets improving the efficiency of the method are obtained. Examples of solving nonlinear programs with the use of the proposed approach are presented.

Parallelization of the global extremum searching process

The parallel algorithm for searching the global extremum of the function of several variables is designed. The algorithm is based on the method of nonuniform coverings proposed by Yu.G. Evtushenko for functions that comply with the Lipschitz condition. The algorithm is realized in the language C and message passing interface (MPI) system. To speed up computations, auxiliary procedures for founding the local extremum are used. The operation of the algorithm is illustrated by the example of atomic cluster structure calculations.

Parallel implementation of Newton’s method for solving large-scale linear programs

Parallel versions of a method based on reducing a linear program (LP) to an unconstrained maximization of a concave differentiable piecewise quadratic function are proposed. The maximization problem is solved using the generalized Newton method. The parallel method is implemented in C using the MPI library for interprocessor data exchange. Computations were performed on the parallel cluster MVC-6000IM. Large-scale LPs with several millions of variables and several hundreds of thousands of constraints were solved. Results of uniprocessor and multiprocessor computations are presented.

Calculation of deformations in nanocomposites using the block multipole method with the analytical-numerical account of the scale effects

Local scale effects for linear continuous media are investigated as applied to the composites reinforced by nanoparticles. A mathematical model of the interphase layer is proposed that describes the specific nature of deformations in the neighborhood of the interface between different phases in an inhomogeneous material. The characteristic length of the interphase layer is determined formally in terms of the parameters of the mathematical model. The local stress state in the neighborhood of the phase boundaries in the interphase layer is examined. This stress can cause a significant change of the integral macromechanical characteristics of the material as a whole if the interphase boundaries are long. Such a situation is observed in composite materials reinforced by microparticles and nanoparticles even when the volume concentration of the inclusions is small. A numerical simulation of the stress state is performed on the basis of the block analytical-numerical multipole method with regard for the local effects related to the special nature of the deformation of the interphase layer in the vicinity of the interface.

New perspective on the theorems of alternative

New general theorems of the alternative are presented. The constructive proofs based on the duality theory are given. From these results many well-known theorems of the alternative are obtained by simple substitutions. Computational applications of theorems of the alternative to solving linear systems, LP and NLP problems are given. A linear systems of possibly unsolvable equalities and inequalities are considered. With original linear system an alternative system is associated such that one and only one of these systems is consistent. If the original system is solvable then numerical method for solving this system consists of minimization of the residual of the alternative inconsistent system. From the results of this minimization the normal solution of the original system is determined.

General lagrange-type functions in constrained global optimization Part I: auxiliary functions and optimality conditions

The paper contains some new results and a survey of some known results related to auxiliary (Lagrange-type) functions in constrained optimization. We show that auxiliary functions can be constructed by means of two-step convolution of constraints and the objective function and present some conditions providing the validity of the zero duality gap property. We show that auxiliary functions are closely related to the so-called separation functions in the image space of the constrained problem under consideration. The second part of the paper (see Evtushenko et al., General Lagrange-type functions in constrained global optimization. Part 11: Exact Auxillary functions. Optimization Methods and Software) contains results related to exact auxiliary functions.