Vitali G. Zhadan

Stable Barrier-projection and Barrier-Newton methods in nonlinear programming

The present paper is devoted to the application of the space transformation techniques for solving nonlinear programming problems. By using surjective mapping the original constrained optimization problem is transformed to a problem in a new space with only equality constraints For the numerical solution of the latter problem the stable version of the gradient-projection and Newtons methods are used. After inverse transformation to the original space a family of numerical methods for solving optimization problems with equality and inequality constraints is obtained. The proposed algorithms are based on the numerical integration of the systems of ordinary differential equations. These algorithms do not require feasibility of starting and current points, but they preserve feasibility. As a result of space transformation the vector fields of differential equations are changed and additional terms are introduced which serve as a Barrier preventing the trajectories from leaving the feasible set. A proof of co...

Stable barrier-projection and barrier-Newton methods in linear programming

The present paper is devoted to the application of the space transformation techniques for solving linear programming problems. By using a surjective mapping the original constrained optimization problem is transformed to a problem in a new space with only equality constraints. For the numerical solution of the latter problem the stable version of the gradient-projection and Newtons methods are used. After an inverse transformation to the original space a family of numerical methods for solving optimization problems with equality and inequality constraints is obtained. The proposed algorithms are based on the numerical integration of the systems of ordinary differential equations. These algorithms do not require feasibility of the starting and current points, but they preserve feasibility. As a result of a space transformation the vector fields of differential equations are changed and additional terms are introduced which serve as a barrier preventing the trajectories from leaving the feasible set. A proof of a convergence is given.

A relaxation method for solving problems of non-linear programming☆

Abstract A RELAXATION method is described for finding the local extrema in the general problem of non-linear programming. The convergence is proved, the convergence rate of the continuous and discrete versions of the method is investigated, and an extension to the case of finding saddle-points is given. The results of numerical computations are quoted.

Stable Barrier-Projection and Barrier-Newton Methods for Linear and Nonlinear Programming

A space transformation technique is used for the reduction of constrained minimization problems to minimization problems without inequality constraints. The continuous and discrete versions of stable barrier-projection method and Newton’s method are applied for solving such reduced LP and NLP problems. The space transformation modifies these methods and introduces additional matrices which play the role of a multiplicative barrier, preventing the trajectories from crossing the boundary of the feasible set. The proposed algorithms are based on the numerical integration of systems of ordinary differential equations. These algorithms do not require feasibility of starting and current points, but they preserve feasibility. Some results about convergence rate analysis for continuous and discrete versions of the methods are presented. We describe primal barrier-projection methods, primal barrier-Newton methods and primal-dual barrier-Newton methods. For LP we develop dual barrier-projection and barrier-Newton methods.

Numerical methods of solving some operational research problems

Abstract METHODS of finding the minima of convex functions with constraints of the recursive type on the range of variation of the argument, are proposed. A generalization of the algorithms for minimizing non-smooth functions is given. The methods are used to solve discontinuous games and to find saddle points. Some results of numerical calculations are presented.

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.

General lagrange-type functions in constrained global optimization part II: Exact auxiliary functions

This paper is a continuation of [13]. For each constrained optimization problem we consider certain unconstrained problems, which are constructed by means of auxiliary (Lagrange-type) functions. We study only exact auxiliary functions, it means that the set of their global minimizers coincides with the solution set of the primal constrained optimization problem. Sufficient conditions for the exactness of an auxiliary function are given. These conditions are obtained without assumption that the Lagrange function has a saddle point. Some examples of exact auxiliary functions are given.

Space-Transformation Technique: The State of the Art

In this paper we give an overview of some current approaches to LP and NLP based on space transformation technique. A surjective space transformation is used to reduce the original problem with equality and inequality constraints to a problem involving only equality constraints. Continuous and discrete versions of the stable gradient projection method and the Newton method are used for treating the reduced problem. Upon the inverse transformation is applied to the original space, a class of numerical methods for solving optimization problems with equality and inequality constraints is obtained. The following algorithms are presented: primal barrier-projection methods, dual barrier-projection methods, primal barrier-Newton methods, dual barrier-Newton methods and primal-dual barrier-Newton methods. Using special space transformation, we obtained asymptotically stable interior-infeasible point algorithms. The main results about convergence rate analysis are given.

Exact Auxiliary Functions in Non-Convex Optimization

A function is said to be on exact auxiliary function (EAP). if the set of global minimizers of this function coincides with the global solution set of initial optimization problem. Sufficient conditions for exact equivalence of constrained minimization problem and minimization of EAP are provided. Paper presents two classes of EAP for a nonlinear programming problem without assumption that the problem has a saddle point of Lagrange function.

DUAL BARRIER-PROJECTION METHODS IN LINEAR PROGRAMMING

A surjective space transformation technique is used to convert an original dual linear programming problem with equality and inequality constraints into a problem involving only equality constraints. Continuous and discrete versions of the stable gradient projection method are applied to the reduced problem. The numerical methods involve performing inverse transformations. The convergence rate analysis for dual linear programming methods is presented. By choosing a particular exponential space-transformation function we obtain the dual a ne scaling algorithm. Variants of methods which have linear local convergence are given.