Yuri G. Evtushenko

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...

Computation of exact gradients in distributed dynamic systems

A new and unified methodology for computing first order derivatives of functions obtained in complex multistep processes is developed on the basis of general expressions for differentiating a composite function. From these results, we derive the formulas for fap automatic differentiation of elementary functions, for gradients arising in optimal control problems, nonlinear programming and gradients arising in discretizations of processes governed by partial differential equations. In the proposed approach we start with a chosqn discretization scheme for the state equation and derive the exact gradient expression. Thus a unique discretization scheme is automatically generated for the adjoint equation For optimal control problems, the proposed computational formulas correspond to the integration of the adjoint system of equations that appears in Pontryagins maximum principle. This technique appears to be very efficient, universal, and applicable to a wide variety of distributed controlled dynamic systems an...

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.

FAD method to compute second order derivatives

We develop a unified methodology for computing second order derivatives of functions obtained in complex multistep processes and derive formulas for Hessians arising in discretization of optimal control problems. Where a process is described by continuous equations, we start with a discretization scheme for the state equations and derive exact gradient and Hessian expressions. We introduce adjoint systems for auxiliary vectors and matrices used for computing the derivatives. A unique discretization scheme is automatically generated for vector and matrix adjoint equations. The structure of the adjoint systems for some approximation schemes is found. The formulas for second derivatives are applied to examples.

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.

Approximating a solution set of nonlinear inequalities

In this paper we propose a method for solving systems of nonlinear inequalities with predefined accuracy based on nonuniform covering concept formerly adopted for global optimization. The method generates inner and outer approximations of the solution set. We describe the general concept and three ways of numerical implementation of the method. The first one is applicable only in a few cases when a minimum and a maximum of the constraints convolution function can be found analytically. The second implementation uses a global optimization method to find extrema of the constraints convolution function numerically. The third one is based on extrema approximation with Lipschitz under- and overestimations. We obtain theoretical bounds on the complexity and the accuracy of the generated approximations as well as compare proposed approaches theoretically and experimentally.

Method of non-uniform coverages to solve the multicriteria optimization problems with guaranteed accuracy

Application of the non-uniform coverage method to the multicriteria optimization problems was considered, and the concept of the ɛ-Pareto set was formulated and studied. An algorithm to construct a ɛ-Pareto set with a guaranteed accuracy ɛ was described. Efficient implementation of this approach was described, and the results of experiments were presented.

A deterministic method for constrained multicriteria optimization

A strictly defined notion of an approximate solution for a multicriteria optimization problem with functional constraints and a deterministic method for obtaining such approximations are presented. Unlike traditional algorithms for constrained multicriteria optimization the proposed method not only generates an approximation but also proves its accuracy.

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.