Affine-scaling Interior-point Methods For Optimization With Linear Constraints

Interior-point methods are very important optimization methods,which origin from linear programming.The methods obtain every iterate point by enforcing a inner point of the feasible region to the optimal point.They are very effective when used in constraint optimization.Nowadays,interior-point methods are widely used in nonliear optimization,combination optimization and complement optimization.Affine-scaling interior-point methods are usually used in nonlinear optimization which proves very effective.In this paper,we discuss optimization problems with linear constraint using affine-scaling interior-point methods.In the algorithm,when defining the trust region subproblem,we do not strictly require x_κ+ s_κin the inner of the boundary at every iterate point,only requiring l≤x_κ+ s_κ≤u.When x_κ+ s_κis on the edge of l < x_κ+ s_κ≤u,in order to ensure the iterate point in the inner of the boundary,we reduce s_κby factorσ_κ,and we give the formulation ofσ_κ.We obtain a trial step s_κthrough resovling trust region subproblem.If the improvement in f(x) has a sufficient reduction from x_κto x_κ+ s_κ,then the trial step s_κis accepted, and x_κ+1 = x_κ+ s_κ.Otherwise,we do not reduce the trust region radiusâ–³_κas usual trust region method,instead we will perform a backtracking line search along s_κsuch that f(x_κ+ωⁱs_κ) < f(x_κ),whereω∈(0,1) is a positive constant and isome positive integer,which will improve the efficiency of the algorithm.In this paper,we obtain global convergence under the standard conditions,and also we obtain local convergence under the rational conditions.At last,we do some numerical experiments with Matlab language.The numerical results shows that the algorithm is very effective.