Um método trust-region para otimização com restrições fazendo uso do método gradiente projetado

Abstract

In this work, we study a trust-region method for solving optimization problems with simple constraints. We are interested in building an algorithm for the following problem: find x 2 such that f(x) f(x), 8x 2 , in which = fx 2 Rn=Li xi Ui; Li; Ui 2 Rg, and f is twice differentiable within the feasible set . Starting from an initial point, the trust-region method generates a sequence fxgk such that lim k!1 xk = x. The sequence is generated by the recursion xk+1 = xk + sk, in which sk is the solution of the following subproblem: sk = arg min kxxkkk LxU f(xk) + D rf(xk); x xk E + 1 2 D x xk;Hk(x xk) E In this expression, Hk is an approximation of the Hessian matrix on the point xk. The projected gradient method is used in order to solve the subproblem, in this way ensuring that all iterations generate feasible solutions.In this work, we study a trust-region method for solving optimization problems with simple constraints. We are interested in building an algorithm for the following problem: find x 2 such that f(x) f(x), 8x 2 , in which = fx 2 Rn=Li xi Ui; Li; Ui 2 Rg, and f is twice differentiable within the feasible set . Starting from an initial point, the trust-region method generates a sequence fxgk such that lim k!1 xk = x. The sequence is generated by the recursion xk+1 = xk + sk, in which sk is the solution of the following subproblem: sk = arg min kxxkkk LxU f(xk) + D rf(xk); x xk E + 1 2 D x xk;Hk(x xk) E In this expression, Hk is an approximation of the Hessian matrix on the point xk. The projected gradient method is used in order to solve the subproblem, in this way ensuring that all iterations generate feasible solutions.