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

Neste trabalho, estudaremos o método Trust-region para resolver um problema de otimizaçãocom restrições simples, ou seja estamos interessados em construir um algoritmopara resolver o seguinte problema: Encontrar x 2tal que f(x) f(x), para todox 2, onde= fx 2 Rn=Li xi Ui; Li; Ui 2 Rg, e a função f é suposta ser duasvezes continuamente diferenciável no conjunto factível, precisamos achar x tal que:x = arg minx2f(x). A partir de um ponto inicial e fazendo uso do método Trust-Region, geraremos uma sequência de pontos fxgk tal que limk!1xk = x, cada ponto éobtido da seguinte maneira, xk+1 = xk + sk e sk é solução do seguinte subproblema:sk = arg minkxxkkkLxUf(xk) +Drf(xk); x xkE+12Dx xk;Hk(x xk)EHk é uma aproximação da matriz hessiana no ponto xk. O método do gradiente projetadovai ser usado para resolver o subproblema, isso vai garantir que as iterações sejam semprepontos factíveis.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