# Convex optimization: convergence of a fixed-point algorithm for a concave objective function

Suppose we have an objective function. $$max_ limits {x} sum_ limits {i} f_i (x_i)$$ with the restriction that $$x_i geq 0, sum_ limits {i} x_i = 1$$.

Each function $$f_i$$ It is continuous and differentiable within the limit, with the following properties: $$f_i (x_i) geq 0$$, $$f_i (0) = 0$$, $$f & # 39; _i (x)> 0$$, $$f & # 39; _i (1)> 0$$, $$f_i & # 39; & # 39; (x_i) <0$$.

By adding a Lagrange term and taking the partial derivative, we obtain

$$frac { partial sum_i f_i (x_i) + lambda (1 – sum_i x_i)} { partial x_i} = f & # 39; _i (x_i) – lambda$$.

By putting the previous expression to zero, moving $$lambda$$ to the right side of the equation and multiplying a $$x_i$$ With each side, we would arrive at a fixed point update.
$$x_i ^ mbox {new} propto x_i cdot f & # 39; _i (x_i)$$.

Would this iterative algorithm monotonously increase the concave objective and is it guaranteed to converge?