# What are the general steps to check if the given solution is optimal for given problem?

Could you describe step-by-step how to check if given vector is an optimal solution to the problem?

For example:

Check if the point [1, 1, 0] is an optimal solution for the problem

minimize $$x^2+2y+(z-1)^2$$

subject to:

$$x^2+y^2\leq2$$

$$x\geq0$$

$$y\geq0$$

$$z\geq0$$

(I just made the problem up, it probably doesn't make much sense. It's just to visualize what I'm talking about)

The first thing to notice is that the problem decomposes into disjoint subproblems: one involving $$x$$ and $$y$$, and one involving $$z$$ only. For the $$z$$ subproblem, you want to minimize $$(z-1)^2$$ subject to $$z \ge 0$$, and $$z=1$$ is the unique optimal solution, so your $$(1,1,0)$$ with $$z=0$$ is not optimal.