Unconstrained Optimization:Why is there no "profit style" function in consumer theory?
The "profit" function for consumer choice does exist; it is the Lagrangean: 
\begin{equation}
\mathcal L(x;\lambda)=u(x)+\lambda[c-g(x)]\,,
\end{equation}
where utility $u(x)$ is to be maximized subject to budget constraint $c\ge g(x)$.

Note that $\max_x \mathcal L(x;\lambda)$ is an unconstrained optimization problem just like $\max_x\pi(x)$.