To reformulate the unconstrained optimization problem

$\min \|v(x)\|_{\infty}$

as a smooth constrained optimization problem, we start by understanding the $l_{\infty}$ norm. The $l_{\infty}$ norm of a vector $v(x) \in \mathbb{R}^m$ is defined as:

$\|v(x)\|_{\infty} = \max_{i} |v_i(x)|$

where $v_i(x)$ denotes the $i$-th component of the vector $v(x)$.

Reformulation Steps

1. Introduce Auxiliary Variable: Let $t$ be a new variable that will represent the maximum absolute value of the components of $v(x)$. Hence, we introduce the constraint:

   $|v_i(x)| \leq t, \quad \forall i = 1, \ldots, m$

2. Formulate the Constrained Problem: The original problem can now be reformulated as:

   $\min t$

   subject to the constraints:

   $|v_i(x)| \leq t \quad \forall i = 1, \ldots, m$

   This formulation maintains smoothness since $v(x)$ is smooth, and the constraints are also well-defined.

Final Formulation

The smooth constrained optimization problem can be stated as:

\text{minimize} & \quad t \\ \text{subject to} & \quad |v_i(x)| \leq t, \quad \forall i = 1, \ldots, m \end{align*}$$ This approach effectively captures the original unconstrained problem using smooth constraints, thus facilitating the use of optimization techniques that rely on constrained formulations. If you have any questions or need further details on this topic, feel free to ask! ### Additional Questions for Exploration: 1. How does the choice of the $$l_{\infty}$$ norm affect the optimization landscape? 2. What methods can be employed to solve the resulting constrained optimization problem? 3. How would you reformulate this problem if $$v(x)$$ were not smooth? 4. What are the implications of introducing auxiliary variables in optimization? 5. Can this formulation be extended to other norms, such as $$l_1$$ or $$l_2$$? ### Tip: When working with constrained optimization problems, ensure that you understand the properties of the constraints and their implications on the feasibility and optimality of solutions.