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Convex optimization is a subfield of optimization that deals with problems where the objective function is convex, and the feasible region is a convex set. This means for any two points within the region, the line segment connecting them lies entirely within the region. A function is convex if its second derivative is always greater or equal to zero—equating to a 'U' shaped curve. These types of problems are particularly useful because they have certain properties that allow for efficient solving techniques.

For example, unlike non-convex optimization problems, a local minimum in a convex optimization problem is also a global minimum. This guarantees that if we find a solution that satisfies the constraints and where the derivative equals to zero, we can be certain that it's the best possible solution. Continuing from the exercise, the given objective function, though not entirely convex due to the mixed norms, can be approached using methods from convex optimization when reformulated with additional variables and constraints.

The Lagrangian is a mathematical device that combines both the objective function and the constraints of the optimization problem into one expression. It introduces new variables, known as Lagrange multipliers, which are associated with each constraint. These multipliers have a physical interpretation as the 'price' you need to pay for a marginal increase in the constraint and they play a central role when transforming a constrained problem into one that can be analyzed using optimization techniques.

In our exercise, the Lagrangian was created by adding the product of the transpose of the Lagrange multipliers \(p_{i}\) with the difference between the new variable \(y_{i}\) and the linear transformation \(A_{i}x + b_{i}\) applied to the decision variable \(x\). This transformation allows us to solve for the dual problem, thereby providing an alternative and sometimes easier way to approach the original problem.

Infimal convolution is an operation used in convex analysis and optimization that combines two convex functions into one. It helps in finding the tightest lower bound—or infimum—for a given expression. Specifically, in the context of dual problem formulation, it allows us to decompose the Lagrangian of a primal problem and find the infimum with respect to each variable separately.

In the solution provided, it was utilized to isolate the variables \(x\) and \(y_i\) to find their respective infimum values. After taking the gradient of the Lagrangian with respect to \(x\) and setting it to zero, one can solve for \(x\), and similarly find the optimal value for \(y_i\) which turns out to be based on the norms of the Lagrange multipliers \(p_i\). This concept is very important because it simplifies the dual function derivation, allowing us to convert a possibly complex problem into a more tractable one.

Equality constraints are conditions that must be exactly met for a solution to be considered feasible in an optimization problem. These constraints are represented as equations that the decision variables must satisfy. In the Lagrangian formulation of an optimization problem, these equality constraints are incorporated using Lagrange multipliers, which adjust the objective function based on how severely a constraint would impact the solution if it were tightened or loosened.

In the provided exercise, the new variables \(y_i\) are related to the decision variable \(x\) through the equality constraints \(y_i = A_i x + b_i\). The presence of these constraints enables us to transform the original minimization problem into an equivalent problem that can be tackled using dual optimization techniques. Moreover, adhering to these constraints ensures that the dual function correctly represents the original problem's feasible region, crucial for finding accurate solutions.