3.6. Objective

Every model has at most one objective, and that objective must evaluate to a scalar. The model sense is minimization by default.

For a first model, it helps to think of the objective as one final scalar built from three layers:

1. a fit term that measures how well the model matches the data

2. a regularization term that expresses what kind of solution you prefer

3. an optional structural term for extra smoothness, barriers, or domain-specific shaping

Objective rules at a glance

Rule	What it means in practice
one objective per model	combine fit, regularization, and other terms into one final scalar expression
scalar-valued expression	aggregate vectorized losses with functions such as sum()
minimization by default	write the quantity you want to minimize directly

3.6.1. Build the Objective in Layers

Start with the simplest fit term that captures the task. Then add regularization only if it helps resolve ambiguity or improves the kind of solution you want. If the model still needs extra structure, add a third term and sum everything into one scalar before calling Model.setObjective().

\[\text{data fit} + \text{regularization} + \text{optional structural term}\]

For example, a common beginner pattern is a least-squares fit with L1 regularization and an optional smoothness term:

fit = 0.5 * admm.sum(admm.square(A @ x - b))
regularization = lam * admm.norm(x, ord=1)
structure = mu * admm.sum(admm.square(D @ x))

model.setObjective(fit + regularization + structure)

If you do not need the third term, leave it out. If your expression is already scalar, such as admm.norm(A @ x - b, ord=1), you can pass it directly to Model.setObjective(). For matrix models, the optional structural term might instead be a barrier or spectral expression such as -admm.log_det(X) + admm.trace(S @ X).

3.6.2. Common Objective Ingredients

Common objective functions are summarized below. In practice, most models choose one fit term from this table and then add one regularizer before exploring more specialized structure.

Typical objective functions

Ingredient family	Example
linear objective	c.T @ x
least-squares data fit	0.5 * admm.sum(admm.square(A @ x - b))
quadratic regularization	0.5 * admm.sum(admm.square(x))
sparsity regularization	lam * admm.norm(x, ord=1)
robust loss	admm.sum(admm.huber(A @ x - b))
classification loss	admm.sum(admm.logistic(z))
matrix barrier or spectral term	-admm.log_det(X) + admm.trace(S @ X)
user-defined proximal term	admm.sum(admm.square(x - y)) + lam * L0Norm(x)

For the full atom catalog, see Supported Building Blocks. For feasibility conditions that accompany the objective, see Constraints.