Lagrange Multipliers for Equality-Constrained Optimization

All flashcards

What is an objective function?

The function whose value is to be maximized or minimized, such as $f(\mathbf{x})$.

What is a feasible set?

The collection of variable values that satisfy all stated constraints.

What does a gradient represent geometrically?

It points toward greatest local increase and is perpendicular to the function's level sets.

What is the Lagrangian for one equality constraint?

A combined function such as $\mathcal{L}(\mathbf{x},\lambda)=f(\mathbf{x})+\lambda g(\mathbf{x})$.

Why differentiate the Lagrangian with respect to the multiplier?

That derivative equals the constraint function, so setting it to zero enforces the constraint.

Quiz questions

1. At a regular constrained extremum with several equality constraints, how is the objective gradient related to the constraint gradients?Answer: It is a linear combination of the constraint gradients.
2. For a constraint map with $M$ equality constraints, what does full row rank of its Jacobian mean?Answer: The $M$ constraint gradients are linearly independent.
3. A point satisfies the Lagrange stationarity equations. What can be true of the Lagrangian in the joint variable-multiplier space?Answer: The stationary point can be a saddle point.
4. Which framework extends multiplier conditions to problems with inequality constraints?Answer: The Karush–Kuhn–Tucker conditions.
5. What does a local sufficient second-order test establish, when its assumptions hold?Answer: A local constrained optimum near the candidate point.