Convex optimization is a branch of optimization that focuses on convex problems, where the objective function and the feasible region are both convex. These problems have unique global minima, and efficient algorithms exist for solving them.

A function is convex if, for any two points in its domain, the line segment connecting them lies above the graph of the function. A set is convex if, for any two points in the set, the line segment connecting them is entirely contained within the set.

Convexity plays an essential role in machine learning because many learning problems can be formulated as convex optimization problems. Examples include linear and logistic regression, support vector machines, and some formulations of neural network training.

There are several optimization algorithms specifically designed for convex problems, which can guarantee convergence to the global minimum.

Gradient descent is a first-order method that uses the gradient of the objective function to iteratively update the parameters. It converges to the global minimum for convex problems.

Newton's method is a second-order method that uses both the gradient and the Hessian of the objective function to update the parameters. It converges faster than gradient descent for convex problems.

Interior point methods, such as the barrier and primal-dual methods, are used for solving convex optimization problems with inequality constraints. They use a barrier function to transform the constrained problem into an unconstrained one.

Subgradient methods generalize gradient descent to non-differentiable convex functions by using subgradients instead of gradients. They have slower convergence rates than gradient-based methods but can handle non-differentiable problems.

Duality is an essential concept in convex optimization that allows one to transform a primal problem into a dual problem. The dual problem often has a simpler structure or can be solved more efficiently. Lagrange multipliers are used to enforce constraints in the dual problem.

Semidefinite programming (SDP) is a subfield of convex optimization that deals with optimization problems involving positive semidefinite matrices. SDP has applications in control theory, combinatorial optimization, and machine learning.