[PDF] first order sufficient condition

What is the first order necessary condition for optimality?

where A is an m × n matrix with m ? n, and b is a vector of length m. Assume that the rank of A is equal to n. We can write down the first-order necessary condition for optimality: If x ? is a local minimizer, then ?f(x ?) = 0. Is this also a sufficient condition? Yes, this is also sufficient.

What is a second order sufficient condition (SOSC)?

In general, the necessary conditions are not sufficient for optimality and additional information is required, such as the Second Order Sufficient Conditions (SOSC). For smooth functions, SOSC involve the second derivatives, which explains its name.

Is f(x ) a sufficient condition for optimality?

Assume that the rank of A is equal to n. We can write down the first-order necessary condition for optimality: If x ? is a local minimizer, then ?f(x ?) = 0. Is this also a sufficient condition? Yes, this is also sufficient. The quick way to see this is simply to see that f(x) is convex, so its extrema are its minimizers.

Which conditions are sufficient for optimality?

The necessary conditions are sufficient for optimality if the objective function of a maximization problem is a differentiable concave function, the inequality constraints are differentiable convex functions, the equality constraints are affine functions, and Slater's condition holds.