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SOSC second-order sufficient condition

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The Second-Order Sufficient Condition (SOSC) is a concept in mathematical optimization that helps determine whether a critical point of a function is a local minimum, maximum, or a saddle point. The SOSC provides a higher level of certainty in characterizing critical points compared to the First-Order Sufficient Condition (FOSC).

Let's consider a function f(x), where x is a vector of variables. To examine the behavior of f(x) at a critical point, we need to analyze the second-order derivatives of f(x). The Hessian matrix H(x) of f(x) is a matrix that contains the second-order partial derivatives of f(x) with respect to each pair of variables.

The SOSC states that if a critical point x* satisfies the following conditions:

  1. The gradient of f(x*) is zero (i.e., ∇f(x*) = 0).
  2. The Hessian matrix H(x*) is positive definite (i.e., all eigenvalues of H(x*) are positive) or negative definite (i.e., all eigenvalues of H(x*) are negative), then x* is a local minimum or maximum, respectively.

The positive definiteness of the Hessian matrix ensures that the function is strictly convex (for a local minimum) or strictly concave (for a local maximum) near the critical point x*. In other words, the function is "bowl-shaped" or "upside-down bowl-shaped" in the vicinity of x*, indicating that it is curving in one direction and that there are no other critical points nearby.

It is important to note that if the Hessian matrix is neither positive definite nor negative definite, the SOSC does not provide conclusive information about the critical point. In this case, the critical point is classified as a saddle point, where the function neither reaches a local minimum nor a local maximum. The Hessian matrix may have both positive and negative eigenvalues, indicating a mixture of concave and convex regions around the critical point.

To summarize, the SOSC gives additional information about the nature of critical points by examining the second-order behavior of the function. If the SOSC is satisfied, it guarantees the existence of a local minimum or maximum. If the conditions are not met, further analysis or additional conditions are required to classify the critical point accurately.