SONC second-order necessary condition

In the context of optimization theory, the second-order necessary condition, also known as the second-order sufficient condition (SONC), is a condition that helps determine whether a critical point of a function is a local minimum or maximum. It provides additional information beyond the first-order necessary condition (the first derivative test) by considering the behavior of the second derivative of the function.

Let's consider a function �(�)f(x) that is twice differentiable on an interval �I containing a critical point �=�x=a where �′(�)=0f′(a)=0 and �′′(�)f′′(a) exists. The SONC states that if �′′(�)>0f′′(a)>0, then �(�)f(a) is a local minimum, and if �′′(�)<0f′′(a)<0, then �(�)f(a) is a local maximum. In other words, if the second derivative at the critical point is positive, the function is concave up, indicating a local minimum. If the second derivative is negative, the function is concave down, indicating a local maximum.

To understand why this condition holds, let's consider a Taylor series expansion of �(�)f(x) around the point �=�x=a:

�(�)=�(�)+�′(�)(�−�)+�′′(�)2!(�−�)2+�′′′(�)3!(�−�)3+…f(x)=f(a)+f′(a)(x−a)+2!f′′(a)​(x−a)2+3!f′′′(a)​(x−a)3+…

Since we are interested in the behavior near the critical point �=�x=a, we can ignore the higher-order terms (terms with degree 3 or higher) and focus on the quadratic term:

�(�)=�(�)+�′′(�)(�−�)22+…f(x)=f(a)+f′′(a)2(x−a)2​+…

Now, let's analyze the behavior of this quadratic term. If �′′(�)>0f′′(a)>0, then the term (�−�)222(x−a)2​ is always positive except when �=�x=a. Thus, for �x sufficiently close to �a, the quadratic term dominates and contributes positively to �(�)f(x). This indicates that �(�)f(x) is increasing around �=�x=a and implies that �(�)f(a) is a local minimum.

On the other hand, if �′′(�)<0f′′(a)<0, then the term (�−�)222(x−a)2​ is always negative except when �=�x=a. For �x sufficiently close to �a, the quadratic term dominates and contributes negatively to �(�)f(x). This indicates that �(�)f(x) is decreasing around �=�x=a and implies that �(�)f(a) is a local maximum.

It's important to note that the SONC only provides a sufficient condition for a local minimum or maximum. In some cases, the second derivative may be zero at a critical point, and further analysis is needed to determine the nature of the critical point using higher-order derivatives or other methods. Additionally, the SONC only considers local behavior and does not guarantee global optimality.

In summary, the second-order necessary condition (SONC) helps determine whether a critical point of a function is a local minimum or maximum based on the sign of the second derivative. If the second derivative is positive, the critical point is a local minimum, and if the second derivative is negative, the critical point is a local maximum.