FONC (first-order necessary condition)

Introduction:

In mathematics, optimization theory deals with finding the best possible solution for a given problem. In optimization theory, we are interested in finding the optimal value of a function, subject to certain constraints. To solve an optimization problem, we need to check the optimality conditions. The optimality conditions can be divided into two types: necessary conditions and sufficient conditions. In this article, we will discuss the first-order necessary condition (FONC) for optimization problems.

First-order necessary condition (FONC):

The first-order necessary condition (FONC) is a necessary condition for a point to be a local minimum or maximum of a function. It tells us that if a function has a minimum or maximum at a point, then the derivative of the function at that point must be equal to zero.

Formally, let f be a function of n variables f(x1, x2, ..., xn), where xi is a real-valued variable. Let x* be a point in the domain of f where f has a local minimum or maximum. Then, if f is differentiable at x*, the gradient of f at x* (denoted by ∇f(x*)) must be equal to zero.

In other words, if f(x) has a local minimum or maximum at x*, then

∇f(x*) = 0

where ∇f(x*) is the gradient of f at x*.

The gradient of a function is a vector that points in the direction of the steepest increase of the function at a given point. In other words, it tells us the direction in which the function increases the fastest. The gradient of a function is given by the partial derivatives of the function with respect to each variable. The gradient of f(x1, x2, ..., xn) is denoted by ∇f(x1, x2, ..., xn) and is defined as

∇f(x1, x2, ..., xn) = (∂f/∂x1, ∂f/∂x2, ..., ∂f/∂xn)

where ∂f/∂xi is the partial derivative of f with respect to xi.

To understand the FONC, let's consider an example.

Example:

Suppose we want to find the minimum value of the function f(x) = x^2. We know that the derivative of this function is f'(x) = 2x. To find the minimum value of f(x), we set the derivative equal to zero:

f'(x) = 2x = 0

Solving this equation, we get x = 0. Therefore, the minimum value of f(x) is f(0) = 0. Note that the FONC is satisfied at x = 0 because f'(0) = 2(0) = 0.

Interpretation of the FONC:

The FONC tells us that if a function has a local minimum or maximum at a point, then the derivative of the function at that point must be equal to zero. This means that at a local minimum or maximum, the function is not changing. In other words, the function is "flat" at a local minimum or maximum.

The FONC is a necessary condition, but it is not sufficient. In other words, if the derivative of a function is equal to zero at a point, it does not necessarily mean that the function has a local minimum or maximum at that point. We need additional conditions to determine whether the point is a local minimum or maximum.

The FONC can also be interpreted geometrically. The gradient of a function at a point is a vector that points in the direction of the steepest increase of the function at that point. When the gradient of a function is equal to zero at a point, it means that there is no direction in which the function is increasing. Therefore, the function is either flat or decreasing in all directions at that point. This suggests that the point could be a local minimum or a maximum.

However, it is possible for a function to have a zero gradient at a point that is neither a local minimum nor a local maximum. For example, consider the function f(x) = x^3. This function has a zero gradient at x = 0, but it is neither a local minimum nor a local maximum. Therefore, we need additional conditions to determine whether a point is a local minimum or maximum.

Second-order necessary condition:

The second-order necessary condition is another necessary condition for a point to be a local minimum or maximum of a function. It tells us that if a function has a minimum or maximum at a point and the second derivative of the function at that point is positive (negative) for a minimum (maximum), then the point is a local minimum (maximum).

Formally, let f be a function of n variables f(x1, x2, ..., xn), where xi is a real-valued variable. Let x* be a point in the domain of f where f has a local minimum or maximum. If f is twice differentiable at x*, then the Hessian matrix of f at x* (denoted by Hf(x*)) must be positive definite (negative definite) for a minimum (maximum).

The Hessian matrix of a function is a matrix of second partial derivatives of the function. The Hessian matrix of f(x1, x2, ..., xn) is denoted by Hf(x1, x2, ..., xn) and is defined as

Hf(x1, x2, ..., xn) = | ∂^2f/∂x1^2 ∂^2f/(∂x1∂x2) ... ∂^2f/(∂x1∂xn) | | ∂^2f/(∂x2∂x1) ∂^2f/∂x2^2 ... ∂^2f/(∂x2∂xn) | | ... ... ... | | ∂^2f/(∂xn∂x1) ∂^2f/(∂xn∂x2) ... ∂^2f/∂xn^2 |

where ∂^2f/(∂xi∂xj) is the second partial derivative of f with respect to xi and xj.

The Hessian matrix is a symmetric matrix, which means that Hf(x*) is equal to its transpose. If the Hessian matrix is positive definite, it means that all the eigenvalues of the matrix are positive. If the Hessian matrix is negative definite, it means that all the eigenvalues of the matrix are negative.

To understand the second-order necessary condition, let's consider an example.

Example:

Suppose we want to find the minimum value of the function f(x) = x^4 - 2x^2 + 1. We know that the first derivative of this function is f'(x) = 4x^3 - 4x, and the second derivative is f''(x) = 12x^2 - 4. To find the minimum value of f(x), we set the first derivative equal to zero:

f'(x) = 4x^3 - 4x = 0

Solving this equation, we get x = 0 or x = ±1. To determine which point is a local minimum, we need to check the second derivative of f(x) at these points. We have:

f''(0) = 12(0)^2 - 4 = -4, which is negative. f''(1) = 12(1)^2 - 4 = 8, which is positive. f''(-1) = 12(-1)^2 - 4 = 8, which is positive.

Since f''(1) and f''(-1) are positive, this means that f(x) has a local minimum at x = ±1. Therefore, the minimum value of f(x) is f(±1) = -2.

Note that the first-order necessary condition alone is not sufficient to determine whether a critical point is a local minimum or maximum. In the above example, the first derivative of f(x) is equal to zero at x = 0, but this point is neither a local minimum nor a local maximum.

Moreover, the second-order necessary condition is also not sufficient to determine whether a critical point is a local minimum or maximum. There are cases where the second derivative is zero at a critical point, but it is neither a minimum nor a maximum. Such points are called inflection points.

Example:

Consider the function f(x) = x^3. The first derivative is f'(x) = 3x^2, and the second derivative is f''(x) = 6x. The critical point of f(x) is x = 0, where both the first and second derivatives are equal to zero. However, this point is neither a local minimum nor a local maximum. Instead, it is an inflection point where the function changes concavity.

In summary, the first-order necessary condition tells us that the gradient of a function must be zero at a local minimum or maximum. The second-order necessary condition tells us that if a function has a minimum or maximum at a point and the second derivative of the function at that point is positive (negative) for a minimum (maximum), then the point is a local minimum (maximum). However, neither condition is sufficient to determine whether a critical point is a local minimum or maximum. Additional analysis is required to determine the nature of the critical point.