Under what conditions does a quadratic function p(𝐱)=𝐱^T K 𝐱-2 𝐱^T 𝐟+c have a finite global maxi

Under what conditions does a quadratic function p(𝐱)=𝐱^T K...

Key Concepts

Concavity and Definiteness

In optimizing quadratic functions, the curvature of the function is governed by the definiteness of the matrix in the quadratic term. For a quadratic function to have a finite global maximum, it must be concave; that is, the matrix multiplying the quadratic term must be negative semidefinite. When this matrix is strictly negative definite, the function is strictly concave, ensuring a unique global maximum. In the case of negative semidefiniteness (not strictly negative), additional conditions are required to prevent directions in which the function might become unbounded.

Stationarity and Critical Points

Finding the maximizer involves calculating the gradient of the quadratic function and setting it equal to zero. This yields a system of linear equations whose solution (if it exists) provides the candidate for the global maximum. For a strictly concave quadratic function, this stationary point is unique and indeed the global maximizer. However, if the matrix is only negative semidefinite, the solution must lie in the range of the matrix to guarantee that the maximum is finite.

Completing the Square

The technique of completing the square is pivotal when optimizing quadratic functions. By rewriting the function in a form that isolates the squared term, one can clearly see how the variable deviations contribute to the function value. This method not only simplifies the process of identifying the maximizer but also helps in computing the maximum value by extracting the constant term and demonstrating that deviations from the critical point decrease the function value when the matrix is negative semidefinite.

Handling Singular or Semidefinite Cases

When the matrix in the quadratic term is not invertible (i.e., is only negative semidefinite and singular), the existence of a finite global maximum depends on the alignment of the linear term with the null space of the matrix. The linear term must vanish in the directions corresponding to the nullspace; otherwise, the function could be unbounded above in those directions. This compatibility condition is essential to ensure that the quadratic function is properly bounded and that a maximizer exists.

Transcript

00:01 Alright, in this question, we need to decide based on this format of an equation, whether the parabola is going to open up.

00:09 Okay, so up might look like a smiley face.

00:13 So this would be a parabola opening up, and this would be a parabola opening down.

00:19 Okay, so let's first take a look at the coefficient a.

00:23 Okay, so if we have an equation that looks like y equals, x squared.

00:33 This has a coefficient, or an a of 1.

00:39 It's a positive coefficient.

00:42 1 is a positive number.

00:44 And so let's take a look at what that graph might look like.

00:48 So if we were to substitute 0 in for our equation, 0 squared is 0.

00:54 So we know that this is one point on the graph.

00:57 What happens if we go in the positive direction? what about 1 squared? well, 1 squared is going to be 1, and 2 squared is going to be 4.

01:10 Okay, let's take a look at the negative direction.

01:13 What happens if we substitute negative 1? okay, so negative 1 squared, hmm, well, that's also 1.

01:20 Negative 2 squared, that's also positive 4.

01:26 That's a little too high.

01:30 Okay, so if we have a equals 1, we're a positive a, okay, positive a in this formula, then our parabola is going to look like this and it's going to open up like a smiley face...

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