Understanding the Gradient and Hessian Matrix in Multivariate Optimization

Key Takeaways

• Multivariate optimization problems require a solution to more than one variable. 

• The gradient is the vector of the first-order derivative while the Hessian matrix represents the matrix of the second-order derivative. 

• CFD tools facilitate the computation of these optimization methods for complex design problems. 

Computational fluid dynamics (CFD) tools are increasingly being used for the design and performance optimization of engineering systems – in particular, for fluid systems.  Many of the problems in fluid dynamics have multiple decision variables that require multivariate optimization to enhance system efficiency. For such optimizations, it is usually easier to evaluate the solution through the gradient and Hessian matrix. In this article, we will explore these methods in detail. 

Understanding the Gradient and Hessian Matrix

The multivariate function in an optimization problem can be expressed as:

 y= f (x1, x2, x3, …., xn)

Note that y is the function of variables x1, x2, and so on. The equation denotes that there is n number of variables that influences the optimization of function y. The gradient and Hessian matrix evaluate the derivatives of the function when more than one variable is involved, i.e., n variables.  

The Gradient 

The gradient indicates the rate of change of a scalar-valued function. For a multivariate function, the changes are required to be analyzed in three dimensions. For n variables, the gradient specifies the change (ascent or descent) along a 3D space.

Mathematically, the gradient can be identified as the derivate of the function of n variables. This can be expressed as:

The solution to this function provides the fastest rate of change (maxima or minima) at a given point. This is an important concept in machine learning, signal processing, etc. 

Hessian Matrix

The Hessian matrix is a square matrix of the second-order partial derivative of the function we get from differentiating the gradient.  For a function f with n variables, the Hessian matrix Hf can be expressed as:

The determinant of the Hessian matrix in two dimensions (x, y) can be written as: 

This determinant is an important identifier of the local maxima/minima of the multivariate function. Here are the general rules to identify maxima/minima for function F and determinant D at a given point.

1. Maxima: F(xx)>0, D >0
2. Minima: F(xx) <0, D>0
3. Saddle point: D < 0
4. The result of D = 0 will require further examination to draw a conclusion. 

These readings are essential to determine the concavity and convexity of the function. 

Computing the Gradient and Hessian Matrix for Complex Optimization Problems

The understanding of local maxima/minima or saddle points and their rate of change – the critical findings of the gradient and Hessian matrix – are essential learnings of multivariate functions. This concept is critical for technologies like machine learning, where the neural network depends on these algorithms and variables for parameter updates.

For complex optimization problems in engineering design through the gradient and Hessian matrix analysis, Cadence’s CFD tools can play a vital role. Subscribe to our newsletter for the latest CFD updates or browse Cadence’s suite of CFD software, including Fidelity and Fidelity Pointwise, to learn more about how Cadence has the solution for you.