However, fcn2optimexpr does not support AD. For example, to use the besselj function, fun2 = y^2),x,y) įcn2optimexpr allows you to use unsupported operations in the problem-based approach. What if you have a black-box function, one for which the underlying code might not even be in MATLAB? Or what if you simply have a function that is not supported for problem-based optimization, such as a Bessel function? In order to include such functions in the problem-based approach, you convert the function to an optimization expression using the fcn2optimexpr function. So far I have talked about automatic differentiation for supported operations. AD is slower when the problem has few variables and has complicated functions. Generally, AD is faster than finite differences when the problem has a large number of variables and is sparse. For complicated expressions, evaluating the automatic derivatives can be even more time-consuming than evaluating finite differences. This is not to say that AD always speeds a solver. With AD and supported functions, solvers do not need to take finite difference steps, so the derivative estimation process takes fewer function evaluations and is more accurate. For problems with a large number of variables, this process requires a large number of function evaluations. The solver evaluates n finite differences of this form by default, where n is the number of problem variables. Without AD, nonlinear solvers estimate gradients by finite differences, such as $(f(x \delta e_1) - f(x))/\delta ,$ where $e_1$ is the unit vector (1,0.,0). What Good is Automatic Differentiation?ĪD lowers the number of function evaluations the solver takes. See Supported Operations on Optimization Variables and Expressions. The list of supported operators includes polynomials, trigonometric and exponential functions and their inverses, along with multiplication and addition and their inverses. To use these rules of differentiation, the software has to have differentiation rules for each function in the objective or constraint functions. Currently, Optimization Toolbox uses only "backward" AD. The details of the process of calculating the gradient are explained in Automatic Differentiation Background, which describes the "forward" and "backward" process used by most AD software. The way that solve and prob2struct convert optimization expressions into code is essentially the same way that calculus students learn, taking each part of an expression and applying rules of differentiation. For example, to minimize the test function $) = $ The problem-based approach to optimization is to write your problem in terms of optimization variables and expressions. What Good is Automatic Differentiation?.Actually, as you'll see later, you have to specify some name-value pairs in order not to have the solver use AD. This process is transparent you do not have to write any special code to use AD. The "general nonlinear" phrase means that automatic differentiation applies to problems that fmincon and fminunc solve, which are general constrained or unconstrained minimization, as opposed to linear programming or least-squares or other problem types that call other specialized solvers.Īutomatic differentiation, also called AD, is a type of symbolic derivative that transforms a function into code that calculates the function values and derivative values at particular points. In a nutshell, as long as your function is composed of elementary functions such as polynomials, trigonometric functions, and exponentials, Optimization Toolbox calculates and uses the gradients of your functions automatically, with no effort on your part. I will explain what all of those words mean. However, with R2020b, the problem-based approach uses automatic differentiation for the calculation of problem gradients for general nonlinear optimization problems. How do you give a gradient to a solver along with the function? Until recently, you had to calculate the gradient as a separate output, with all the pain and possibility of error that entails. Optimization Toolbox algorithms are based on more sophisticated algorithms than this, yet these more sophisticated algorithms also benefit from a gradient. This is easy to understand: the gradient points uphill, so if you travel in the opposite direction, you generally reach a minimum. You may know that solving an optimization problem, meaning finding a point where a function is minimized, is easier when you have the gradient of the function. This column is written by Alan Weiss, the writer for Optimization Toolbox documentation.
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