Cocalc Tutorial 22a Optimization Optim Ipynb

This notebook uses the Optim.jl package which has general purpose routines for optimization. (As alternatives, consider the NLopt.jl and JuMP.jl) For linear-quadratic problems (mean-variance, least squares, etc), it is probably more efficient use specialized routines. This is discussed in another notebook. finds the x value (in the interval [a,b]) that minimizes fn1(x,0.5). The x->fn1(x,0.5) syntax makes this a function of x only.

The output (Sol) contains a lot of information. If you prefer to give a starting guess c instead of an interval, then supply it as as a vector [c]. Until now, you've always used Gradient Descent to update the parameters and minimize the cost. In this notebook, you will learn more advanced optimization methods that can speed up learning and perhaps even get you to a better final value for the cost function. Having a good optimization algorithm can be the difference between waiting days vs. just a few hours to get a good result.

Gradient descent goes "downhill" on a cost function JJJ. Think of it as trying to do this: Notations: As usual, ∂J∂a=\frac{\partial J}{\partial a } = ∂a∂J​= da for any variable a. To get started, run the following code to import the libraries you will need. A simple optimization method in machine learning is gradient descent (GD). When you take gradient steps with respect to all mmm examples on each step, it is also called Batch Gradient Descent.

Lecture slides for UCLA LS 30B, Spring 2020 Be able to explain the significance of optimization in biology, and give several examples. Be able to describe the main biological process that underlies all optimization problems in biology. Be able to distinguish local maxima and local minima of a function from global extrema. Know the significance of local maxima in evolution. The methods learned in Chapter 4 of the text for finding extreme values have practical applications in many areas of life.

In this lab, we will use SageMath to help with solving several optimization problems. The following strategy for solving optimization problems is outlined on Page 264 of the text. Read and understand the problem. What is the unknown? What are the given quantities and conditions? Draw a picture.

In most problems it is useful to draw a picture and identify the given and required quantities in the picture. Introduce variables. Asign a symbol for the quantity, let us call it QQQ, that is to be maximized or minimized. Also, select symbols for other unknown quantities. Use suggestive notation whenever possible: AAA for area, hhh for height, rrr for radius, etc. All material moved to the more comprehensive CoCalc Manual

CoCalc is a cloud-based service that provides infrastructure and services that are useful for running courses based on Jupyter Notebooks. It is used for teaching by Universities around the world. All material moved to the more comprehensive CoCalc Manual For a list of authors see the contributors section. of linear-quadratic problems, using the OSQP.jl package. An alternative, Clarabel.jl is discussed briefly at the end of the notebook.

The methods illustrated here are well suited for cases when the objective is linear-quadratic, for instance, the portfolio variance (w′Σww'\Sigma ww′Σw) or when the estimation problem is based on minimizing the sum of squared... The OSQP.jl package is tailor made for solving linear-quadratic problems (with linear restrictions). It solves problems of the type min⁡0.5θ′Pθ+q′θ\min 0.5\theta' P \theta + q' \thetamin0.5θ′Pθ+q′θ subject to l≤Aθ≤ul \leq A \theta \leq ul≤Aθ≤u, where θ\thetaθ is a vector of choice variables.

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