Cocalc Numerical Optimization Ipynb

This notebook explores numerical optimization techniques available in SageMath, from finding minima and maxima of functions to solving linear and integer programming problems. The history of optimization is rich and spans millennia. Ancient Greeks studied isoperimetric problems (finding shapes with maximum area for fixed perimeter). Isaac Newton and Gottfried Leibniz developed calculus in the 17th century, providing tools for finding extrema via derivatives. The simplex algorithm, revolutionary for linear programming, was developed by George Dantzig in 1947. Leonid Kantorovich and Tjalling Koopmans won the Nobel Prize in Economics (1975) for their work on optimal resource allocation.

Modern optimization combines classical analysis, linear algebra, and computational algorithms. Optimization problems generally take the form: Unconstrained optimization: No constraints on xxx Linear programming (LP): fff, gig_igi​, hjh_jhj​ all linear 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. By the end of this comprehensive tutorial, you will: Master linear programming fundamentals and mathematical formulation

Understand the geometric interpretation of LP problems and feasible regions Apply duality theory and perform sensitivity analysis Solve real-world optimization problems in production, transportation, and finance 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]. In this notebook I implement the classical mean-variance optimization (MVO) algorithm that finds the optimal weights of a portfolio.

For a theoretical background see modern portfolio theory. Probability density of the tangency portfolio In the following I define the investment horizon of 1 month and the risk free monthly return Rf. I also load the time series of daily prices from filename, and print the names of the stocks considered in the analysis. Since the number of stocks is small, it could be useful to plot the normalized time series and the correlation matrix of the log-returns. Let us indicate the stock price at time ttt with StS_tSt​.

We can now recall some important definitions: This notebook contains Part 4 from the main SageMath_Calculus_Derivatives_Optimization notebook. For the complete course, please refer to the main notebook: SageMath_Calculus_Derivatives_Optimization.ipynb Identify the quantity to optimize and constraints Set up variables and express the objective function Find the domain of the objective function

This class, Optimization, is the eighth of eight classes in the Machine Learning Foundations series. It builds upon the material from each of the other classes in the series -- on linear algebra, calculus, probability, statistics, and algorithms -- in order to provide a detailed introduction to training machine... Through the measured exposition of theory paired with interactive examples, you’ll develop a working understanding of all of the essential theory behind the ubiquitous gradient descent approach to optimization as well as how to... You’ll also learn about the latest optimizers, such as Adam and Nadam, that are widely-used for training deep neural networks. Over the course of studying this topic, you'll: Discover how the statistical and machine learning approaches to optimization differ, and why you would select one or the other for a given problem you’re solving.

Understand exactly how the extremely versatile (stochastic) gradient descent optimization algorithm works, including how to apply it This notebook contains Part 3 from the main SageMath_Calculus_Derivatives_Optimization notebook. For the complete course, please refer to the main notebook: SageMath_Calculus_Derivatives_Optimization.ipynb Critical Point: Where f′(x)=0f'(x) = 0f′(x)=0 or f′(x)f'(x)f′(x) is undefined Local Maximum: f(c)≥f(x)f(c) \geq f(x)f(c)≥f(x) for all xxx near ccc Local Minimum: f(c)≤f(x)f(c) \leq f(x)f(c)≤f(x) for all xxx near ccc

Write a function called newton which takes input parameters fff, x0x_0x0​, hhh (with default value 0.001), tolerance (with default value 0.001) and max_iter (with default value 100). The function implements Newton's method to approximate a solution of f(x)=0f(x) = 0f(x)=0. In other words, compute the values of the recursive sequence starting at x0x_0x0​ and defined by Use the central difference formula with step size hhh to approximate the derivative f′(xn)f'(x_n)f′(xn​). The desired result is that the method converges to an approximate root of f(x)f(x)f(x) however there are several possibilities: The sequence reaches the desired tolerance ∣f(xn)∣≤tolerance|f(x_n)| \leq \mathtt{tolerance}∣f(xn​)∣≤tolerance and newton returns the value xnx_nxn​.

The number of iterations exceeds the maximum number of iterations max_iter, the function prints the statement "Maximum iterations exceeded" and returns None. A zero derivative is computed f′(xn)=0f'(x_n) = 0f′(xn​)=0, the function prints the statement "Zero derivative" and returns None.

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