Find the second derivative

Now that you have the derivative of your objective function, you can calculate the second order derivative. You'll do so symbolically using sympy.

The values for dc_dq and q_opt, and the objective function c with the quantity variable q from the previous exercise are available to you.

Cet exercice fait partie du cours

Introduction to Optimization in Python

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Exercice interactif pratique

Essayez cet exercice en complétant cet exemple de code.

# Find the second derivative
d2c_dq2 = ____

# Substitute the optimum into the second derivative
sol = ____(____)
print(f"The 2nd derivative at q_opt is: {sol}")

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Cet exercice fait partie du cours

Introduction to Optimization in Python

IntermédiaireNiveau de compétence

4.8+

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This chapter introduces optimization, its core components, and its wide applications across industries and domains. It presents a quick, exhaustive search method for solving an optimization problem. It provides a mathematical primer for the concepts required for this course.

Exercise 1: Introduction to mathematical optimization Exercise 2: Understanding mathematical optimization Exercise 3: Applying an objective function Exercise 4: Exhaustive search method Exercise 5: Univariate optimization Exercise 6: Finding the derivative Exercise 7: Find the second derivative

Exercice en cours

Exercise 8: Multivariate optimization Exercise 9: Partial derivatives with SymPy Exercise 10: Limitations of differentiation

This chapter covers solving unconstrained and constrained optimization problems with differential calculus and SymPy, identifying potential pitfalls. SciPy is also introduced to solve unconstrained optimization problems, in single and multiple dimensions, numerically, with a few lines of code. The chapter goes on to solve linear programming in SciPy and PuLP.

Exercise 1: Unconstrained optimization Exercise 2: Finding the maxima Exercise 3: Multivariate optimization Exercise 4: Bound-constrained optimization Exercise 5: Working with Bounds Exercise 6: Handling hard inequalities Exercise 7: Linear programming Exercise 8: What is linear programming?Exercise 9: PuLP for linear optimization Exercise 10: Handling multiple elements

This chapter introduces convex-constrained optimization problems with different constraints and looks at mixed integer linear programming problems, essentially linear programming problems where at least one variable is an integer.

Exercise 1: Convex constrained-optimization Exercise 2: Interpreting indifference curves Exercise 3: Visualize the indifference curve Exercise 4: Utility maximization Exercise 5: Convex-constrained optimization with inequality constraints Exercise 6: Interior or corner?Exercise 7: Linear constrained biscuits Exercise 8: Nonlinear constrained biscuits Exercise 9: Mixed integer linear programming (MILP)Exercise 10: Adjusting MILP Exercise 11: Understanding integrality

This chapter covers finding the global optimum when multiple good solutions exist. We will conduct sensitivity analysis and learn linearization techniques that reduce non-linear problems to easily solvable ones with SciPy or PuLP. In terms of applications, we will solve an HR allocation with training costs problem and capital budgeting with dependent projects.

Exercise 1: Transforming non-linear problems Exercise 2: Solving the capital budgeting problem Exercise 3: Allocating resources Exercise 4: Global optimization in SciPy Exercise 5: Find the global optimum Exercise 6: Using callback Exercise 7: Sensitivity analysis in PuLP Exercise 8: Slack and shadow price Exercise 9: Shadow prices with juice Exercise 10: The diet problem revisited Exercise 11: Congratulations!