Pyomo Workshop

Pyomo Workshop#

Welcome to the Pyomo Workshop - your comprehensive guide to optimization modeling in Python!

What is Pyomo?#

Pyomo (Python Optimization Modeling Objects) is a powerful, open-source software package for formulating and solving large-scale optimization problems. Developed at Sandia National Laboratories, Pyomo provides a collection of Python software packages that support a diverse set of optimization capabilities for formulating optimization models [].

Pyomo enables users to:

  • Model optimization problems using algebraic expressions in Python

  • Define optimization problems with linear, nonlinear, mixed-integer, and stochastic programming formulations

  • Solve problems using commercial and open-source optimization solvers

  • Analyze and visualize optimization results

Installation Instructions#

Before diving into Pyomo fundamentals, ensure you have the necessary software installed:

Python and Pyomo Installation#

  1. Install Python (version 3.8 or higher recommended)

    • Download from python.org or use a package manager like conda

  2. Install Pyomo

    pip install pyomo

  3. Install additional packages for this workshop

    pip install matplotlib numpy pandas openpyxl jupyter

Solver Installation#

Pyomo requires optimization solvers to solve mathematical models. For this workshop:

  1. GLPK (GNU Linear Programming Kit) - Open-source linear programming solver

    • Windows: Download from GLPK website

    • macOS: brew install glpk

    • Linux: sudo apt-get install glpk-utils (Ubuntu/Debian)

  2. CBC (Coin-or Branch and Cut) - Open-source mixed-integer programming solver

    • Windows: Download from COIN-OR website

    • macOS: brew install coin-or-tools/coinor/cbc

    • Linux: sudo apt-get install coinor-cbc

  3. IPOPT - Open-source nonlinear programming solver

    • Can be installed via conda: conda install -c conda-forge ipopt

Verification#

Test your installation by running:

import pyomo.environ as pyo
model = pyo.ConcreteModel()
print("Pyomo successfully installed!")

Mathematical Programming Fundamentals#

Mathematical programming (optimization) involves finding the best solution to a problem from a set of feasible alternatives. Every optimization problem consists of three essential components:

1. Decision Variables#

Variables represent the unknowns in your optimization problem - the values you want the solver to determine. In Pyomo, variables are defined with domains, bounds, and initial values.

Example:

  • In a production planning problem: How many units of each product to manufacture?

  • In a portfolio optimization: What fraction of budget to invest in each asset?

Mathematical notation: Typically denoted as \(x_1, x_2, \ldots, x_n\) or vectors \(\mathbf{x}\)

2. Objective Function#

The objective function quantifies what you want to optimize - either maximize (profit, efficiency) or minimize (cost, time, risk). It expresses the goal of your optimization problem as a mathematical function of the decision variables.

Mathematical form:

  • Minimize: \(\min f(\mathbf{x})\)

  • Maximize: \(\max f(\mathbf{x})\)

Examples:

  • Minimize total production cost: \(\min \sum_{i} c_i x_i\)

  • Maximize portfolio return: \(\max \sum_{i} r_i x_i\)

3. Constraints#

Constraints define the limitations and requirements that any feasible solution must satisfy. They restrict the values that decision variables can take.

Types of constraints:

  • Equality constraints: \(g(\mathbf{x}) = 0\) (must be satisfied exactly)

  • Inequality constraints: \(h(\mathbf{x}) \leq 0\) or \(h(\mathbf{x}) \geq 0\)

  • Bound constraints: \(x_{\text{min}} \leq x \leq x_{\text{max}}\)

Examples:

  • Resource limitations: Total material usage ≤ Available material

  • Demand requirements: Production ≥ Minimum demand

  • Logical constraints: Binary variables for yes/no decisions

Types of Optimization Problems#

Understanding different problem types helps you choose appropriate solution methods:

  1. Linear Programming (LP): Linear objective and constraints

  2. Mixed-Integer Programming (MIP): Includes integer variables

  3. Nonlinear Programming (NLP): Nonlinear objective or constraints

  4. Mixed-Integer Nonlinear Programming (MINLP): Combines integer and nonlinear elements

Why Use Pyomo?#

  • Flexibility: Model complex optimization problems with intuitive Python syntax

  • Solver Integration: Interface with 20+ commercial and open-source solvers

  • Extensibility: Build custom solution algorithms and extensions

  • Open Source: Free to use with active community support

  • Scalability: Handle problems from small examples to large-scale industrial applications

Workshop Structure#

This workshop is organized into progressive sections that build your Pyomo expertise:

References#

For deeper understanding of optimization modeling and Pyomo:

  • [] - Comprehensive Pyomo reference

  • [] - Mathematical programming fundamentals

  • [] - Convex optimization theory

  • [] - GLPK solver documentation

  • [] - CBC solver documentation

Contents