djabelo712/optimization-tools

SVD image compression and nonlinear optimization with Gekko, Pyomo, and CasADi , AIMS Ghana 2025 # Mathematical Optimization: Tools and Techniques **AIMS Ghana — Block Course, December 2025** *Ounimborbitibou Djabon* ## Overview This repository covers two major topics in applied mathematical optimization, developed during the course "Mathematical Optimization: Theory, Methods and Applications" taught by Prof. Dr. Abebe Geletu W. Selassie (AIMS Rwanda) at AIMS Ghana in December 2025. The first topic is image compression using Singular Value Decomposition (SVD), a fundamental technique in numerical linear algebra. The second topic is the formulation and solution of nonlinear optimization problems, both unconstrained and constrained, using three Python-based optimization frameworks: Gekko, Pyomo, and CasADi. ## Repository Structure ``` optimization-tools/ │ ├── notebooks/ │ └── optimization_methods.ipynb # Full annotated notebook: SVD + Gekko + Pyomo + CasADi │ ├── data/ │ ├── Timbuktu.jpg # Input image used for SVD compression │ └── gray_image.txt # Grayscale pixel matrix of the image (text format) │ ├── images/ # Figures exported from the notebook │ ├── .gitignore ├── requirements.txt └── README.md ``` ## Part 1: Image Compression using SVD A grayscale image is simply a 2D matrix of pixel intensity values. The Singular Value Decomposition factorizes this matrix as: $$A = U \Sigma V^T$$ By keeping only the $k$ largest singular values and discarding the rest, we obtain a rank-$k$ approximation of the original image that requires far less storage. This is the Eckart-Young theorem: the rank-$k$ truncation is the best possible approximation of $A$ in both the 2-norm and the Frobenius norm. We apply this to a photograph of the Great Mosque of Djenné in Timbuktu and compare three compression levels. ### Results | Compression | Singular values kept | Data retained | |---|---|---| | 30% | 30 out of 600 | ~5% of original | | 50% | 50 out of 600 | ~9% of original | | 80% | 80 out of 600 | ~14% of …