Mathematical Optimization

When to use

• Formulating a real-world problem as a mathematical optimization model
• Choosing between LP, MIP, convex, CSP, or combinatorial approaches
• Selecting the right gradient descent variant for a machine learning or fitting problem
• Solving scheduling, assignment, routing, or bin packing problems
• Applying SAT/SMT solvers or constraint programming for logical constraint problems
• Evaluating solver options (PuLP, CVXPY, OR-Tools, Gurobi, Z3)

Core principles

1. Formulate before you code — identify variables, objective, and constraints explicitly before touching a solver
2. Tight LP relaxation = faster MIP — every improvement to the relaxation bound shrinks the branch-and-bound tree
3. Convexity is the dividing line — convex problems are reliably solvable; non-convex require restarts and heuristics
4. Adam is not always the answer — L-BFGS beats Adam on smooth well-conditioned problems with moderate dimension
5. CP-SAT over custom backtracking — Google OR-Tools CP-SAT handles constraint propagation and search orders better than hand-rolled solvers