Decision Variables are the unknowns in an optimization problem whose values must be determined to optimize the objective function while satisfying constraints. They represent controllable decisions such as: production quantities, resource allocations, assignment choices, routing decisions, or timing. Variables can be: continuous (real-valued), integer (discrete), or binary (0/1 for yes/no decisions). Proper variable definition is critical for effective modeling - variables should capture all decision freedom while avoiding redundancy. Considerations: dimensionality, domains, symmetry, and implied constraints. Preprocessing techniques like variable fixing, bound tightening, and aggregation can reduce problem size and improve computational performance.

Duality theory establishes that every linear programming problem (the primal) has an associated dual problem. The optimal value of the primal equals the optimal value of the dual (strong duality theorem). If the primal is: maximize c^T x subject to Ax ≤ b, x ≥ 0, then the dual is: minimize b^T y subject to A^T y ≥ c, y ≥ 0. Duality provides economic interpretations (shadow prices), bounds on optimal values, and alternative solution methods. Complementary slackness conditions link primal and dual optimal solutions.

Dynamic Programming is a method for solving complex problems by breaking them down into simpler overlapping subproblems. The approach is applicable when the problem exhibits: (1) Optimal substructure - optimal solution contains optimal solutions to subproblems; (2) Overlapping subproblems - same subproblems are solved multiple times. DP stores solutions to subproblems (memoization) to avoid redundant computation. Key applications include: shortest path problems, resource allocation, inventory management, production planning, and the knapsack problem. The Bellman equation characterizes optimal solutions recursively.