Game Theory analyzes strategic interactions where multiple decision-makers' choices affect each other's outcomes. Relevant to OR in: competitive facility location, supply chain coordination, pricing strategies, and auction design. Key concepts: (1) Nash Equilibrium - no player can improve by unilaterally changing strategy; (2) Zero-sum games - one player's gain equals another's loss, solved via linear programming; (3) Cooperative games - players form coalitions, using solution concepts like Shapley value. Applications: pricing competition, resource allocation in shared infrastructure, and mechanism design for efficient outcomes. Game-theoretic models inform: competitive strategy, contract design, and understanding market equilibria. Modern topics include: algorithmic game theory and optimization in multi-agent systems.

GAMS is a high-level modeling system for mathematical programming and optimization. Designed for complex, large-scale modeling applications, GAMS provides: algebraic model representation, separation of model and data, support for various problem types (LP, NLP, MIP, MINLP), and interfaces to numerous solvers. Features include: conditional statements and loops, dollar control options, model libraries (GAMSlib with hundreds of examples), and comprehensive documentation. GAMS syntax emphasizes sets and indexed operations, making it powerful for multi-dimensional problems. The IDE includes debugging tools, solution analysis, and visualization. Used extensively in energy systems modeling, economic equilibrium models, supply chain optimization, and policy analysis.

Genetic Algorithms are evolutionary optimization methods inspired by natural selection and genetics. The algorithm maintains a population of candidate solutions (chromosomes) and evolves them through: (1) Selection - choosing fitter individuals based on fitness function; (2) Crossover - combining genetic information from parents to create offspring; (3) Mutation - introducing random changes to maintain diversity; (4) Replacement - forming new generation. GAs are particularly effective for: combinatorial optimization, nonlinear problems with multiple local optima, problems with discrete and mixed variables, and when gradients are unavailable. Common variants include: steady-state GA, elitist GA, and multi-objective GA (NSGA-II).

Gradient Descent is a first-order iterative optimization algorithm for finding local minima of differentiable functions. Starting from an initial point x₀, it updates: xₖ₊₁ = xₖ - αₖ∇f(xₖ), where αₖ is the step size (learning rate) and ∇f is the gradient. Variants include: batch gradient descent (uses entire dataset), stochastic gradient descent (SGD - uses single samples), mini-batch gradient descent, and momentum-based methods. Convergence rate depends on function properties: linear for strongly convex smooth functions. Widely used in machine learning, deep learning, and large-scale optimization.

Greedy algorithms make locally optimal choices at each step, hoping to find a global optimum. The approach: (1) select the best immediate option without reconsidering previous choices; (2) never backtrack; (3) build solution incrementally. Greedy algorithms are optimal for problems with greedy choice property and optimal substructure (e.g., minimum spanning tree with Kruskal's or Prim's algorithm, Huffman coding, fractional knapsack). However, for many problems (0/1 knapsack, traveling salesman), greedy approaches yield only approximate solutions. Advantages include: simplicity, efficiency (often polynomial time), and effectiveness as constructive heuristics for complex problems.

Gurobi is a state-of-the-art mathematical programming solver known for speed and robustness. It solves: linear programming, mixed-integer linear programming, quadratic programming, mixed-integer quadratic programming, and quadratically constrained programming. Key features: presolve and cut generation, parallel algorithms for multi-core processors, distributed optimization for cluster computing, parameter tuning tool, and interactive shell for model building. Gurobi supports multiple interfaces: Python (gurobipy), MATLAB, R, Java, C++, and modeling languages (AMPL, GAMS). Performance innovations include: advanced branching strategies, cutting planes, heuristics, and solution polishing. Widely adopted in academia and industry for its balance of performance, usability, and support.