In the field of mathematical optimization, stochastic programming is a framework for modeling optimization problems that involve uncertainty. Whereas deterministic optimization problems are formulated with known parameters, real world problems almost invariably include some unknown parameters. Stochastic programming has applications in a broad range of areas including finance, economics, biology, transportation, and energy.
Integer and Combinatorial Optimization (together sometimes called Discrete Optimization) involves solutions sets that are discrete in nature, such as, subsets of edges in a graph, points with integer coordinates in a convex set, all possible partitions of a sample set, or a family of subsets of a universal finite set. There are fascinating connections with convex geometry, number theory and functional analysis on the theoretical side, and very impactful applications in astronomy, statistical learning, scheduling problems, and routing problems, to name a few.
Continuous optimization is a branch of optimization in applied mathematics. As opposed to integer and combinatorial optimization, the variables used in the objective function are required to be continuous variables, i.e., they are chosen from the set of real numbers. Both theoretical and algorithmic contributions from continuous optimization are important, as witnessed by their use in a wide range of engineering areas that include real-time model predictive control, computer vision, model prediction, energy, network design, and data assimilation.
Convex optimization is a subfield of optimization that studies the problem of minimizing convex functions over convex sets. Convexity imparts a rich set of properties for the optimization problem that is utilized to prove strong theoretical results and design efficient algorithms. For example, convex optimization problems have the property that all local minimizers are, in fact, global minimizers. Convex optimization is commonly used in application areas such as computer vision, machine learning, signal processing, and circuit design.