Optimizer

An optimizer is a mathematical algorithm or procedure that seeks to find the best values for a set of decision variables by maximizing or minimizing an objective function, possibly subject to constraints. In optimization problems, the objective expresses the goal, the decision variables specify choices, and constraints limit feasible solutions. Optimizers are central to operations research, economics, engineering, and machine learning, where they tune parameters, schedules, or designs to improve performance or efficiency.

Problems are categorized into unconstrained or constrained; continuous or discrete; linear or nonlinear; convex or nonconvex; deterministic or stochastic. Common classes include linear programming, integer programming, nonlinear programming, convex optimization, combinatorial optimization, and stochastic optimization. The distinction between global and local optima matters; convex problems admit a unique global optimum under mild conditions.

Algorithms include gradient-based methods (gradient descent, conjugate gradient), second-order methods (Newton, quasi-Newton), interior-point and barrier methods for constrained problems, simplex methods for linear programs, and branch-and-bound for discrete problems. Optimality conditions such as the Karush-Kuhn-Tucker conditions provide certificates of optimality under regularity assumptions.

In machine learning, optimizers adjust model parameters to minimize a loss function, with popular choices including stochastic gradient descent and adaptive methods like Adam or RMSprop. Real-world optimization also involves trade-offs among speed, accuracy, and robustness, handling noise, constraints, and large-scale data. Software tools implement various optimizers within scientific computing and ML frameworks.