optimizationover

Optimizationover is a term used in optimization theory to describe the operation of computing the best value of a function by optimizing over a subset of its variables, often while treating the remaining variables as parameters. In mathematical notation, for a function f(x,y) defined on a domain X × Y, optimization over x given y produces g(y) = inf_{x in X} f(x,y) (or sup for maximization). The result is a reduced problem where x has been eliminated, yielding a new objective g that depends only on y. This pattern appears in two-stage and bilevel optimization, where an inner problem is solved for x as a function of y, and an outer problem optimizes over y using the inner solution.

Common scenarios include convex inner problems where the inner optimum is unique and can be expressed as a function x*(y), enabling reformulation g(y) = f(x*(y), y). There are also nonconvex cases requiring numerical methods such as alternating optimization or block coordinate descent, which iteratively optimize over blocks of variables. The inner-outer structure can facilitate decomposition, reduce dimensionality, or expose problem separability for specialized solvers.

Algorithmic considerations: differentiating through the inner optimization can be challenging; approaches include implicit differentiation, surrogate models, or bi-level reformulations. Convergence depends on the smoothness of f, the geometry of X and Y, and the stability of the inner optimum with respect to y. In practice, exact inner solutions may be replaced with approximate solves to improve tractability.

Applications span machine learning for hyperparameter tuning, control systems, economics and game theory, operations research, and engineering design. Although not a standard term in every field, optimizationover appears as a useful descriptive label for problems that repeatedly optimize over a subset of variables within a larger decision framework. Proper specification of domains, objectives, and optimization directions is essential for clarity.