minimiproblem

Minimiproblem is a term used in optimization theory to denote a problem whose objective is to find the values of decision variables that minimize a given objective function subject to a set of constraints. It is the counterpart to maximization problems and appears in a wide range of disciplines.

Formal representation typically takes the form: minimize f(x) subject to x in X, where f: R^n -> R is the objective function and X is the feasible region defined by equalities and inequalities. Minimiproblems are categorized as linear, convex, non-convex, or combinatorial, depending on the structure of f and X. When f is convex and X is convex, any local minimum is also a global minimum.

Existence and uniqueness depend on problem structure. If X is compact and f is continuous, a minimizer exists. Uniqueness is often guaranteed by strict convexity of f on a convex feasible set, though additional conditions may be required for more complex cases.

Solution methods vary with problem type. Analytic solutions apply to simple forms, while numerical approaches include gradient-based methods for differentiable f, subgradient methods for non-differentiable f, and interior-point or active-set methods for convex problems. For discrete or large-scale non-convex minimiproblems, branch-and-bound, cutting planes, or metaheuristics are common.

Examples include linear programming, which minimizes a linear objective over a polyhedral feasible region, and constrained quadratic programming. Minimiproblems are central to fields such as operations research, economics, engineering, and machine learning, where efficient and reliable optimization is essential. See also optimization, minimization problem, linear programming, convex optimization.