Objective function

An objective function assigns a numerical value to each candidate decision, expressing what an optimization problem seeks to minimize or maximize.

Optimization problem: A mathematical problem that seeks the best feasible decision according to a specified criterion. It combines an objective function with constraints to define what counts as the best decision.
Scalar function: A function that maps inputs to single numerical values. An objective function is scalar-valued, even when its input is a vector of decisions.
Empirical risk minimization: A learning principle that selects a model by minimizing its average loss on observed examples. The average training loss serves as the objective for fitting model parameters.
Constraint programming: A problem-solving approach that specifies allowed variable combinations through constraints. It can seek any feasible solution without assigning each one a scalar objective.
Decision variable: A variable whose value represents a choice made within a mathematical model. The objective function evaluates choices encoded by these variables.
Argmax: The input or inputs at which a function attains its greatest value. An objective's argmax identifies decisions that achieve its maximum.
Maximum likelihood estimation: A method that estimates model parameters by maximizing the probability of observed data. Its likelihood, or log-likelihood, is the quantity being optimized.
Pareto efficiency: A state in which no objective can improve without worsening at least one other objective. With conflicting objectives, solutions may be compared by dominance rather than one score.
Feasible region: The set of variable assignments that satisfy every constraint in an optimization problem. Only decisions in this region are candidates for optimizing the objective.
Argmin: The input or inputs at which a function attains its smallest value. An objective's argmin identifies decisions that achieve its minimum.

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