Scalarization

Scalarization is the process of converting a vector or set of criteria into a scalar quantity for comparison or optimization. In mathematics and operations research, it maps a vector of objective values f(x) = (f1(x), ..., fm(x)) to a single real number φ(f(x)). This enables standard single-objective methods and decision rules. Different scalarization schemes yield different Pareto-optimal solutions; the choice of scheme affects which parts of the Pareto frontier can be detected, especially for nonconvex fronts. Common methods include the weighted-sum approach φ_w(f) = ∑ w_i f_i, with nonnegative weights; Chebyshev or L∞ scalarization φ∞(f) = max_i w_i f_i; and lexicographic or epsilon-constraint variants that impose priorities or thresholds on some objectives.

In physics, scalarization refers to the appearance or coupling of a scalar field in addition to the metric. In scalar-tensor theories of gravity, matter can source a scalar field, and compact objects such as neutron stars may undergo spontaneous scalarization, developing a nontrivial scalar configuration that alters their structure and gravitational interaction. Dynamical scalarization can occur during binary evolution, affecting gravitational-wave emission. These effects are constrained by pulsar timing and gravitational-wave observations.

In broader terms, scalarization also appears in computer science and decision analysis as a general term for turning multiple outputs or criteria into a single scalar score for ranking, selection, or learning. See also multiobjective optimization, scalar-tensor gravity, and Pareto efficiency.