Nonlinear Scalarization in Optimization & Physics

• Nonlinear scalarization is a method that converts vector-valued or field-theoretic outputs into a single scalar using nonlinear maps, overcoming the limitations of linear scalarization.
• In optimization, it employs nonlinear scalarizing functionals—such as hypervolume scalarizers and cone-based maps—to reveal hidden non-convex regions of the Pareto front.
• In gravitational physics, it triggers scalar hair formation in black holes by converting stable configurations into distinct scalarized states through nonlinear instability mechanisms.

Searching arXiv for recent and relevant papers on nonlinear scalarization across optimization, multiobjective learning, and gravitational physics. arxiv_search query: "^^^^^^^^1all: \1^^^^^^^^"nonlinear scalarization\"" Nonlinear scalarization denotes the passage from a vector-valued, set-valued, or field-theoretic structure to a scalar quantity by a nonlinear mechanism. In optimization, it is a scalarization method that converts vector optimization problems into scalar optimization problems and includes nonlinear scalarizing functionals, hypervolume scalarizers, and cone-based scalarization maps (&&&^^^^1all: \1^^^^&&&, Bouza et al., 2021, Zhang, 2023). In gravitational physics, the same expression denotes the formation of scalar hair through a genuinely nonlinear instability, typically in theories where the scalar-free solution is linearly stable and no tachyonic linear trigger is present (Doneva et al., 2021, Zhang et al., 2024). This suggests that the term is discipline-dependent: its common core is the replacement of a linear reduction principle by a nonlinear one, but its mathematical and physical content varies sharply across fields.

1. Terminological scope and basic contrast

In vector and multiobjective optimization, scalarization is introduced to compare vector outcomes through a real-valued functional. The standard linear scalarizer is PRESERVED_PLACEHOLDER_^^^^1all: \1^^^^, but several arXiv works emphasize that linear scalarization is geometrically restrictive: it cannot recover non-convex or concave regions of the Pareto front, and it does not exhaust the classes of monotone scalarizing functionals used in vector and set optimization (Zhang, 2023, Bouza et al., 2021). In this usage, “nonlinear scalarization” means a nonlinear map $\Psi:Y\to\mathbb{R}$ or $s_\lambda:\mathbb{R}^k\to\mathbb{R}$ that preserves an order, a domination relation, or a preference relation more faithfully than a weighted sum.

In gravitational theory, the contrast is instead between spontaneous scalarization and nonlinear scalarization. Spontaneous scalarization is tied to a linear tachyonic instability of a scalar-free configuration. Nonlinear scalarization occurs when the scalar-free black hole remains linearly stable, but sufficiently large finite perturbations probe higher-order terms in the coupling and trigger a scalarized branch (Doneva et al., 2021, Zhang et al., 2024). In this literature, the central distinction is therefore not linear versus nonlinear objective reduction, but linear versus nonlinear instability channels.

2. Ordered spaces, cone metrics, and scalarizing functionals

A central optimization framework uses a cone $K\subseteq Y$ to induce a partial order. In that setting, a continuous scalarizing functional $\Psi:Y\to\mathbb{R}$ is required to satisfy monotonicity and representability properties. The relevant notions include $K$-monotonicity, strict $K$-monotonicity, and strong $K$-monotonicity, together with the inclusions

$$\{y\in Y:\Psi(y)\le 0\}\subseteq -K,\qquad \{y\in Y:\Psi(y)<0\}\subseteq -\operatorname{int}(K).$$

Within this framework, the relationships among Gerstewitz, Hiriart-Urruty, and Drummond-Svaiter scalarizations are completely determined, and under the stated assumptions one has

$$\Omega_{GW}\subseteq \Omega_{HU}\subseteq \Omega_{DS},$$

with generally strict inclusions. The larger quasidifferentiable positively homogeneous class $\Psi:Y\to\mathbb{R}$^^^^1all: \1^^^^ extends this hierarchy to

$\Psi:Y\to\mathbb{R}$1

and the Gerstewitz class is minimal in this sense (Bouza et al., 2021).

The same nonlinear-scalarization vocabulary also appears in topological vector space valued cone metrics. If $\Psi:Y\to\mathbb{R}$2 is a topological vector space with cone $\Psi:Y\to\mathbb{R}$3 and $\Psi:Y\to\mathbb{R}$4, the nonlinear scalarization map is

$\Psi:Y\to\mathbb{R}$5

Given a cone metric $\Psi:Y\to\mathbb{R}$6, one defines

$\Psi:Y\to\mathbb{R}$7

The key metrizability statement is that the topology induced by the topological vector space valued cone metric coincides with the topology induced by the metric obtained via this nonlinear scalarization function; hence any topological vector space valued cone metric space is metrizable (&&&^^^^1all: \1^^^^&&&). In this setting, nonlinear scalarization is a structural tool that transfers convergence, completeness, and fixed-point arguments from cone-valued distances to ordinary metric spaces.

3. Multiobjective learning and nonlinear preferences

In multiobjective bandits and Bayesian optimization, nonlinear scalarization is used to explore Pareto fronts that linear weighted sums miss. The hypervolume scalarizer

$\Psi:Y\to\mathbb{R}$8

is introduced precisely because $\Psi:Y\to\mathbb{R}$9 cannot recover non-convex or concave regions of the Pareto front. A key representation is that dominated hypervolume can be written as the expectation of the maximized hypervolume scalarization under uniformly random weights, and uniformly random hypervolume scalarization achieves an optimal sublinear hypervolume regret bound of $s_\lambda:\mathbb{R}^k\to\mathbb{R}$^^^^1all: \1^^^^, with matching lower bounds (Zhang, 2023). In multiobjective stochastic linear bandits, the same paper gives a non-Euclidean analysis leading to $s_\lambda:\mathbb{R}^k\to\mathbb{R}$1-style bounds and introduces ExploreUCB.

A related but distinct development appears in multi-objective reinforcement learning with nonlinear preferences over trajectories. There the target is the expected scalarized return,

$s_\lambda:\mathbb{R}^k\to\mathbb{R}$2

which differs from the scalarized expected return $s_\lambda:\mathbb{R}^k\to\mathbb{R}$3. Because $s_\lambda:\mathbb{R}^k\to\mathbb{R}$4 is nonlinear, Bellman optimality must be extended to depend on state, current accumulated reward, and time remaining. The resulting value function is parameterized by $s_\lambda:\mathbb{R}^k\to\mathbb{R}$5, and Reward-Aware Value Iteration discretizes $s_\lambda:\mathbb{R}^k\to\mathbb{R}$6 via

$s_\lambda:\mathbb{R}^k\to\mathbb{R}$7

For smooth scalarization functions with a constant number of rewards, the algorithm computes an approximately optimal non-stationary policy in pseudopolynomial time, with runtime

$s_\lambda:\mathbb{R}^k\to\mathbb{R}$8

in the model-based setting (Peng et al., 2023). Here nonlinear scalarization is not merely a surrogate objective; it changes the state description needed for optimal control.

4. Black-hole scalarization beyond the tachyonic channel

The modern gravitational usage of nonlinear scalarization was sharpened by scalar-Gauss-Bonnet studies in which the scalar-free black hole is linearly stable. A central construction imposes

$s_\lambda:\mathbb{R}^k\to\mathbb{R}$9

so that the linearized scalar equation reduces to the free massless wave equation and no tachyonic instability can occur. In that case Schwarzschild can still be unstable against nonlinear scalar perturbations, and sufficiently large perturbations can drive the system toward a scalarized black-hole phase (Doneva et al., 2021). The resulting stable scalarized branch is not continuously connected to Schwarzschild, so scalarization and descalarization occur with a jump.

This mechanism was developed explicitly in Einstein-scalar-Gauss-Bonnet gravity for couplings satisfying $K\subseteq Y$^^^^1all: \1^^^^. For $K\subseteq Y$1, $K\subseteq Y$2, and $K\subseteq Y$3, Schwarzschild is linearly stable because $K\subseteq Y$4, but finite-amplitude perturbations can produce a nonlinear instability when the coupling includes terms higher than $K\subseteq Y$5. For the coupling

$K\subseteq Y$6

$K\subseteq Y$7 is the main parameter controlling how strongly the nonlinear scalarized branches differ, whereas $K\subseteq Y$8 plays a supplementary role, and the first law $K\subseteq Y$9 is verified numerically (Zhang et al., 2024).

Rotating black holes exhibit the same distinction. In the decoupling-limit study of Kerr black holes, the coupling

$\Psi:Y\to\mathbb{R}$^^^^1all: \1^^^^

satisfies $\Psi:Y\to\mathbb{R}$1, so there is no tachyonic instability. Scalarization then occurs only above a finite perturbation threshold, and there is no probe limit with zero scalar charge at finite mass: bald and hairy Kerr solutions are separated by a gap and only connect when the mass goes to zero together with the charge (Doneva et al., 2022). In Einstein-Maxwell-Scalar theory, the coupling

$\Psi:Y\to\mathbb{R}$2

realizes mixed scalarization: the quadratic term can trigger linear instability, while the quartic term supports or opposes additional nonlinear hair formation (Belkhadria et al., 2023).

5. Branch structure, quenching, and phase transitions in compact objects

Once scalarized solutions exist, their organization is typically branch-like rather than perturbative. In the mixed Einstein-Maxwell-Scalar model, the domain of existence is described by an existence or bifurcation line, a turning line in the nonlinear-dominated case, and a critical line where $\Psi:Y\to\mathbb{R}$3. Nonlinear scalarization displays a two-branch structure: a cold branch starting from extremal Reissner-Nordström and generally unstable, and a hot branch extending into the over-extremal regime and associated with the nonlinear scalarized phase. The spontaneous channel usually dominates, while positive $\Psi:Y\to\mathbb{R}$4 or positive $\Psi:Y\to\mathbb{R}$5 act as counter-scalarization terms that suppress or quench scalarization (Belkhadria et al., 2023).

The same suppression pattern appears when a scalar potential is added in scalar-Gauss-Bonnet gravity. For

$\Psi:Y\to\mathbb{R}$6

a nonzero mass or positive quartic self-interaction suppresses or quenches scalarization, whereas a negative quartic self-interaction enhances it. The existence domain of scalarized black holes can change significantly, but the presence and size of the jump between stable bald and stable scalarized black holes are much less sensitive to the scalar potential (Pombo et al., 2023).

Multi-scalar Gauss-Bonnet gravity shows that nonlinear scalarization is compatible with vanishing scalar charge. With target-space geometries $\Psi:Y\to\mathbb{R}$7, $\Psi:Y\to\mathbb{R}$8, and $\Psi:Y\to\mathbb{R}$9, and couplings

$K$^^^^1all: \1^^^^

the scalar field decays as $K$1, so scalar charge and scalar dipole radiation vanish. Thermodynamically, lower branches are likely unstable, whereas the branch with the largest scalar field at fixed mass is the likely stable one when it exists (Staykov et al., 2022).

Neutron stars provide an even closer analogue of first-order transitions. In scalar-Gauss-Bonnet gravity, nonlinear scalarization produces disconnected neutron-star branches and a jump-like transition between a GR branch and a scalarized branch. The effect is described as a gravitational phase transition, and the analysis indicates that it can occur over a wide range of parameters, so no fine-tuning is needed (&&&2^^^^1all: \1^^^^&&&).

6. Spin-induced, cosmological, and toy-model extensions

Rotation can itself generate the nonlinear trapping mechanism. In Einstein-scalar-Gauss-Bonnet gravity with

$K$2

Kerr is linearly stable, but sufficiently rapid rotation creates a negative near-horizon polar region in the Gauss-Bonnet invariant. This geometric trapping mechanism becomes effective above the threshold spin $K$3. Fully backreacted scalarized solutions occupy a finite low-mass high-spin wedge in the spin-mass plane; toward the low-spin boundary the solutions approach a weak-hair limit as $K$4, while toward the high-spin edge they approach a near-extremal regime (Lai et al., 30 Apr 2026).

Nonlinear scalarization also admits rigorous cosmological formulations. In flat FLRW Einstein-scalar-Gauss-Bonnet cosmology with quadratic coupling

$K$5

global existence and singularity-free solutions have been proved, together with a rigorous proof of nonlinear spontaneous scalarization triggered by a tachyonic instability induced by the Gauss-Bonnet term. The argument is built on decoupled differential inequalities for the Hubble parameter derived from a structural identity called the power identity (He et al., 21 Jul 2025). A plausible implication is that the gravitational notion of nonlinear scalarization is not limited to compact objects: it also functions as a dynamical selection mechanism in cosmological evolution.

Analytic toy models clarify why nonlinear terms matter. In Maxwell-scalar models for a charged conducting sphere, the exactly linearizing coupling

$K$6

produces scalarized configurations but also runaway behavior, whereas nonlinear continuations such as

$K$7

and

$K$8

can heal this behavior, producing stable scalarized solutions that are dynamically preferred over the Coulomb one (Herdeiro et al., 2020). This suggests that, across gravitational models, nonlinear scalarization is frequently less about the onset of instability than about the existence of a stable nonlinear endpoint.