Quartic Inhomogeneous Polynomial Optimization

• Quartic inhomogeneous polynomial optimization involves minimizing or maximizing fourth-degree polynomials with lower-order terms, forming the simplest NP-hard polynomial case beyond quadratic functions.
• Advanced computational methods such as SDP relaxations, proximal alternating minimization, and canonical duality are employed to tackle nonconvexity and guarantee convergence.
• This problem has broad applications in quantum mechanics, control systems, and engineering design, making it a critical topic in modern optimization theory and practice.

A quartic inhomogeneous polynomial optimization problem concerns minimizing (or maximizing) a function that is a general polynomial of degree four with nonhomogeneous (lower-degree) terms, often subject to algebraic constraints such as norm or sphere constraints. Such problems pervade mathematical optimization, quantum mechanics, control, and computational chemistry, with both convex and nonconvex instances arising depending on the data and structure. This class represents the lowest-degree polynomial optimization—beyond the quadratic—that is generically NP-hard, supporting rich geometric, algorithmic, and duality theory.

1. Formal Structure and Problem Classes

A general unconstrained quartic inhomogeneous polynomial optimization problem has the form

$\min_{x \in \mathbb{R}^n} \; p(x),$

where $p(x)$ is a real polynomial of degree at most four, most generally written as: $$p(x) = c_0 + c_1^\top x + \frac{1}{2} x^\top Q x + \frac{1}{6}T[x]^3 + \frac{1}{24} \sum_{i=1}^n a_i^4 + \text{lower-degree terms},$$ with coefficients $c_0 \in \mathbb{R}$, $c_1 \in \mathbb{R}^n$, $Q \in \mathbb{R}^{n\times n}$ symmetric, $T$ a third-order symmetric tensor, and possible quartic (homogeneous or not) contribution.

Frequently encountered subclasses include:

• Structured convex quartics (e.g., quartic-plus-linear: $f(x) = P_4(x) + b^T x$, with $P_4$ homogeneous convex quartic) (Dragomir et al., 2023).
• Spherical quartic/quadratic problems: minimize $$f(z) = \frac{1}{2} z^*Az + \frac{\beta}{2} \sum_{k=1}^n |z_k|^4$$ subject to $\|z\|_2=1$ (zhang et al., 2019, Chen et al., 31 Dec 2025).
• Cubic-quartic regularization subproblems: $$M(s) = f_0 + g^T s + \frac{1}{2} s^T H s + \frac{\beta}{6}\|s\|^3 + \frac{\sigma}{4}\|s\|^4$$, as in higher-order adaptive regularization frameworks (Zhu et al., 2023, Zhu et al., 28 Apr 2025, Zhou et al., 31 Oct 2025).
• Canonical dual/quartic double-well-type: e.g. sums of shifted quadratic forms squared, generating multiwell landscapes (Gao et al., 2011, Chen et al., 2013).

The "inhomogeneity" refers specifically to the presence of terms of degree less than four, resulting in rich geometric structure and nontrivial duality properties.

2. Optimality Conditions and Duality Theory

For unconstrained quartic polynomials, the first-order necessary condition for a minimizer is stationarity: $\nabla p(x^*) = 0.$ Second-order sufficient/necessary conditions involve positive semidefiniteness of the Hessian (and higher-derivative tensors for nonconvex cases) at the candidate minimizer.

For succinct quartic-quadratic forms on the sphere, the KKT system (using, e.g., Wirtinger derivatives for complex variables) reduces to

$$\text{grad}\,f(z) = 0 \implies [A + 2\beta \operatorname{Diag}(|z|^2)]z = 2\lambda z$$

with spherical constraints $\|z\|_2=1$, and tangent-space projected second-order tests (zhang et al., 2019).

Quartic inhomogeneous polynomials often admit tractable dual characterizations when expressed through the canonical duality framework. This involves:

• Forming a canonical measure $\Lambda(x)$ (quadratic mappings),
• Introducing a dual variable $\varsigma$ via Legendre dualization,
• Writing the total complementary function and deriving the dual program.

The triality theory establishes that when the primal and dual have matching dimensions, strong min–max duality holds: global minimizers correspond to global maximizers of the dual, with local extremizers pairing under curvature sign consistency (Gao et al., 2011). If the dimensions disagree, the double-min duality holds only on a suitable subspace.

The canonical dual approach thereby precisely captures inhomogeneity by incorporating shifts and constant terms at the dual-variable level, and yields explicit conditions for classifying global minima versus local extrema. All duality gaps vanish under mild regularity (Gao et al., 2011, Chen et al., 2013).

3. Algorithmic Frameworks and Computational Methods

Recent advances have spurred the development of specialized algorithms for quartic inhomogeneous polynomial problems, exploiting their structure:

• SDP Relaxations for Cubic-Quartic Models: For models $$M(s) = f_0 + g^Ts + \frac{1}{2}s^THs + \frac{\beta}{6}\|s\|^3 + \frac{\sigma}{4}\|s\|^4$$, an exact semidefinite programming (SDP) relaxation can be formulated involving a lifted matrix of moments and sum-of-squares (SOS) structure for the univariate norm (Zhou et al., 31 Oct 2025). The SDP is tight if and only if $\|s^*\|(\beta + 3\sigma \|s^*\|)\geq 0$ for the global minimizer $s^*$.
• Proximal Alternating Minimization (PAM): When the quartic polynomial can be represented via symmetric tensors, as for forms $f(x) = \frac{\theta}{2}\sum x_i^4 + x^T B x$ on the sphere, the problem can be transformed into a multilinear optimization problem (MOP) and solved by a proximal block-coordinate descent cycling over four blocks (PAM), with provable convergence to stationary points (Chen et al., 31 Dec 2025).
• Cubic-Quartic Regularization (CQR) and Diagonal Tensor Methods (DTM): Adaptive regularization methods for higher-order Taylor polynomials (e.g., in third-order tensor optimization) utilize local models of cubic plus quartic form and exploit secular root-finding (via nonlinear eigenvalue problems) or surrogates based on tensor diagonalization (Zhu et al., 2023, Zhu et al., 28 Apr 2025).
• Accelerated Higher-Order Descent: In convex cases, higher-order Taylor models plus quartic regularization admit efficient, accelerated methods (e.g., Tensor Descent, Homogenized Gradient), achieving superior complexity bounds $O(n^{1/5}\log^{O(1)}(1/\epsilon))$ for structured quartics and $\ell_4$-regression (Bullins, 2018, Dragomir et al., 2023).
• Canonical Dual Algorithms: For multiwell and double-well quartics, the canonical dual provides a direct minimization route: solve the dual algebraic system (often by root-finding for a small system), then recover the primal global (and largest local) optimizers analytically (Gao et al., 2011, Chen et al., 2013).

Table: Algorithmic Guarantees for Key Quartic Inhomogeneous Classes

Problem Structure	Method	Global/Local Minima Characterization
Unconstrained cubic-quartic	SDP relaxation (Zhou et al., 31 Oct 2025)	Tight if $\|s^*\|(\beta + 3\sigma \|s^*\|)\geq 0$
Spherical quartic-quadratic	Convexification/sorting	O(n log n) if diagonal, no spurious minima (zhang et al., 2019)
Homogenized convex quartic + linear	Hom-GD/Accel-GD (Dragomir et al., 2023)	Rate $O(\kappa^2/k^2)$, condition number reduction
General inhom. quartic polynomial	Triality dual (Gao et al., 2011)	Global min ↔ dual max, local extrema ↔ dual pairs

4. Geometric and Landscape Properties

The geometric properties of the objective heavily dictate both the difficulty of the problem and the tractability of algorithmic approaches. For real or complex spherical quartic-quadratic problems, the landscape is characterized in terms of:

• Lack of spurious local minima in diagonal (separable) cases.
• Strict-saddle property: every stationary point is either a strict local minimum, has large gradient, or supports a negative curvature direction. For $\beta \gg n^{3/2}$ (or $\beta \ll 1$), this ensures that generic second-order methods efficiently escape saddles (zhang et al., 2019).
• Explicit solution structure: global minimizers can sometimes be represented and computed in closed-form (modulo phases), e.g., via projection onto the simplex (zhang et al., 2019).
• KL exponent: On the sphere, the Kurdyka-Łojasiewicz exponent is $\theta = 1/4$ for a broad range of stationary points, guaranteeing global convergence and (sub)linear rates for descent methods (zhang et al., 2019).

Convex quartic-plus-linear problems are uniformly convex of degree four with respect to suitable norms, allowing accelerated convergence and efficient preconditioning via generalized Lewis weights (Dragomir et al., 2023). Careful regularization (e.g., large enough quartic coefficient) ensures sum-of-squares convexity and closes the gap between necessary and sufficient conditions (Zhu et al., 28 Apr 2025).

5. Special Cases and Explicit Solutions

Important explicit cases include:

• Diagonal Quadratic/Quartic Terms (Sphere): Leads to a convex quadratic program over the simplex, solved in $O(n \log n)$ by sorting, and guarantees the absence of spurious local minima. Global minimizers correspond to all assignments $|z_k|^2 = u_k^*$ with arbitrary phase, where $u^* = \operatorname{Proj}_{\Delta_n} (-a/2\beta)$ (zhang et al., 2019).
• Rank-One Quadratic Terms: When the quadratic term is $A = a a^*$, and $\|a\|_\infty \leq \frac{1}{2}\|a\|_1$, uniform phase choices can generically nullify the quadratic term, yielding global minima with all components equal modulus (zhang et al., 2019).
• Structured Convex Quartics: For forms $$f(x) = c^T x + x^T G x + T[x,x,x] + (1/24)\|A x\|_4^4$$, the tensor step subproblem admits efficient approximation (with O($n^{1/5}$) evaluation complexity) via accelerated higher-order methods (Bullins, 2018).

In all such cases, the particular algebraic structure simplifies both analysis and computation, and sharpens optimality characterizations.

6. Applications and Practical Relevance

This class models phenomena such as:

• Quantum mechanics and chemistry: Spherical quartic-quadratic forms appear in mean-field theory, Bose–Einstein condensates, and energy landscape analysis (zhang et al., 2019, Chen et al., 31 Dec 2025).
• High-order numerical optimization: Adaptive regularization methods for smooth nonconvex programs, leveraging local cubic-quartic models to obtain optimal evaluation complexity (Zhu et al., 2023, Cartis et al., 2023, Zhu et al., 28 Apr 2025).
• Regression and signal processing: $\ell_4$-regression and higher-order moment estimation, where structured quartic objective functions arise (Bullins, 2018, Dragomir et al., 2023).
• Engineering design: Multiwell quartics for modeling phase transitions, structural stability, and more, leveraging triality duality for global to local solution correspondence (Gao et al., 2011, Chen et al., 2013).

Algorithmic frameworks typically combine classical root-finding, convex optimization, block coordinate methods, and semidefinite relaxations, with complexity (and success) determined by the problem's algebraic and geometric structure. These problems act as critical subproblems within higher-order tensor algorithms used in large-scale machine learning and scientific computation.

7. Open Directions and Theoretical Significance

Despite the substantial progress, key open areas include:

• Extension of efficient global solution methods (e.g., SDP relaxations) to broader quartic inhomogeneous regimes lacking particular structure or “tightness” conditions.
• Unified theory reconciling SoS-convexity, numerical tractability, and duality gaps for general nonconvex quartic polynomials.
• Complexity bounds and landscape analysis for mixed-sphere/linear constraint settings and for wider classes of inhomogeneous polynomials not reducible to explicit forms.
• Further exploration of the Kurdyka–Łojasiewicz exponents and strict-saddle landscape regimes to improve algorithmic rates for nonconvex and mixed-convex cases.

The quartic inhomogeneous polynomial optimization problem thus sits at a juncture between algebraic geometry, numerical analysis, and optimization theory, supporting a diverse set of rigorous methodologies and deep structural results. This area integrates canonical duality/triality, convex tensor analysis, semidefinite/sum-of-squares relaxations, and block-coordinate/alternating minimization, providing a foundation for addressing both theoretical questions and concrete computational challenges (zhang et al., 2019, Chen et al., 31 Dec 2025, Zhou et al., 31 Oct 2025, Zhu et al., 2023, Bullins, 2018, Cartis et al., 2023, Gao et al., 2011, Chen et al., 2013, Dragomir et al., 2023, Zhu et al., 28 Apr 2025).