Explicit Analytic Quadratic Projections

• Explicit analytic quadratic projections are methods to find orthogonal projections onto quadratic constraint sets using eigen-decomposition and closed-form root-finding.
• They reduce high-dimensional projection problems into one-dimensional secular equations that ensure globally optimal solutions, even in degenerate cases.
• Applications include optimization, statistical inference, and computational geometry, with use cases spanning ellipsoids, paraboloids, and Wasserstein projections.

An explicit analytic quadratic projection is the process of finding the orthogonal projection (in the sense of Euclidean or other norms) of a point onto a surface or subset defined by a quadratic constraint, with the projection formula or algorithm given in wholly explicit, closed-form, or one-dimensional root-finding terms. Such surfaces include quadric hypersurfaces (ellipsoids, hyperboloids, paraboloids), quadratic manifolds, and general quadratic constraint sets encountered across optimization, statistical inference, and applied mathematics. The development of these projections leverages spectral theory, Lagrange multipliers, canonical transformations, and root-finding theory to yield globally optimal solutions—including all degenerate or non-unique cases—in analytic or semi-analytic form.

1. Canonical Analytic Reduction of Quadratic Projection Problems

The central object is a quadratic constraint of the form

$x^\top A x + b^\top x + c = 0,$

with $A\in\mathbb{R}^{n\times n}$ real symmetric and (non)singular, $b\in\mathbb{R}^n$, and $c\in\mathbb{R}$. The projection of $y\in\mathbb{R}^n$ onto the corresponding hypersurface involves solving

$$\min_x \|x-y\|_2^2 \quad \textrm{s.t. } x^\top A x + b^\top x + c = 0.$$

By eigen-decomposing $A = P\Lambda P^\top$ and introducing a shifted variable $d = -\frac{1}{2}A^{-1}b$, the problem is reduced via $x = Pu + d$, $y^0 = P^\top(y-d)$ to a form with decoupled constraints: $$\min_u \|u - y^0\|_2^2 \quad \textrm{s.t. } \sum_{i=1}^{n}\lambda_i u_i^2 = 1,$$ where $\{\lambda_i\}_{i=1}^n$ are the eigenvalues of $A$. This reduction, universal for central quadrics, transforms the high-dimensional projection problem into a lower-dimensional or even one-dimensional parameter root-finding task (Hoorebeeck et al., 2022).

2. Lagrange Multiplier Structure and the Secular Equation

The use of Lagrange multipliers produces a necessary condition (Karush-Kuhn-Tucker) for the minimizer. For each $i$,

$$u_i(1 + \mu\lambda_i) = y^0_i, \qquad \Rightarrow \quad u_i(\mu) = \frac{y^0_i}{1 + \mu \lambda_i}, \qquad \mu \neq -1/\lambda_i,$$

which, enforced in the quadratic constraint, yields the secular equation: $$f(\mu) := \sum_{i=1}^n \lambda_i\frac{(y^0_i)^2}{(1+\mu\lambda_i)^2} - 1 = 0.$$ This is a strictly monotone function in most regimes (ellipsoid, hyperboloid, or degenerate cases), with precisely characterized admissible intervals determined by the spectrum of $A$:

• For an ellipsoid ($\forall\,\lambda_i>0$) the admissible interval is $(-1/\lambda_1, +\infty)$ with unique root.
• For indefinite $A$ (mixed-signed eigenvalues), the interval is defined by the poles corresponding to the maximal/minimal reciprocal eigenvalues (Hoorebeeck et al., 2022).

3. Explicit Solutions and Degenerate Cases

Once the Lagrange multiplier $\mu^*$ is found,

$$u^*_i = \frac{y^0_i}{1 + \mu^* \lambda_i}, \quad \forall i,$$

and the explicit projected point in the original coordinates is

$$x^* = d + P\,\mathrm{diag}\left( \frac{1}{1+\mu^* \lambda_i} \right) P^\top (y-d)$$

giving a fully explicit mapping up to the root-finding step in $\mu$. In the presence of degeneracies (such as $y_i^0=0$ for some $i$ or multiple eigenvalues), the solution set may contain additional finitely many “axis-intersection” candidates associated with these coordinates; all must be checked, and the true minimum selected accordingly. For specific quadratic constraints with repeated zero eigenvalues, an entire $(m-1)$-sphere of solutions in the nullspace is possible, a direct result of the geometry of the intersection (Hoorebeeck et al., 2022).

4. General Quadratic Constraints and Affine-Equivalence Methodology

For the problem

$$\min_x\,\frac12(x-x_0)^\top Q(x-x_0) \quad \textrm{s.t.}\; x^\top A x + 2a^\top x + \alpha = 0,$$

with $Q$ symmetric positive (semi-)definite, affine equivalence is exploited to diagonalize both $Q$ and $A$, reducing the primary problem to canonical coordinates with block-diagonal or block-scalar structure (Albers et al., 2017). Critical steps include:

• Affine canonicalization: $Q\to I_n$, $A\to$ block-diagonal,
• Dimensionality reduction: retaining only one coordinate per distinct eigenvalue block,
• Direct secular equation in Lagrange multiplier, with at most three solution branches (interior root or two boundary cases).

Each conceivable case (perfect solution, essentially perfect, projected but imperfect) can be characterized, and algebraic solution forms are provided for each branch, guaranteeing completeness and global optimality of the approach (Albers et al., 2017).

5. Quadratic Projections onto Specific Non-Ellipsoidal Quadratics

• Parabola and Paraboloid: For the planar parabola $\mathcal{P} = \{ (x, y) : y = ax^2 + bx + c \}$, the distance minimizer from $(u,v)$ is given by stationary points of a quartic $D(x)$, so the projection abscissa $x^*$ is a root of the associated cubic derivative. Explicit solutions are constructed using Cardano's formulas for the cubic, including all cases (single or triple real roots). The projection onto radial parabolas in higher dimensions is similarly reduced to a quartic in a scalar variable (Aragón-Artacho et al., 27 Dec 2025).
• Rectangular Hyperbolic Paraboloid: The constraint $\langle x, y\rangle = \alpha\gamma$ in $\mathbb{R}^n\times\mathbb{R}^n\times\mathbb{R}$ is reduced, by an orthogonal change of variables, to a single quintic (generic) or cubic (degenerate directions) in a Lagrange multiplier. The function in question is strictly monotone on $(-1,1)$, ensuring unique solutions except in strictly characterized degenerate regimes, where a continuum arises (Bauschke et al., 2022).

6. Quadratic Projections in Stochastic and Wasserstein Contexts

The explicit characterization extends to probabilistic projections, notably quadratic Wasserstein projections between measures under convex order. For Gaussian measures, one obtains fully explicit expressions for the mean, covariance, and optimal transport map for the quadratic Wasserstein projection onto the set of laws dominated by another in the convex order. The explicit result uses orthogonal diagonalization so that in appropriate coordinates the covariance matrices are diagonalized, and the optimal covariance is given by the coordinatewise minimum or maximum of the variances (Alfonsi et al., 30 Jun 2025).

The solution map exhibits non-expansiveness in the source law and Hölder-1/2 regularity in the target; all such formulae are achievable explicitly due to the Gaussian structure.

7. Algorithmic and Computational Aspects

All analytic projections onto quadratics described above—central quadrics, ellipsoids, paraboloids—ultimately reduce to one-dimensional root-finding (secular/quintic/cubic) in a Lagrange multiplier associated with the KKT system. For the generic central quadric,

• The dominant computational cost is the eigendecomposition of the symmetric matrix ($O(n^3)$),
• The root-finding (Newton, bisection, or safeguarded double-Newton) exploits strict monotonicity on well-isolated intervals,
• Explicit numerical safeguards prevent singularity at poles and preserve stability in degenerate coordinates (Hoorebeeck et al., 2022, Bauschke et al., 2022).

For projections with multiple constraints (e.g., quadratic programming over ellipsoids), augmented-Lagrangian approaches allow decomposition into a sequence of secular equation solves and orthogonal projections, with explicit expressions throughout and convergence typically ensured by robust root-finding algorithms (Phan et al., 2017).

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Explicit analytic quadratic projections thus refer to the class of projection problems for which the solution can be written explicitly (up to a scalar root in one variable) using canonicalization, eigen-decomposition, and Lagrangian duality. The literature provides a complete classification of cases, explicit formulas, root-separation intervals, and degeneration analysis, covering all quadrics and many infinite-dimensional analogues (Hoorebeeck et al., 2022, Albers et al., 2017, Phan et al., 2017, Aragón-Artacho et al., 27 Dec 2025, Bauschke et al., 2022, Alfonsi et al., 30 Jun 2025).