Quadratic Cone Programs Overview

• Quadratic cone programs are optimization problems blending quadratic objectives with conic constraints to facilitate convex reformulations and tractable solving.
• They extend classical quadratic programming by incorporating second-order and semidefinite conic structures to achieve exact convex hull representations.
• Recent advances include scalable algorithms, parametric and online formulations, and differentiable optimization approaches for real-time and embedded applications.

to=arxiv_search 六和彩 content='query: "quadratic cone programs" OR "quadratic cone programming" OR "conic programs with quadratic objectives"; max_results: 10; sort_by: relevance' to=arxiv_search ＿影音先锋 content='query: "(Santana et al., 2018)" OR "(Goulart et al., 2024)" OR "(Chari et al., 16 Mar 2025)" OR "(Li et al., 2014)" OR "(Barrios et al., 2015)" OR "(Sheen et al., 2019)"; max_results: 10; sort_by: relevance' Quadratic cone programs are optimization problems in which quadratic structure and conic structure interact in a mathematically essential way. In the most common convex conic form, they are problems with objective $\frac{1}{2}x^\top P x + q^\top x$ subject to affine conic constraints $Ax+s=b$, $s\in K$, where $P\succeq 0$ and $K$ is a closed convex cone; this viewpoint includes quadratic programs, second-order cone programs, and semidefinite variants (Goulart et al., 2024). Closely related usages treat quadratic programs over specific cones such as simplicial or polyhedral cones, or use second-order cone representability to describe exact convex hulls and tight relaxations of nonconvex quadratic sets arising in QCQP (Barrios et al., 2015, Santana et al., 2018). The topic therefore spans convex conic optimization, cone-constrained quadratic programming, exact convexification, and algorithmic frameworks for both offline and online computation.

1. Canonical formulations and scope

The literature uses the phrase in several closely connected senses. In the convex conic programming sense, the primal-dual pair is

$$\min_{x,s}\; \frac{1}{2}x^\top P x + q^\top x \quad \text{subject to}\quad Ax + s = b,\;\; s\in K,$$

and

$$\max_{x,z}\; -\frac{1}{2}x^\top P x - b^\top z \quad \text{subject to}\quad Px + A^\top z = -q,\;\; z\in K^*,$$

with KKT conditions

$$Ax+s=b,\qquad Px+A^\top z=-q,\qquad s^\top z = 0,\qquad (s,z)\in K\times K^*.$$

Within this template, a quadratic program is obtained by taking cones such as $\{0\}$, $\mathbb{R}_+$, and second-order cones, while a linear cone program is recovered when $Ax+s=b$0 (Goulart et al., 2024, Healey et al., 24 Aug 2025).

Usage in the literature	Representative formulation	Role of the cone
Convex conic QP/QCP	$Ax+s=b$1 s.t. $Ax+s=b$2	Feasible set is an affine slice of a convex cone
Cone-constrained QP	$Ax+s=b$3 s.t. $Ax+s=b$4	Variables are constrained to a simplicial or polyhedral cone
SOCP convexification of quadratic sets	Exact or tight SOC representations of lifted quadratic sets	Cone constraints encode convex hulls or relaxations

A second established meaning is a quadratic program over a fixed cone. A simplicial cone constrained convex quadratic program takes the form

$Ax+s=b$5

where $Ax+s=b$6 is symmetric positive definite and $Ax+s=b$7 is nonsingular. The cone $Ax+s=b$8 is then a closed, pointed, full-dimensional polyhedral cone, and its dual satisfies

$Ax+s=b$9

This formulation turns optimality over a cone into a complementarity problem and then into a nonsmooth equation (Barrios et al., 2015).

A third classical viewpoint appears in cone regression, where the feasible set is a convex polyhedral cone

$s\in K$0

and the problem is the weighted projection

$s\in K$1

Because $s\in K$2 is closed and convex, the projection exists and is unique. The same model covers isotonic regression, concave regression, and ANOVA under partial orderings, and it admits primal, dual, LCP, and proximal interpretations (Dimiccoli, 2015).

2. Geometric representability and exact convex hulls

A central structural result for quadratic cone programming is that certain nonconvex quadratic sets admit exact second-order cone descriptions after convexification. For

$s\in K$3

with $s\in K$4 symmetric and $s\in K$5 a polytope, the exact convex hull satisfies

$s\in K$6

No special structure is assumed on the quadratic form beyond symmetry of $s\in K$7; it may be indefinite, mixed-sign, and fully general. This gives an exact SOCP description of the convex hull of a single quadratic equality intersected with any bounded polyhedron (Santana et al., 2018).

The constructive proof proceeds by diagonalizing the quadratic form via the spectral theorem,

$s\in K$8

using the change of variables $s\in K$9, completing squares, and reducing the equation to a canonical form with positive-square variables, negative-square variables, linear variables, and variables absent from the quadratic equation. Affine invariance,

$P\succeq 0$0

preserves SOC representability under these transformations. The argument then studies extreme points of the quadratic surface inside the bounding polytope and uses the disjunctive identity

$P\succeq 0$1

Two geometric cases arise: either the surface is a union of two convex pieces, or it is a ruled surface, in which case interior points of the polytope cannot be extreme points and the proof recurses on facets of the polytope (Santana et al., 2018).

This exact SOC representability is nontrivial for two reasons. First, the standard sum-of-squares / Lasserre hierarchy would produce SDP relaxations. Second, even the cone of $P\succeq 0$2 PSD matrices is not SOC-representable. The result therefore does not imply that every SDP relaxation can be replaced by SOCP; it identifies a specific geometric class whose exact convex hull is SOC-representable. It is also essential that the theorem describes the convex hull $P\succeq 0$3, not the original nonconvex set $P\succeq 0$4, and the resulting SOC formulation may require an exponential number of variables (Santana et al., 2018).

A related geometric framework appears in completely positive reformulations of quadratic and polynomial optimization. There, a nonconvex conic program over

$P\succeq 0$5

is analyzed using a nonconvex cone $P\succeq 0$6, a face $P\succeq 0$7 of $P\succeq 0$8, and a translated supporting hyperplane $P\succeq 0$9. Under a moderate geometric assumption, the problem over $K$0 is equivalent to a convex conic program over $K$1, which explains geometrically why certain lifted quadratic problems admit exact convex conic reformulations (Kim et al., 2019).

3. Convex relaxations for nonconvex QCQP and structured mixed-integer models

Much of the modern theory concerns how nonconvex QCQP can be strengthened or exactly convexified using second-order cones. One approach decomposes each indefinite quadratic constraint

$K$2

by writing

$K$3

introducing an auxiliary variable, and relaxing the resulting two SOC-type relations. Products of these SOC constraints with linear constraints are then linearized to obtain the Generalized SOC-RLT families GSRT-A and GSRT-B. In the stated hierarchy,

$K$4

and similarly for GSRT-B; the paper also extends the construction to SOC $K$5 SOC products and Kronecker-product LMIs (Jiang et al., 2016).

A complementary line of work exploits sparsity directly in SOCP relaxations of QCQP. In the lifted SOCP relaxation, every $K$6 principal submatrix PSD condition is a second-order cone constraint. If only edges in the aggregate sparsity graph are retained, the sparse feasible set

$K$7

is sufficient, and the optimal value of the sparse SOCP relaxation equals that of the full SOCP relaxation: $K$8 The completion procedure is also simpler than in sparse SDP: setting unspecified off-pattern entries to zero maximizes the SOCP analogue of determinant $K$9 among all feasible completions (Sheen et al., 2019).

For mixed-integer convex quadratic programs with dynamic state evolution and indicators, state elimination produces a reduced model

$$\min_{x,s}\; \frac{1}{2}x^\top P x + q^\top x \quad \text{subject to}\quad Ax + s = b,\;\; s\in K,$$0

where $$\min_{x,s}\; \frac{1}{2}x^\top P x + q^\top x \quad \text{subject to}\quad Ax + s = b,\;\; s\in K,$$1 is positive definite and factorizable in the scalar case or block-factorizable in the vector case. Using closed-form inverse decompositions of principal submatrices, the exact convex hull of the mixed-integer epigraph is characterized, and the resulting formulation can be written as a tight SOCP with $$\min_{x,s}\; \frac{1}{2}x^\top P x + q^\top x \quad \text{subject to}\quad Ax + s = b,\;\; s\in K,$$2 conic constraints. In the unconstrained case, the support set is encoded as a path in a DAG, which yields a shortest-path reformulation with complexity $$\min_{x,s}\; \frac{1}{2}x^\top P x + q^\top x \quad \text{subject to}\quad Ax + s = b,\;\; s\in K,$$3, followed by an $$\min_{x,s}\; \frac{1}{2}x^\top P x + q^\top x \quad \text{subject to}\quad Ax + s = b,\;\; s\in K,$$4 shortest-path computation on the DAG (Lee et al., 2024).

Sparse nonconvex quadratic minimization over the unit hypercube also admits exact SOC descriptions under graph conditions. For

$$\min_{x,s}\; \frac{1}{2}x^\top P x + q^\top x \quad \text{subject to}\quad Ax + s = b,\;\; s\in K,$$5

a continuous extension of the Reformulation Linearization Technique is combined with perspective-type inequalities. If the nodes with plus loops form a stable set, then the convex hull $$\min_{x,s}\; \frac{1}{2}x^\top P x + q^\top x \quad \text{subject to}\quad Ax + s = b,\;\; s\in K,$$6 is SOC-representable. If, in addition, there exists a tree decomposition such that each bag contains at most one plus-loop node, the treewidth is bounded by $$\min_{x,s}\; \frac{1}{2}x^\top P x + q^\top x \quad \text{subject to}\quad Ax + s = b,\;\; s\in K,$$7, and the spread of each plus-loop node is $$\min_{x,s}\; \frac{1}{2}x^\top P x + q^\top x \quad \text{subject to}\quad Ax + s = b,\;\; s\in K,$$8, then $$\min_{x,s}\; \frac{1}{2}x^\top P x + q^\top x \quad \text{subject to}\quad Ax + s = b,\;\; s\in K,$$9 has a polynomial-size SOC-representable formulation. Under these conditions, the optimal value of the original nonconvex quadratic program equals the optimal value of a polynomial-size SOCP (Dey et al., 25 Aug 2025).

4. Algorithms and solver architectures

For cone-constrained convex QPs, one influential route is reformulation as a nonsmooth equation. In the simplicial-cone model, the central equation is

$$\max_{x,z}\; -\frac{1}{2}x^\top P x - b^\top z \quad \text{subject to}\quad Px + A^\top z = -q,\;\; z\in K^*,$$0

Any solution $$\max_{x,z}\; -\frac{1}{2}x^\top P x - b^\top z \quad \text{subject to}\quad Px + A^\top z = -q,\;\; z\in K^*,$$1 yields a solution of the original quadratic program, namely $$\max_{x,z}\; -\frac{1}{2}x^\top P x - b^\top z \quad \text{subject to}\quad Px + A^\top z = -q,\;\; z\in K^*,$$2. A semi-smooth Newton method is then applied to

$$\max_{x,z}\; -\frac{1}{2}x^\top P x - b^\top z \quad \text{subject to}\quad Px + A^\top z = -q,\;\; z\in K^*,$$3

using generalized Jacobian element

$$\max_{x,z}\; -\frac{1}{2}x^\top P x - b^\top z \quad \text{subject to}\quad Px + A^\top z = -q,\;\; z\in K^*,$$4

and iteration

$$\max_{x,z}\; -\frac{1}{2}x^\top P x - b^\top z \quad \text{subject to}\quad Px + A^\top z = -q,\;\; z\in K^*,$$5

If $$\max_{x,z}\; -\frac{1}{2}x^\top P x - b^\top z \quad \text{subject to}\quad Px + A^\top z = -q,\;\; z\in K^*,$$6, the sequence converges $$\max_{x,z}\; -\frac{1}{2}x^\top P x - b^\top z \quad \text{subject to}\quad Px + A^\top z = -q,\;\; z\in K^*,$$7-linearly from any starting point with rate

$$\max_{x,z}\; -\frac{1}{2}x^\top P x - b^\top z \quad \text{subject to}\quad Px + A^\top z = -q,\;\; z\in K^*,$$8

and the numerical results reported convergence in very few iterations, typically around three, even for dimensions up to $$\max_{x,z}\; -\frac{1}{2}x^\top P x - b^\top z \quad \text{subject to}\quad Px + A^\top z = -q,\;\; z\in K^*,$$9 (Barrios et al., 2015).

For broader convex quadratic conic programming, the Schur complement based semi-proximal ADMM constructs a convergent multi-block splitting for problems with a single coupling linear equality constraint and an objective equal to the sum of two proper closed convex functions plus an arbitrary number of convex quadratic or linear functions. The method is particularly suitable for quadratic semidefinite programming with linear equalities, a positive semidefinite cone, and a simple convex polyhedral set. Under the stated constraint qualification and positive-definiteness conditions on the proximalized blocks, the iterates are well defined and the full sequence converges; numerically, the method is intended for low-to-medium accuracy and is often used to generate a good initial point for a later high-accuracy method (Li et al., 2014).

Interior-point solvers increasingly treat the quadratic objective natively rather than converting it to an epigraph and extra SOC constraints. Clarabel solves

$$Ax+s=b,\qquad Px+A^\top z=-q,\qquad s^\top z = 0,\qquad (s,z)\in K\times K^*.$$0

through a homogeneous embedding specialized to the quadratic objective, supports symmetric cones $$Ax+s=b,\qquad Px+A^\top z=-q,\qquad s^\top z = 0,\qquad (s,z)\in K\times K^*.$$1, $$Ax+s=b,\qquad Px+A^\top z=-q,\qquad s^\top z = 0,\qquad (s,z)\in K\times K^*.$$2, $$Ax+s=b,\qquad Px+A^\top z=-q,\qquad s^\top z = 0,\qquad (s,z)\in K\times K^*.$$3, $$Ax+s=b,\qquad Px+A^\top z=-q,\qquad s^\top z = 0,\qquad (s,z)\in K\times K^*.$$4 and nonsymmetric cones such as the exponential and power cones, and includes chordal decomposition methods for semidefinite cones. QOCO and QOCOGEN focus on quadratic-objective SOCPs with product cones of nonnegative orthants and second-order cones, using a primal-dual interior-point method with Mehrotra’s predictor-corrector, Nesterov–Todd scaling, quasidefinite regularization, and custom linear algebra; the generated solvers use static memory allocation only and target embedded repeated-solve settings (Goulart et al., 2024, Chari et al., 16 Mar 2025).

5. Parametric, online, and differentiable quadratic cone programs

In control and online optimization, regularity of the solution map is often as important as raw solve time. One conservative QP-to-SOCP reformulation starts from

$$Ax+s=b,\qquad Px+A^\top z=-q,\qquad s^\top z = 0,\qquad (s,z)\in K\times K^*.$$5

and replaces the original polyhedron by the ball

$$Ax+s=b,\qquad Px+A^\top z=-q,\qquad s^\top z = 0,\qquad (s,z)\in K\times K^*.$$6

with

$$Ax+s=b,\qquad Px+A^\top z=-q,\qquad s^\top z = 0,\qquad (s,z)\in K\times K^*.$$7

equivalently

$$Ax+s=b,\qquad Px+A^\top z=-q,\qquad s^\top z = 0,\qquad (s,z)\in K\times K^*.$$8

The resulting SOCP has the closed-form solution

$$Ax+s=b,\qquad Px+A^\top z=-q,\qquad s^\top z = 0,\qquad (s,z)\in K\times K^*.$$9

and, under the stated unit-norm row assumption, Lipschitz assumptions, and existence of a feasible Lipschitz selector $\{0\}$0, the minimizer is unique, belongs to the original feasible set $\{0\}$1, and is Lipschitz. The construction is conservative because $\{0\}$2 (Agrawal et al., 25 Aug 2025).

Differentiation through a quadratic cone program is now treated directly at the level of the homogeneous embedding. For the primal-dual pair with data $\{0\}$3, the solution map

$\{0\}$4

is analyzed by introducing a homogeneous primal-dual embedding, writing complementarity through projections via the Minty parameterization, and defining a normalized residual map $\{0\}$5. When the derivative $\{0\}$6 is invertible and the relevant cone projections are differentiable, the implicit function theorem yields

$\{0\}$7

and hence the derivative of the primal-dual solution map. The implementation diffqcp is designed around matrix-free JVPs and VJPs, supports CPU and GPU execution, and on a reported test problem with $\{0\}$8, $\{0\}$9 achieved 44.20 s/iteration for CuClarabel and diffqcp versus 96.86 s/iteration for Clarabel and diffcp, a 2.19× speedup (Healey et al., 24 Aug 2025).

For repeatedly solved online problems, conditioning can dominate algorithmic performance. A hypersphere preconditioner for strongly convex QCPs begins with the Cholesky factorization

$\mathbb{R}_+$0

uses the change of variables

$\mathbb{R}_+$1

and transforms the objective to $\mathbb{R}_+$2. After block row-normalization of the constraints, the associated KKT matrix has a condition number minimized by

$\mathbb{R}_+$3

where $\mathbb{R}_+$4 is the smallest eigenvalue of $\mathbb{R}_+$5. The paper also derives the key parameter relation

$\mathbb{R}_+$6

for the proportional-integral projected gradient method. On a nonconvex multi-phase rocket landing guidance problem solved via sequential conic optimization in 6 SeCO iterations, the optimal scaling reduced both KKT condition numbers and the number of PIPG iterations required for convergence (Kamath et al., 24 Jan 2025).

6. Related conic frameworks, exactness questions, and broader variants

Quadratic cone programming also interfaces with exact conic reformulations over cones stronger than the second-order cone. A standard quadratic program

$\mathbb{R}_+$7

admits the exact completely positive reformulation

$\mathbb{R}_+$8

Replacing $\mathbb{R}_+$9 by the doubly nonnegative cone $Ax+s=b$00 gives the DNN relaxation

$Ax+s=b$01

with $Ax+s=b$02. Exactness is characterized by

$Ax+s=b$03

and positive-gap instances are characterized by

$Ax+s=b$04

This places exactness questions for quadratic optimization within copositive and completely positive cone geometry (Gokmen et al., 2020).

A different cone-based tractability theory arises from the quadratic Graver cone. For the integer quadratic problem

$Ax+s=b$05

polynomial-time solvability is obtained, given the Graver basis $Ax+s=b$06, when the quadratic matrix lies in the dual quadratic Graver cone

$Ax+s=b$07

This framework covers some nonconvex quadratic integer programs and some convex ones, including all separable convex quadratics through the diagonal dual cone $Ax+s=b$08. It also establishes that $Ax+s=b$09 and the PSD cone are incomparable: neither contains the other (Lee et al., 2010).

These related theories clarify a common misconception. “Quadratic cone program” does not denote a single invariant model class. In one strand it means convex conic optimization with a quadratic objective; in another it refers to QPs over cones such as simplicial or polyhedral cones; in another it designates exact or tight SOCP, SDP, or completely positive convexifications of nonconvex quadratic sets. What unifies these usages is not a single syntax, but the role of cone geometry in making quadratic structure representable, convexifiable, differentiable, or computationally tractable (Goulart et al., 2024, Barrios et al., 2015, Santana et al., 2018).