Dual Cone: Theory and Applications

• Dual cone is a fundamental construct in convex analysis defined as the set of vectors yielding nonnegative inner products with all elements in the original cone.
• Its structural properties enable primal-dual relationships, underpinning separation theorems and intersection-sum identities in optimization.
• Applications span constructing feasibility certificates, efficient algorithms for polynomial optimization, and forming tensor product frameworks in operator systems.

A dual cone is a fundamental construct in convex analysis, optimization, and operator systems, providing a polar perspective on cone membership via inner product inequalities. Given a closed convex cone $K \subseteq V$ within a finite-dimensional real vector space $V$, its dual cone $K^* \subseteq V^*$ is the set of all linear functionals on $V$ that take nonnegative values on $K$. Under an inner product identification $V \cong V^*$, this is frequently written $$K^* = \{ y \in V : \langle y, x \rangle \ge 0 \ \forall x \in K \}$$. Dual cones characterize primal-dual relationships, facilitate proofs in separation and optimization theory, underlie tensor product constructions, and appear as domains for self-concordant barriers, which are crucial in polynomial and matrix optimization.

1. Formal Definition and Basic Properties

The dual cone of a closed convex $K \subseteq V$ is: $$K^* = \left\{ \phi \in V^* \mid \langle \phi, x \rangle \ge 0 \quad \forall x \in K \right\}$$ or, using an inner product identification,

$$K^* = \left\{ y \in V \mid \langle y, x \rangle \ge 0 \ \forall x \in K \right\}$$

Key properties include:

• $K \subseteq K^{**}$, with equality iff $K$ is closed and convex.
• If $K$ is proper (closed, pointed, full-dimensional), so is $K^*$.
• The classical dual-cone calculus identity for closed convex cones $K_1, K_2$ in $V$ is:

$(K_1 \cap K_2)^* = K_1^* + K_2^*$

where $+$ denotes Minkowski sum (Hauser, 2013). This formula generalizes in two dimensions to non-convex sets under homogenization-convexity.

2. Dual Cones in Cone-Systems

Let $\mathcal{S} \colon \mathsf{C} \to \mathsf{FVec}$ be a "stem" assigning finite-dimensional real vector spaces to objects in an indexing category $\mathsf{C}$ (e.g., simplices, Hilbert spaces). An $\mathcal{S}$-system on $X$ is a functor $\mathcal{G} : \mathsf{C} \to \mathsf{FCone}$ with compatibility: for each $c$, $\mathcal{G}(c) \subseteq X \otimes \mathcal{S}(c)$ is a closed convex cone, and for every morphism $\varphi : c \to d$, the map $\mathrm{id}_X \otimes \mathcal{S}(\varphi)$ sends $\mathcal{G}(c)$ into $\mathcal{G}(d)$. The dual system $\mathcal{G}^\vee$ is defined level-wise: $$\mathcal{G}^\vee(c) = \left( \mathcal{G}(c^*) \right)^* \subseteq (X \otimes \mathcal{S}(c^*))^* \cong X \otimes \mathcal{S}(c)$$ Self-duality for a system occurs when $\mathcal{G} = \mathcal{G}^\vee$ (Netzer, 14 Aug 2024). This unifies dual cones into categorical tensor product and system frameworks.

3. Dual Cone Calculus and Intersection-Sum Relations

Dual cones encode polar relationships and facilitate structural theorems. For closed convex cones, the intersection-sum formula $(K_1 \cap K_2)^* = K_1^* + K_2^*$ ensures representation of functionals nonnegative on $K_1 \cap K_2$ as sums from $K_1^*$ and $K_2^*$ (Hauser, 2013). Extensions to nonconvex or nonclosed sets in $\mathbb{R}^2$ apply when homogenization-convexity holds.

In optimization, vectors in the dual cone traditionally certify non-membership in $K$ via separation (hyperplanes with negative inner products). Recent theory has established that certain dual cone vectors can serve as constructive certificates for $b \in K$, leveraging barrier functions and exploiting the interior geometry of $K^*$ (Lee et al., 13 Jun 2025).

4. Applications: Certificates, Tensor Product Structure, and Polynomial Cones

Dual cone elements underpin the computation of primal feasibility certificates: given a suitable logarithmically-homogeneous self-concordant barrier (LHSCB) on $K^*$, one constructs $B$-certificates of membership, utilizing the Hessian and gradient of the barrier: $B(y) = H(y) + g(y) g(y)^T$ If $B(y)^{-1} b \in K$ for some $y \in (K^*)^\circ$, then $b \in K$. This enables closed-form primal solution extraction from dual iterates. In polynomial optimization, the cones for SOS, SONC, and SAGE polynomials admit explicit dual cone representations and barrier structures enabling efficient dual certification and primal recovery (Dressler et al., 2018, Lee et al., 13 Jun 2025).

Tensor product constructions of cones use duality: the minimal tensor product $K \otimes_{\min} L$ is the smallest convex cone containing all $x \otimes y$ for $x \in K$, $y \in L$. The maximal tensor product is the polar of the minimal tensor product of polars: $K \otimes_{\max} L = (K^* \otimes_{\min} L^*)^*$ Minimal and maximal tensor products are polar-dual pairs, with explicit formulas dictating their structure. In quantum bipartite cones, the cone of separable states vs. block-positive matrices exemplifies the distinction, with the full positive semidefinite cone sitting as a self-dual intermediate (Netzer, 14 Aug 2024).

5. Concrete Dual Cones: Copositive, Completely Positive, and SONC Structures

The copositive cone $C$ of symmetric $n \times n$ matrices comprises those $A$ with $x^T A x \ge 0$ for all $x \ge 0$; its dual $C^*$ is the cone of completely positive matrices, convex hulls of $xx^T, x \ge 0$ (Lasserre, 2010). Nested hierarchies of spectrahedral (semidefinite representable) approximations converge to these cones from inside and outside, enabling practical computational schemes.

For SONC polynomials supported on $\mathcal{A} \subseteq \mathbb{N}_0^n$, the dual cone $(C_{\mathrm{sonc}(\mathcal{A})})^*$ is defined by explicit nonnegativity conditions on components indexed by exponents and by existence of slack and entropy certificates ensuring relative-entropy inequalities across "circuits". These provide optimization and duality characterizations for conic programming relaxations (Dressler et al., 2018).

6. Dual cone identities in Operator Systems and Extension Theorems

In operator systems and tensor product categories, dual cone structures enable the existence of self-dual tensor products and systems. If a cone system $\mathcal{G}$ meets $\mathcal{G} \subseteq \mathcal{G}^\vee$, a maximal self-dual enlargement $\mathcal{E}$ exists such that $$\mathcal{G} \subseteq \mathcal{E} = \mathcal{E}^\vee \subseteq \mathcal{G}^\vee$$. The construction leverages order-theoretic methods via Zorn's lemma, closure properties under system operations, and boundary arguments establishing self-duality of the enlargement (Netzer, 14 Aug 2024). This framework generalizes the operational calculus for cones to functorial and categorical realms.

7. Computational Frameworks and Practical Usage

Numerical algorithms for polynomial optimization, convex feasibility, and conic programming exploit dual cone structure for constructing certificates and recovering primal solutions. Algorithms such as short-step interior-point methods use the geometry of $K^*$ under barrier functions to generate iterates $y$ that serve as membership certificates in $K$ through matrix-based preimage tests. For linear images $K_A = \{ Ax \mid x \in K \}$, explicit reconstruction formulas yield $x \in K$ with $A x = b$ precisely, conferring robust verification and error bounds tied to dual optimality (Lee et al., 13 Jun 2025).

This dual cone theory delivers strong guarantees for convergence, verifiability, and expressive certificate generation across cones relevant to matrix analysis, optimization, operator theory, and polynomial nonnegativity, with deep links to classical separation, tensor calculus, and modern computational frameworks.