Carathéodory-Based KKT Set Representation

• Carathéodory-Based KKT Set Representation is a decomposition method that leverages Carathéodory’s theorem to express KKT multipliers as sparse vectors determined by the rank of constraint matrices.
• It partitions the KKT set into a union of branches with explicit partial Lagrange-multiplier expressions, simplifying the analysis of polynomial generalized Nash equilibrium problems.
• The approach enables efficient global solutions via the Moment–SOS hierarchy, allowing complete certification of generalized Nash equilibria or their nonexistence.

A Carathéodory-based KKT set representation is a structured decomposition technique for the Karush-Kuhn-Tucker (KKT) sets arising in polynomial generalized Nash equilibrium problems (GNEPs) with quasi-linear (affine-in-own-strategy) constraints. Utilizing Carathéodory’s theorem for conic combinations, the approach expresses any KKT multiplier as a sparse vector with support size confined by the rank of the corresponding constraint matrix. This yields a union-of-branches description: the KKT set is represented as the union of finitely many smaller sets, each associated with a specific constraint support and given explicitly by partial Lagrange-multiplier expressions (pLMEs), leading to polynomial systems tractable by contemporary polynomial optimization solvers. This representation underpins numerically efficient global solution methods via the Moment–SOS hierarchy, allowing for complete computation or certification of nonexistence of generalized Nash equilibria in a broad class of polynomial GNEPs with many linear constraints (Choi et al., 2024).

1. Carathéodory’s Theorem in Multiplier Sparsity

Carathéodory’s theorem (conic case) states that, for any $y$ in the conic hull of $V\subset\mathbb{R}^n$, there exists a conic representation with at most $n$ vectors of $V$. Its crucial role in the KKT framework is to ensure that, for each player $i$ in a GNEP, the KKT multiplier $\lambda_i$ (associated with their linear constraints) can be replaced by an equivalent vector with at most $\mathrm{rank}(A_i)$ nonzero entries, where $A_i$ is the player’s constraint matrix. Thus, the support of multipliers in the KKT system can always be reduced to subsets of at most $r_i := \mathrm{rank}(A_i)$ coordinates, drastically reducing the complexity of the stationary points to consider (Choi et al., 2024).

2. KKT Conditions for Polynomial GNEPs with Quasi-Linear Constraints

Given an $N$-player GNEP where each player $i$ solves

$$\min_{x_i\in \mathbb{R}^{n_i}} f_i(x_i, x_{-i}), \quad \text{subject to} \quad g_i(x_i, x_{-i}) := A_i x_i - b_i(x_{-i}) \geq 0,$$

with $f_i$ a polynomial and $g_i$ affine in $x_i$, the standard KKT system consists of:

• Stationarity: $\nabla_{x_i} f_i(x) - A_i^\top \lambda_i = 0$,
• Primal feasibility: $A_i x_i - b_i(x_{-i}) \geq 0$,
• Dual feasibility: $\lambda_i \geq 0$,
• Complementarity: $\lambda_i^\top (A_i x_i - b_i(x_{-i})) = 0$.

The collection of all such conditions across players yields a polynomial system in $(x, \lambda)$ whose real solutions describe the KKT set $\mathcal{K}$, encompassing all potential GNEs (Choi et al., 2024).

3. Partial Lagrange Multiplier Expressions and Carathéodory Decomposition

Defining $\mathcal{P}_i$ as the family of $r_i$-element subsets $J_i \subseteq \{1,\dots,m_i\}$ for which the submatrix $A_{i,J_i}$ has full rank, one establishes, for each such $J_i$, a partial Lagrange-multiplier expression (pLME): $$\lambda_{i,J_i}(x) = (A_{i,J_i} A_{i,J_i}^\top)^{-1} A_{i,J_i} \nabla_{x_i} f_i(x),$$ with $\lambda_{i,j} = 0$ for $j \notin J_i$. Substituting the pLME into the remaining feasibility and complementarity conditions yields the "branch" KKT set

$$\mathcal{K}_J = \{x \in X:\ \lambda_{i,J_i}(x) \geq 0,\; (A_{i,J_i} x_i - b_{i,J_i}(x_{-i})) \geq 0,\; \lambda_{i,j}(x) (A_{i,j} x_i - b_{i,j}(x_{-i})) = 0\;\forall j \in J_i,\forall i\},$$

with $J = (J_1, ..., J_N)$. The union-of-branches theorem states

$$\mathcal{K} = \bigcup_{J \in \mathcal{P}_1 \times \cdots \times \mathcal{P}_N} \mathcal{K}_J,$$

guaranteeing completeness: each KKT point lies in some branch $\mathcal{K}_J$, and every solution to a branch system arises from a valid KKT point (Choi et al., 2024).

4. Structured Polynomial Optimization for Each Branch

Once $J$ is fixed, $\mathcal{K}_J$ is carved out by explicit polynomial equalities and inequalities in $x$. To identify KKT points in $\mathcal{K}_J$, one minimizes a strictly convex polynomial objective

$\Theta(x) = [x]_{2}^\top E [x]_{2},$

where $E$ is positive definite and $[x]_{2}$ denotes the collection of monomials up to degree 2 in $x$. The branch problem is

$$\min\ \Theta(x) \quad \text{subject to} \quad x \in \mathcal{K}_J,$$

which—if feasible—yields a unique KKT point per branch. To enumerate all KKT points in $\mathcal{K}_J$, this program is repeatedly solved with exclusion constraints to successively peel off isolated solutions. GNE status is certified by $N$ polynomial subproblems checking optimality for each player at candidate KKT points (Choi et al., 2024).

5. Global Solution via Moment–SOS Hierarchy

Each branch subproblem is a polynomial optimization task of the form $\min f(x)$ subject to given polynomial equalities $\{h_i(x) = 0\}$ and inequalities $\{g_j(x) \geq 0\}$. The standard Lasserre (Moment–SOS) hierarchy is applied:

• A moment matrix $M_k[y] \succeq 0$ for measure representation,
• Localizing matrices $L_{g_j}^{(k)}[y] \succeq 0$ for inequalities,
• Linear constraints from equality conditions.

Under archimedean and genericity premises, the hierarchy will reach finite convergence. Satisfying a flat-extension condition on the rank of $M_k[y]$ allows extraction of all real minimizers. Feasibility failure at any branch immediately certifies the emptiness of that $\mathcal{K}_J$ (Choi et al., 2024).

6. Correctness, Completeness, and Computational Characteristics

The theoretical guarantees underlying the approach include:

• Equivalence of the KKT system and the union of branch systems: every KKT point arises in some branch and vice versa (Theorem 3.2).
• The explicit enumeration of active constraint supports, via Carathéodory's theorem, reduces a potentially enormous KKT system into a finite—often tractable—collection of smaller polynomial systems (Theorem 3.3).
• Moment–SOS relaxation delivers finite-order global solutions for each branch subproblem under genericity (Theorem 5.2).
• The methodology yields a certified mechanism for finding all real GNEs or demonstrating their nonexistence in polynomial-GNEPs with quasi-linear constraints.

7. Illustrative Example

In a two-player GNEP with $m_1 = m_2 = 4$ (number of linear constraints each) and $\mathrm{rank}(A_1) = \mathrm{rank}(A_2) = 2$, one computes $\mathcal{P}_i$ as all two-element subsets of $\{1,2,3,4\}$, yielding $6 \times 6 = 36$ branches. For $J_1 = \{1,4\}$, $J_2 = \{1,2\}$, the explicit pLMEs are constructed, and substitution into the branch conditions leads to a polynomial system with four equations and eight inequalities in four variables. Solving via the described SDP hierarchy produces a singleton branch KKT set at $(18/49,\, 3/49,\, 0,\, 62/49)$, which is the unique GNE. This demonstrates how only reduced-support multipliers—guaranteed by Carathéodory’s theorem—need to be considered, allowing decomposition of large KKT systems into manageable polynomial branch systems and facilitating complete, globally certified solution by Moment-SOS and SDP tools (Choi et al., 2024).