Complementarity AKKT2 in Optimization

• Complementarity AKKT2 (CAKKT2) is a sequential optimality condition that integrates both complementarity and curvature information to ensure local optimality in nonlinear conic problems.
• It is applied to complex scenarios such as nonlinear semidefinite programming, second-order cone programming, and mathematical programs with cardinality constraints.
• CAKKT2 provides necessary optimality conditions without requiring constraint qualifications, thereby enhancing convergence analysis in degenerate and combinatorial settings.

Complementarity AKKT2 (CAKKT2) is a second-order sequential optimality condition in nonlinear optimization that is particularly relevant for problems with complementarity-type constraints, including nonlinear semidefinite optimization, second-order cone programming, mathematical programs with cardinality constraints, and tensor complementarity formulations. The CAKKT2 condition refines first-order sequential stationarity concepts by incorporating both complementarity and second-order “curvature” information, yielding necessary conditions for local optimality that do not require any constraint qualification (CQ). CAKKT2 guarantees that accumulation points of iterates from suitable algorithms satisfy both the complementarity and second-order conditions, facilitating a rigorous convergence theory even in degenerate scenarios.

1. Mathematical Formulation of CAKKT2

The CAKKT2 condition is a sequence-based necessary optimality criterion for nonlinear conic programs. In the nonlinear semidefinite programming (NSDP) setting, consider: $$\begin{array}{ll} \text{minimize} & f(x) \ \text{subject to} & h(x) = 0, \ & G(x) \succeq 0, \end{array}$$ where $f, h$ are smooth (twice continuously differentiable) and $G(x)$ is a symmetric matrix constraint.

The CAKKT2 condition at a feasible point $\bar{x}$ stipulates the existence of sequences $(x^k, \mu^k, \Omega_k, \epsilon_k)$ such that:

• $x^k \to \bar{x}$, $\mu^k \to \bar{\mu}$, $\Omega_k \to \bar{\Omega}$, $\epsilon_k \to 0$
• First-order CAKKT: $\nabla_x L(x^k, \mu^k, \Omega_k) \to 0$, $(G(x^k) \circ \Omega_k) \to 0$
• Second-order CAKKT2: for all directions $d$ in the perturbed critical subspace $S(x^k, \bar{x})$,

$$d^\top \left[ \nabla^2_{xx} L(x^k, \mu^k, \Omega_k) + \sigma(x^k, \Omega_k) \right] d \geq -\epsilon_k \|d\|^2$$

where $L$ is the Lagrangian, $\sigma(x, \Omega)$ is a problem-dependent curvature correction (e.g., in NSDP, this relates to conic curvature), and $S(x^k, \bar{x})$ encodes the current active constraints.

This extends naturally to SOCPs and other conic programs by replacing the conic constraint and Lagrangian accordingly (Fukuda et al., 2023).

2. Context and Motivation: Sequential Optimality Beyond CQs

Classical Karush-Kuhn-Tucker (KKT) conditions provide first- and second-order necessary criteria only under suitable CQs (e.g., Robinson’s CQ, LICQ). In many algorithms—especially for degenerate, singular, or combinatorially constrained problems—iterates may converge to points where CQs fail and classical KKT, or classical strong stationarity, need not be satisfied. AKKT and CAKKT introduce sequential (approximate) variants to sidestep this obstacle:

• AKKT is a first-order sequential stationarity condition using limit points of primal-dual sequences.
• CAKKT augments this with (approximate) complementarity: e.g., for conic constraints, $G(x^k) \circ \Omega_k \to 0$.
• CAKKT2 includes both of the above and further enforces a limiting second-order (curvature) inequality in critical directions.

CAKKT2 is thus necessary at any local minimizer, regardless of CQ, and directly targets what can be achieved by modern penalty algorithms operating in degenerate regimes (Li et al., 29 Jul 2025).

3. Formal CAKKT2 Conditions in Nonlinear Semidefinite and Second-Order Cone Programming

Table: Core Elements of the CAKKT2 Framework in NSDP and SOCP

Component	NSDP (Li et al., 29 Jul 2025)	SOCP (Fukuda et al., 2023)
Conic Constraint	$G(x) \succeq 0$	$g_i(x) \in \mathcal{K}_i$
Lagrangian	$L(x, \mu, \Omega)$	$L(x, \mu, \omega)$
Curvature/“Sigma” Term	$\sigma(x, \Omega)$	$\sigma(x, \omega)$
Subspace for Quadratic Test	$S(x^k, \bar{x})$ (perturbed critical)	All $d \in \mathbb{R}^n$; partitions by cone structure
CAKKT2 Second-order Inequality	$d^\top[\cdots]d \geq -\epsilon_k \|d\|^2$	$d^\top[\cdots]d \geq 0$ for all $d$ (plus other terms)

The role of the “sigma” term and the structure of the subspace reflect the particular geometry of the relevant conic constraints. For SOCPs, this includes additional terms reflecting the partitioning of indices and cone faces.

4. Algorithmic Strategies and Practical Attainment of CAKKT2 Points

Modern penalty and augmented Lagrangian methods for conic programs are designed so that their iterates, under mild assumptions, generate CAKKT2 sequences. Key elements are:

• Penalty Methods: Iteratively minimize

$$\phi_\rho(x) = f(x) + \frac{\rho}{2} \left( \|h(x)\|^2 + \| \Pi_{\mathbb{S}_+}(-G(x)) \|_F^2 \right)$$

with $\rho \to \infty$, then extract Lagrange and matrix multipliers from first-order conditions as $\mu^k = -\rho h(x^k)$, $\Omega_k = \rho \Pi_{\mathbb{S}_+}( -G(x^k) )$. Approximate second-order stationarity for the penalty subproblems, transferred via chain rules, yields the desired CAKKT2 sequence (Li et al., 29 Jul 2025).

• Twice Continuously Differentiable Penalty Functions: Employ a smooth penalty of the form

$$F(x; v, M, \rho, \sigma, \tau) = \rho f(x) + (\sigma \tau/2)\| (1/\tau)v - g(x) \|^2 + (\sigma \tau/4) \operatorname{tr}([(1/\tau)M - G(x)]_+^4)$$

with algorithmic updates and proper approximate minimization ensuring both the first- and second-order conditions from CAKKT2. Smoothness ensures all directional derivatives/Hessians are computable and stable, thus facilitating the use of Newton-type and trust-region methods (Yamakawa, 24 Sep 2025).

• Augmented Lagrangian and SQP: Both can be analyzed and tuned (via, e.g., parameter selection, stabilization terms, and multiplier updates) to yield CAKKT2 accumulation points, as proven for SOCPs in (Fukuda et al., 2023).

5. Relationship to Classical and Weak Second-Order Conditions

CAKKT2 strictly generalizes the classical second-order necessary condition. When Robinson’s CQ and the weak constant rank (WCR) property hold, any point satisfying CAKKT2 (or AKKT2) also satisfies the weak second-order necessary condition (WSONC): $$d^\top (\nabla^2_{xx} L(\bar{x}, \bar{\mu}, \bar{\Omega}) + \sigma(\bar{x}, \bar{\Omega})) d \geq 0, \quad \forall d \in S(\bar{x})$$ where $S(\bar{x})$ is the critical subspace with respect to the active constraints at $\bar{x}$. This embedding provides a direct connection to classical optimality theory, showing that the sequential approach converges to and refines established results when additional regularity is available (Li et al., 29 Jul 2025, Fukuda et al., 2023).

6. Extensions and Relevance in Broader Complementarity and Degenerate Regimes

CAKKT2 is applicable far beyond NSDPs and SOCPs. In mathematical programs with cardinality constraints (MPCaC), CAKKT2 refines the CAKKT and AW-stationarity conditions by enforcing stronger sequential and complementarity properties in the face of degeneracy. The implication chain CAKKT2 $\implies$ CAKKT $\implies$ AW-stationarity ensures that algorithms targeting CAKKT2 points resolve a broad class of degenerate behaviors (Krulikovski et al., 2020).

For nonlinear complementarity (NCP) and tensor complementarity problems (TCP), the column-adequate tensor framework guarantees ω-uniqueness, i.e., uniqueness of the dual (complementarity) variable for any X-solution. This property is directly relevant for ensuring the stability and uniqueness of the complementarity multipliers in CAKKT2-based algorithms (Dutta et al., 2022). Such uniqueness is critical for guaranteeing convergence and for extracting meaningful information from dual sequences.

7. Implications for Numerical Algorithms and Convergence Theory

The adoption of CAKKT2 as a target for optimization algorithms leads to rigorously justifiable convergence properties:

• Accumulation points of trust-region, Newton-type, penalty, or augmented Lagrangian methods can be guaranteed to satisfy both first- and second-order stationarity—including complementarity—for problems lacking any CQ.
• Smooth penalty functions (especially those twice continuously differentiable) facilitate the use of higher-order methods and ensure transfer of KKT-type conditions from the penalty subproblem to the original constrained problem, which is not possible with nonsmooth penalty terms (Yamakawa, 24 Sep 2025).
• The explicit sequence-based requirements of CAKKT2 allow for effective algorithmic stopping criteria, diagnostics, and robust global convergence analysis even in degenerate or combinatorial settings where constraint qualifications fail or are not checkable.

In summary, the CAKKT2 condition plays a pivotal role in the current theory and practice of degenerate and conic-constrained optimization, unifying sequential optimality conditions, complementarity structure, and second-order necessary conditions within a framework that is both theoretically complete and practically implementable.