analyze kkt conditions

This work proposes a reference-set-free adaptive convergence metric for multi-objective optimization that addresses the scalability limitations of existing indicators when the true Pareto front is unknown. By leveraging the Karush–Kuhn–Tucker (KKT) optimality conditions, the method integrates an entropy-inspired stationarity measure with a quantile normalization mechanism to enhance robustness against heterogeneous residual distributions. While preserving the intrinsic interpretability of KKT-based analysis, the proposed metric significantly improves stability and applicability in both many-objective and high-dimensional scenarios, thereby overcoming the scalability bottlenecks inherent in conventional convergence indicators.

This paper investigates the computational complexity of computing Karush–Kuhn–Tucker (KKT) points for nonconvex quadratic programming over the unit hypercube $[0,1]^n$. Although KKT conditions constitute only first-order necessary optimality conditions—and are traditionally regarded as computationally easier than global optimization—we establish, for the first time, that computing KKT points is intrinsically hard: the problem is complete for the complexity class CLS (Continuous Local Search). Via a carefully constructed polynomial-time reduction from a CLS-complete problem to KKT point computation, and by integrating analytical tools from both PPAD and PLS, we rigorously prove that KKT point computation is computationally equivalent in difficulty to global optimization. This result breaks the conventional complexity paradigm focused solely on global optima, providing the first tight computational lower bound for first-order critical point computation and revealing the inherent computational robustness—i.e., intractability—of necessary optimality conditions in nonconvex optimization.

研究了在约束值从样本估计时，增广原始-对偶动力学的稳定性和收敛性问题，通过递归估计约束值来解决均衡偏差。

This work addresses the ill-conditioning and numerical instability inherent in traditional primal-dual interior-point methods for quadratic programming, which arise from explicitly enforcing complementarity conditions. To overcome this limitation, the authors propose a novel approach that implicitly satisfies the Karush–Kuhn–Tucker (KKT) complementarity conditions. By introducing auxiliary variables, employing a retraction mapping, and replacing the exponential map with the softplus function, the method ensures spectral boundedness of the KKT system, thereby fundamentally mitigating severe ill-conditioning near the solution. Coupled with a linear solver strategy that avoids matrix refactorization at each iteration, the proposed framework not only supports high-accuracy solutions and low-precision arithmetic but also opens new avenues for decomposition-free or indirect solution techniques tailored to large-scale quadratic programming problems.

For real-time parametric optimization problems (e.g., model predictive control), this paper proposes an end-to-end self-supervised neural iterative solver: a neural network first generates high-quality initial points, which are then refined by a differentiable primal-dual iterative module. The key contributions are twofold: (i) the design of the first KKT-based, label-free loss function, whose global minima are theoretically guaranteed to coincide exactly with KKT points; and (ii) a local convexification approximation strategy for non-convex problems, extending convergence guarantees to non-convex settings. The method requires no ground-truth labels and enables purely self-supervised training. Evaluated on two canonical non-convex benchmark tasks, it achieves a 10× speedup over IPOPT while attaining solution accuracy orders of magnitude higher than existing learning-based approaches.

This work addresses a class of nonconvex constrained optimization problems where both the objective and inequality constraints are compositions of convex Lipschitz outer functions with smooth inner mappings. The authors propose a smoothed proximal linear augmented Lagrangian method, reformulating the original problem as a nonsmooth nonconvex-concave minimax problem by restricting dual variables to a compact set. A finite-step mechanism is designed to map stationary points of the truncated minimax problem to KKT points of the original problem. Under a local cone regularity condition, they show that the artificial dual truncation automatically deactivates near feasible points, thereby establishing explicit convergence rates for the KKT residual: a global rate of $O(K^{-1/3})$ under dual regularization, which improves to $O(K^{-1/2})$ when the outer functions are piecewise linear and a local dual error bound holds.

本文提出了一种基于KKT点拉格朗日乘子特征的非凸优化分类方法，定义了五种操作模式，并通过数值实验验证了理论预测。

This study addresses whether the linear dependence of stochastic complexity on the condition number is necessary in nonconvex-strongly-concave minimax optimization. By constructing nonconvex zero-chains, dual gradient routing, and Moreau envelope stationarity criteria, this work develops lower bound instances that match the upper bound of the SAPD+ algorithm. It provides the first rigorous proof that this linear dependence cannot be improved, establishing a worst-case complexity of Θ(κLGσ²ε⁻⁴) for zero-respecting algorithms. Furthermore, it derives a combined stochastic-deterministic lower bound of Ω(LΔ(√κε⁻²+κσ²ε⁻⁴)), verifying optimal convergence rates and closing existing theoretical gaps.

本文针对光滑单调变分不等式问题，提出了一种新的二阶方法及更高阶方法，达到了最优收敛速度，解决了现有方法收敛速度低于理论下界的问题。

This study investigates whether accumulation points of bounded mirror descent sequences must necessarily be KKT stationary points. Focusing on Shannon entropy-induced mirror descent over the non-negative orthant (ℝ₊ⁿ, n≥3) and the probability simplex (Δₙ, n≥4), the authors construct smooth objective functions and corresponding iterates that yield, for the first time, counterexamples where accumulation points include non-KKT points—thereby challenging the prevailing assumption that all accumulation points are KKT points. By leveraging entropy-based relative smoothness, a Bregman geometric boundary degeneracy mechanism, and asymptotic step sizes αₖ ∼ k⁻ᵝ with β∈(1/2,1), they generate sequences whose objective values are monotonically non-increasing and whose accumulation sets form smooth boundary arcs containing non-stationary points. This reveals that mirror descent may converge to non-stationary points near the boundary of the feasible region.