We first prove the second part of the Theorem. Now, we will prove the first part of the Theorem. The proof is by induction. The following auxiliary proposition is crucial to prove the theorem. A for the first attribute.

That is, we simultaneously both solve for the DEQ fixed point and optimize over network inputs, all within a single "augmented" DEQ model that jointly encodes both the original network and the optimization

Origami-inspired mechanisms can transform flat sheets into functional three-dimensional dynamic structures that are lightweight, compact, and capable of complex motion. These properties make origami increasingly valuable in robotic and deployable systems. However, accurately simulating their folding behavior and interactions with the environment remains challenging. To address this, we present a design framework for origami mechanism simulation that utilizes MuJoCo's deformable-body capabilities. In our approach, origami sheets are represented as graphs of interconnected deformable elements with user-specified constraints such as creases and actuation, defined through an intuitive graphical user interface (GUI). This framework allows users to generate physically consistent simulations that capture both the geometric structure of origami mechanisms and their interactions with external objects and surfaces. We demonstrate our method's utility through a case study on an origami catapult, where design parameters are optimized in simulation using the Covariance Matrix Adaptation Evolution Strategy (CMA-ES) and validated experimentally on physical prototypes. The optimized structure achieves improved throwing performance, illustrating how our system enables rapid, simulation-driven origami design, optimization, and analysis.

Constrained non-convex optimization is fundamentally challenging, as global solutions are generally intractable and constraint qualifications may not hold. However, in many applications, including safe policy optimization in control and reinforcement learning, such problems possess hidden convexity, meaning they can be reformulated as convex programs via a nonlinear invertible transformation. Typically such transformations are implicit or unknown, making the direct link with the convex program impossible. On the other hand, (sub-)gradients with respect to the original variables are often accessible or can be easily estimated, which motivates algorithms that operate directly in the original (non-convex) problem space using standard (sub-)gradient oracles. In this work, we develop the first algorithms to provably solve such non-convex problems to global minima. First, using a modified inexact proximal point method, we establish global last-iterate convergence guarantees with $\widetilde{\mathcal{O}}(\varepsilon^{-3})$ oracle complexity in non-smooth setting. For smooth problems, we propose a new bundle-level type method based on linearly constrained quadratic subproblems, improving the oracle complexity to $\widetilde{\mathcal{O}}(\varepsilon^{-1})$. Surprisingly, despite non-convexity, our methodology does not require any constraint qualifications, can handle hidden convex equality constraints, and achieves complexities matching those for solving unconstrained hidden convex optimization.

Kernel power $k$-means (KPKM) leverages a family of means to mitigate local minima issues in kernel $k$-means. However, KPKM faces two key limitations: (1) the computational burden of the full kernel matrix restricts its use on extensive data, and (2) the lack of authentic centroid-sample assignment learning reduces its noise robustness. To overcome these challenges, we propose RFF-KPKM, introducing the first approximation theory for applying random Fourier features (RFF) to KPKM. RFF-KPKM employs RFF to generate efficient, low-dimensional feature maps, bypassing the need for the whole kernel matrix. Crucially, we are the first to establish strong theoretical guarantees for this combination: (1) an excess risk bound of $\mathcal{O}(\sqrt{k^3/n})$, (2) strong consistency with membership values, and (3) a $(1+\varepsilon)$ relative error bound achievable using the RFF of dimension $\mathrm{poly}(\varepsilon^{-1}\log k)$. Furthermore, to improve robustness and the ability to learn multiple kernels, we propose IP-RFF-MKPKM, an improved possibilistic RFF-based multiple kernel power $k$-means. IP-RFF-MKPKM ensures the scalability of MKPKM via RFF and refines cluster assignments by combining the merits of the possibilistic membership and fuzzy membership. Experiments on large-scale datasets demonstrate the superior efficiency and clustering accuracy of the proposed methods compared to the state-of-the-art alternatives.

I n light of this, we introduce MA TAI, a generalist ML framework for alloy property prediction and inverse design. Unlike task - specific models, MA TAI integrate s domain knowledge from diverse alloy systems and support s multi - objective, constraint - aware optimization across broad compositional spaces . The framework consists of four core components: 1) a holistic alloy database containing over 10,000 experimentally verified compositions, aggregated from open databases, literature, and in - house experiments; 2) foundational property predictor s capable of estimating multiple alloy properties such as density, yield strength (YS), ultimate tensile s trength (UTS), and elongation directly from alloy compositions; 3) a generalist alloy designer that performs constrained optimization over multiple objectives, enabling the discovery of promising alloy candidates without exhaustive searches; and 4) an iterative AI - experiment feedback loop that continuously refines the model through experimental validation of AI - generated candidates . To demonstrate the effectiveness and robustness of MA TAI, we apply the framework to the titanium (Ti) - based alloys, a canonical aerospace alloy system valued for its low density with high strength . Using MA TAI, we identifi ed novel compositions that achieve high strength (>1000 MPa) and moderate elongation (>5%) while retaining a low density (< 4.45 g/cm

This paper introduces the distributed Halpern Peaceman--Rachford (dHPR) method, an efficient algorithm for solving distributed convex composite optimization problems with non-smooth objectives, which achieves a non-ergodic $O(1/k)$ iteration complexity regarding Karush--Kuhn--Tucker residual. By leveraging the symmetric Gauss--Seidel decomposition, the dHPR effectively decouples the linear operators in the objective functions and consensus constraints while maintaining parallelizability and avoiding additional large proximal terms, leading to a decentralized implementation with provably fast convergence. The superior performance of dHPR is demonstrated through comprehensive numerical experiments on distributed LASSO, group LASSO, and $L_1$-regularized logistic regression problems.