Paul Jünger | Searcharxiv

arXiv2026

Sparse Cuts for the Positive Semidefinite Cone

We consider optimization problems containing nonconvex quadratic functions for which semidefinite programming (SDP) relaxations often yield strong bounds. We investigate linear inequalities that outer approximate the positive semidefinite cone and are sparse in the sense that they are supported only on the variables corresponding to products of variables present in quadratic functions. We show that these sparse linear inequalities yield an LP relaxation that gives the same bound as the SDP relaxation. We demonstrate how to identify these inequalities via a separation procedure that involves solving a structured ``projection'' SDP. In a computational study, we find that the sparse LP relaxations defined by these inequalities can accelerate branch-and-bound methods for globally solving nonconvex optimization problems.

math.OC↗

arXiv2026

Self-Augmenting Retrieval for Diffusion Language Models

Discrete diffusion language models generate text by iteratively denoising an entire response in parallel. At each step, they predict tentative tokens for every masked position, committing the confident predictions to the output and discarding the unconfident ones. We show that the discarded tokens are in fact a useful lookahead signal for retrieval-augmented generation: even low-confidence tokens often surface salient entities early in the denoising trajectory, enabling retrieval of stronger evidence before the output is finalized. We exploit this through Self-Augmenting Retrieval for Diffusion Language Models (SARDI), a dynamic RAG framework that uses these lookahead tokens to guide retrieval during denoising. SARDI is training-free, retriever-agnostic, and applicable to any reasoning-capable discrete diffusion language model. Across five multi-hop QA benchmarks, SARDI outperforms current training-free diffusion and autoregressive retrieval baselines at up to $8\times$ higher throughput.

cs.CL↗