Local Search

Autoresearch agents tackle open-ended problems by repeatedly proposing candidate solutions, evaluating them, and using feedback to guide subsequent experiments. We show that independent runs of the same agent on the same task often plateau at substantially different scores, with gaps that persist even after considerable additional compute. Embedding their candidate artifacts by functional similarity provides further evidence that trajectories remain in localized regions of the solution space, which we call idea basins. To help agents escape these basins, we study a simple periodic intervention, fork-and-flush. Our method forks the agent into parallel trajectories, each inheriting the accumulated workspace but starting with a fresh chat context. After running each trajectory for a fixed horizon, the agent continues from the highest-scoring one. Across 13 long-horizon research and engineering tasks, with individual agent runs lasting up to several days, fork-and-flush outperformed the single-run and best-of-N baselines by a relative improvement of 66.0% and 44.4%, respectively, on the min-max normalized average score under an equal compute budget.

Lloyd's algorithm and the discrete local (D-local) optimization method (Li et al., 2025) for $k$-means provide only weak local-optimality guarantees, and their solution quality remains sensitive to initialization. In this paper, we introduce $r$-point local optimality, under which no reassignment of at most $r$ samples decreases the objective function, and focus on $r=2$. The main computational obstacle is the $\mathcal{O}(n^2(k^2+d))$ cost of exhaustive two-point certification for $n$ samples in $d$ dimensions and $k$ clusters. To address this challenge, we prove that (i) every improving two-point move of a D-local optimum must involve a cluster shared by both reassignments, and (ii) only certificate-defined boundary points can participate in an improving pair. Exploiting this structure, we propose Boundary-Point-Screened Two-Point Local Search (BPS-2PLS), which terminates at a two-point local optimum. For fixed $k,d$ and nonvanishing cluster occupancy, the number $m$ of retained candidates satisfies $m=\mathcal{O}_{\mathbb{P}}(\log n)$ under i.i.d. sampling from a bounded-support distribution with bounded density or from a Gaussian mixture. Across twelve benchmarks, BPS-2PLS attains the lowest available mean WCSS on ten. In a subsampling study, screening retains 0.10% to 2.81% of samples on average at the largest tested sizes. The code is available at https://github.com/lwl-learning/BPS-2PLS.

Heuristic design for combinatorial optimization remains heavily reliant on expert knowledge, while existing large language model (LLM)-enhanced evolutionary methods typically evolve isolated algorithmic components, even when one determines the search state on which another operates. This paper proposes LLM-driven Heuristic Components Joint Generation (LLM-HCJG), a population-based framework that jointly generates and co-evolves interdependent heuristic components under a shared design blueprint. Applied to guided local search (GLS), LLM-HCJG couples solution initialization with penalty construction and embeds the generated pair into an enhanced online search mechanism. The resulting form is further transferred from the traveling salesman problem (TSP) to the capacitated vehicle routing problem (CVRP). Theoretical analysis establishes the non-separable state-transition effects between the two components and the advantage in generation consistency. Across synthetic instances and 41 public TSPLIB/CVRPLIB benchmarks, LLM-HCJG attains consistently low optimality gaps, including best or tied-best results on 28 of 29 TSPLIB instances and all 12 CVRPLIB instances. Ablation and structural analyses further indicate that these gains are associated with cross-component compatibility and alignment rather than isolated-component recombination. These results support effective cross-instance transfer within the evaluated routing settings under limited-sample, modest-cost training.

Constrained single-objective numerical optimization requires a careful balance among feasibility, objective convergence, and computational efficiency under a fixed function-evaluation budget. This paper proposes DE-2LS, a late-stage, locally search-enhanced variant of differential evolution built on the RDEx framework. The proposed method preserves the original RDEx components, including mutation and crossover operators, success-history adaptation, archive mechanism, population-size reduction, and $ε$-based constraint handling. A lightweight coordinate-pattern local search is added as a guarded polishing component around the current best solution. It is activated only in the late stage of the run, uses a small evaluation budget, and accepts candidates through a feasibility-aware comparison rule. Ablation results show that the finalized DE-2LS configuration achieves the best U-score among all tested variants, confirming that controlled late-stage refinement is more effective than aggressive or premature local search. In the direct comparison with RDEx, DE-2LS achieves a 5.58% gain in U-score. In the four-algorithm comparison, DE-2LS obtains the highest overall U-score of 80968 and the best total rank of 48 among RDEx, CL-SRDE, and UDE-III. These results indicate that DE-2LS improves the exploitation capability of the RDEx-based search framework while preserving its speed advantage under the combined speed-accuracy scoring criterion. The source code of DE-2LS is available at https://github.com/ChauhanDikshit?tab=repositories.

Unconstrained single-objective numerical optimization requires a careful balance among global exploration, late-stage exploitation, and function-evaluation efficiency. This paper presents DE-2LS, a late-stage, local-search-enhanced differential evolution framework built on RDEx for unconstrained single-objective optimization with variable bounds. The proposed method preserves the original RDEx evolutionary search engine and introduces two conservative refinements: a smoothed exploitation-biased branch-rate update in the late search stage and a guarded coordinate-pattern local-search that serves as a budget-aware refinement mechanism. Since the considered setting is unconstrained apart from variable bounds, all selection and local-search acceptance decisions are based solely on objective values. To determine the final algorithm configuration, we conduct a staged ablation study by testing multiple settings of the EB-rate smoothing mechanism, the initial EB-rate, the standard-branch Gaussian sampling scale, the selection-pressure parameters, and the local-search coefficients. The final configuration is selected using a U-score-based evaluation that jointly reflects solution quality and convergence speed. Experimental results show that DE-2LS consistently improves the original RDEx in direct head-to-head comparison. In particular, DE-2LS increases the U-score from $33602.0$ to $37448.0$, corresponding to an improvement of $11.45\%$. Moreover, compared with several competitive and IEEE CEC-winning algorithms, DE-2LS achieves the best overall U-score of $178966.5$, outperforming the others by $34.43\%$. These results show that a carefully designed late-stage local-search strategy can improve both convergence speed and the final objective quality of the algorithm. The source code of DE-2LS is available at https://github.com/ChauhanDikshit?tab=repositories.

Local search is a well-known heuristic method used in optimization. In this thesis, we explore its capabilities on the vertex coloring problem, an $NP$-hard problem with relevance in both theoretical analysis and practical application. To recognize limitations in the applicability of local search of the vertex coloring problem, we analyze local search landscapes on differently-structured bipartite graphs. We identify structures that ensure only global optima can exist as well as ones that enable the existence of non-global local optima, showing that on general bipartite graphs, it is possible for local search to return arbitrarily bad results. Further, we analyze the capabilities of local search on graphs where a local optimum can be found. To do so, we introduce a gray-box local search mutation operator that removes less frequent colors with higher probability and prove that it finds an optimal coloring on complete bipartite graphs in an expected run time of $Θ(n \log n)$. This is a drastic improvement to the exponential tun time of the black-box Random Local Search, showing that gray-box mutation operators can improve the run time of local search.

Neural combinatorial optimization has recently achieved strong results on the Euclidean Traveling Salesman Problem (TSP) using generative models such as diffusion and consistency models. State-ofthe-art approaches like FT2T combine fast consistency-based prediction with gradient-based inference time refinement. However, gradient search often incurs significant computational overhead and may not align with the discrete structure of feasible solutions. We introduce Projected Consistency Inference (PCI), a plug-and-play, retraining-free alternative that replaces gradient refinement with structure-aware projections: PCI decodes valid Hamiltonian tours from the consistency model output and applies a lightweight local search (e.g., 2-opt). PCI achieves an average optimality gap (OG) of 0.17% on TSP with 500 cities, and 0.31% on TSP with 1000 cities, outperforming FT2T best settings (OG 0.22% and 0.36%, respectively) while reducing the inference time up to 30 to 40%. PCI also exhibits lower variance and memory usage, and can surpass classical heuristics such as LKH3 in rapid solution generation. Our results demonstrate that structure-aware inference time operations provide a practical and principled path for neural TSP solvers, complementing training time objectives.

We present HNTL (Hierarchical No-pointer Tangent-Local), the core vector indexing and candidate generation framework of the Aperon vector memory system. Proximity graphs (e.g., HNSW) incur a heavy pointer tax in memory overhead and induce irregular memory accesses that stall CPU pipelines. HNTL resolves this by partitioning the high-dimensional space into local, coherent grains, representing vectors as low-dimensional coordinates on local tangent spaces, and scanning them sequentially using a pointerless Block-SoA (Structure-of-Arrays) layout. On anisotropic manifold data (d=768, N=10,000), local PCA captures 96.3% of the variance, allowing HNTL to achieve a final Rerank Recall@10 of 1.0000 with a candidate pool size of only C=20 vectors. Hardware profiling via Apple kperf CPU Performance Monitoring Unit (PMU) counters demonstrates a 3.61x speedup (4.137 ns/vector vs. 14.951 ns/vector) for our NEON auto-vectorized C++ Block-SoA scan engine over standard pointer-chasing graph traversals, driven by a 3.59x IPC (Instructions Per Cycle) and near-zero L1/L2 data cache misses.

We analyze the two combinatorial problems of Dominating Set and Vertex Coloring regarding what kind of local optima are present for various instances. For a variety of graph classes each, we determine whether the induced landscapes are unimodal, plateau-unimodal (all optima are just one plateau), equimodal (all local optima are global) or truly multimodal. We do this for two different neighborhood operators, one based on making only a single change and one also allowing swaps (interchanging two parts of the solution).

Bayesian optimization (BO) is a popular and effective approach for tuning expensive, noisy experiments, but requires the formulation of an explicit objective function. Preferential BO (PBO) removes this requirement by learning from pairwise human feedback, yet existing methods struggle to efficiently optimize beyond low- and medium-dimensional problems due to their global search approaches. We address this limitation by developing a family of local PBO methods that transfer key ideas from high-dimensional BO to the preferential setting. In particular, we introduce local PBO methods which adapt trust-region and derivative-informed local search to pairwise preference feedback, where the latter exploits first- and second-order derivatives of the Laplace-approximated GP posterior. Our benchmark on GP sample paths, standard optimization benchmark functions, and policy-search tasks shows that local PBO methods are especially effective in high-dimensional and complex landscapes with steep optima. Compared with global preference-based baselines, they can substantially reduce cumulative regret, making them particularly useful for real-world preference-based optimization tasks such as policy search.

Applying local search algorithms to combinatorial optimization problems is not an easy feat. Typically, human intervention is required to compile the constraints to input data for some metaheuristic algorithm. In this paper, we establish a link between symmetry properties of constraint optimization problems and local search neighborhoods, and we use this link to automatically generate neighborhoods from a constraint specification in the context of the IDP system. We evaluate the obtained neighborhoods for six classical optimization problems. The resulting observations support the viability of this technique.

We introduce the penalised Saito functional $\mathfrak S_{λ,β}(\mathcal{A};d_1,d_2)$ for a reduced arrangement $\mathcal{A}$ of $n$ lines and a prescribed pair $d_1+d_2=n-1$. It measures the alignment of a candidate Saito determinant with the defining polynomial while penalising the failure of the candidate derivations to be logarithmic. We prove that the functional takes values in $[0,1]$, vanishes exactly when $\mathcal{A}$ is free with exponents $(1,d_1,d_2)$, and lies strictly between $0$ and $1$ otherwise. For fixed $(d_1,d_2)$, it is upper semicontinuous on the reduced configuration space, continuous at arrangements free with the prescribed pair, and converges as $λ\to\infty$ to the corresponding binary freeness test. We use a numerical approximation of this functional, together with a small $b_2$-shell term, to guide fixed-cardinality line-replacement searches over $\mathbb{Q}$ and selected quadratic extensions. Numerical values are used only to select candidates; every reported arrangement is certified in exact arithmetic using Saito's criterion. At the current snapshot, the certified database contains $6{,}146$ representatives with distinct Weisfeiler--Leman fingerprints and cardinalities up to $n=28$. Among them, $3{,}012$ have multiplicity gap $ε(\mathcal{A})=d_1-m(\mathcal{A})\geq2$, including lower-bound-extremal examples with $ε=7$. These non-supersolvable arrangements provide test cases for studying realisation spaces and the persistence of freeness among realisations of the same intersection lattice, in connection with Terao's conjecture.

Discretizing raw features into bucketized attribute representations is a popular step before sharing a dataset. It is, however, evident that this step can cause significant bias in data and amplify unfairness in downstream tasks. In this paper, we address this issue by introducing the unbiased binning problem that, given an attribute to bucketize, finds its closest discretization to equal-size binning that satisfies group parity across different buckets. Defining a small set of boundary candidates, we prove that unbiased binning must select its boundaries from this set. We then develop an efficient dynamic programming algorithm on top of the boundary candidates to solve the unbiased binning problem. Finding an unbiased binning may sometimes result in a high price of fairness, or it may not even exist, especially when group values follow different distributions. Considering that a small bias in the group ratios may be tolerable in such settings, we introduce the epsilon-biased binning problem that bounds the group disparities across buckets to a small value epsilon. We first develop a dynamic programming solution, DP, that finds the optimal binning in quadratic time. The DP algorithm, while polynomial, does not scale to very large settings. Therefore, we propose a practically scalable algorithm, based on local search (LS), for epsilon-biased binning. The key component of the LS algorithm is a divide-and-conquer (D&C) algorithm that finds a near-optimal solution for the problem in near-linear time. We prove that D&C finds a valid solution for the problem unless none exists. The LS algorithm then initiates a local search, using the D&C solution as the upper bound, to find the optimal solution.