Automated Valet Parking (AVP) is a crucial component of advanced autonomous driving systems, focusing on the endpoint task within the "human-vehicle interaction" process to tackle the challenges of the "last mile".The perception module of the automated parking algorithm has evolved from local perception using ultrasonic radar and global scenario precise map matching for localization to a high-level map-free Birds Eye View (BEV) perception solution.The BEV scene places higher demands on the real-time performance and safety of automated parking planning tasks. This paper proposes an improved automated parking algorithm based on the A* algorithm, integrating vehicle kinematic models, heuristic function optimization, bidirectional search, and Bezier curve optimization to enhance the computational speed and real-time capabilities of the planning algorithm.Numerical optimization methods are employed to generate the final parking trajectory, ensuring the safety of the parking path. The proposed approach is experimentally validated in the commonly used industrial CARLA-ROS joint simulation environment. Compared to traditional algorithms, this approach demonstrates reduced computation time with more challenging collision-risk test cases and improved performance in comfort metrics.

Local Bayesian optimization is a promising practical approach to solve the high dimensional black-box function optimization problem. Among them is the approximated gradient class of methods, which implements a strategy similar to gradient descent. These methods have achieved good experimental results and theoretical guarantees. However, given the distributional properties of the Gaussian processes applied on these methods, there may be potential to further exploit the information of the Gaussian processes to facilitate the BO search. In this work, we develop the relationship between the steps of the gradient descent method and one that minimizes the Upper Confidence Bound (UCB), and show that the latter can be a better strategy than direct gradient descent when a Gaussian process is applied as a surrogate. Through this insight, we propose a new local Bayesian optimization algorithm, MinUCB, which replaces the gradient descent step with minimizing UCB in GIBO. We further show that MinUCB maintains a similar convergence rate with GIBO. We then improve the acquisition function of MinUCB further through a look ahead strategy, and obtain a more efficient algorithm LA-MinUCB. We apply our algorithms on different synthetic and real-world functions, and the results show the effectiveness of our method. Our algorithms also illustrate improvements on local search strategies from an upper bound perspective in Bayesian optimization, and provides a new direction for future algorithm design.

The multidimensional knapsack problem (MKP) is an NP-hard combinatorial optimization problem whose solution is determining a subset of maximum total profit items that do not violate capacity constraints. Due to its hardness, large-scale MKP instances are usually a target for metaheuristics, a context in which effective feasibility maintenance strategies are crucial. In 1998, Chu and Beasley proposed an effective heuristic repair that is still relevant for recent metaheuristics. However, due to its deterministic nature, the diversity of solutions such heuristic provides is insufficient for long runs. As a result, the search for new solutions ceases after a while. This paper proposes an efficiency-based randomization strategy for the heuristic repair that increases the variability of the repaired solutions without deteriorating quality and improves the overall results.

In this paper, we introduce a local search algorithm for hierarchical clustering. For the local step, we consider a tree re-arrangement operation, known as the {\em interchange}, which involves swapping two closely positioned sub-trees within a tree hierarchy. The interchange operation has been previously used in the context of phylogenetic trees. As the objective function for evaluating the resulting hierarchies, we utilize the revenue function proposed by Moseley and Wang (NIPS 2017.) In our main result, we show that any locally optimal tree guarantees a revenue of at least $\frac{n-2}{3}\sum_{i < j}w(i,j)$ where is $n$ the number of objects and $w: [n] \times [n] \rightarrow \mathbb{R}^+$ is the associated similarity function. This finding echoes the previously established bound for the average link algorithm as analyzed by Moseley and Wang. We demonstrate that this alignment is not coincidental, as the average link trees enjoy the property of being locally optimal with respect to the interchange operation. Consequently, our study provides an alternative insight into the average link algorithm and reveals the existence of a broader range of hierarchies with relatively high revenue achievable through a straightforward local search algorithm. Furthermore, we present an implementation of the local search framework, where each local step requires $O(n)$ computation time. Our empirical results indicate that the proposed method, used as post-processing step, can effectively generate a hierarchical clustering with substantial revenue.

Large Language Models (LLMs) can generate useful code, but often the code they generate cannot be trusted to be sound. In this paper, we present VerMCTS, an approach to begin to resolve this issue by generating verified programs in Dafny and Coq. VerMCTS uses a logical verifier in concert with an LLM to guide a modified Monte Carlo Tree Search (MCTS). This approach leverages the verifier to gain intermediate feedback inside the search algorithm by checking partial programs at each step to estimate an upper bound on the value function. To measure the performance of VerMCTS, we develop a new suite of multi-step verified programming problems in Dafny and Coq. In terms of pass@T, a new metric which computes the pass rate given a budget of T tokens sampled from the LLM, VerMCTS leads to more than a 30% absolute increase in average pass@5000 across the suite over repeated sampling from the base language model.

Recent advances in automated theorem proving leverages language models to explore expanded search spaces by step-by-step proof generation. However, such approaches are usually based on short-sighted heuristics (e.g., log probability or value function scores) that potentially lead to suboptimal or even distracting subgoals, preventing us from finding longer proofs. To address this challenge, we propose POETRY (PrOvE Theorems RecursivelY), which proves theorems in a recursive, level-by-level manner in the Isabelle theorem prover. Unlike previous step-by-step methods, POETRY searches for a verifiable sketch of the proof at each level and focuses on solving the current level's theorem or conjecture. Detailed proofs of intermediate conjectures within the sketch are temporarily replaced by a placeholder tactic called sorry, deferring their proofs to subsequent levels. This approach allows the theorem to be tackled incrementally by outlining the overall theorem at the first level and then solving the intermediate conjectures at deeper levels. Experiments are conducted on the miniF2F and PISA datasets and significant performance gains are observed in our POETRY approach over state-of-the-art methods. POETRY on miniF2F achieves an average proving success rate improvement of 5.1%. Moreover, we observe a substantial increase in the maximum proof length found by POETRY, from 10 to 26.

Adaptive-submodularity is a widely used framework in stochastic optimization and machine learning [GK11, GK17, EKM21, ACN22]. Here, an algorithm makes sequential decisions while (partially) observing uncertainty. We study a basic problem in this context: covering an adaptive-submodular function at the minimum expected cost. We show that the natural greedy algorithm for this problem has approximation ratio at least 1.3 (1 + ln Q), where Q is the maximal function value. This is in contrast to special cases such as deterministic submodular cover or (independent) stochastic submodular cover, where the greedy algorithm achieves a tight (1 + ln Q) approximation ratio.

Active search formalizes a specialized active learning setting where the goal is to collect members of a rare, valuable class. The state-of-the-art algorithm approximates the optimal Bayesian policy in a budget-aware manner, and has been shown to achieve impressive empirical performance in previous work. However, even this approximate policy has a superlinear computational complexity with respect to the size of the search problem, rendering its application impractical in large spaces or in real-time systems where decisions must be made quickly. We study the amortization of this policy by training a neural network to learn to search. To circumvent the difficulty of learning from scratch, we appeal to imitation learning techniques to mimic the behavior of the expert, expensive-to-compute policy. Our policy network, trained on synthetic data, learns a beneficial search strategy that yields nonmyopic decisions carefully balancing exploration and exploitation. Extensive experiments demonstrate our policy achieves competitive performance at real-world tasks that closely approximates the expert's at a fraction of the cost, while outperforming cheaper baselines.

Combinatorial optimization (CO) is naturally discrete, making machine learning based on differentiable optimization inapplicable. Karalias & Loukas (2020) adapted the probabilistic method to incorporate CO into differentiable optimization. Their work ignited the research on unsupervised learning for CO, composed of two main components: probabilistic objectives and derandomization. However, each component confronts unique challenges. First, deriving objectives under various conditions (e.g., cardinality constraints and minimum) is nontrivial. Second, the derandomization process is underexplored, and the existing derandomization methods are either random sampling or naive rounding. In this work, we aim to tackle prevalent (i.e., commonly involved) conditions in unsupervised CO. First, we concretize the targets for objective construction and derandomization with theoretical justification. Then, for various conditions commonly involved in different CO problems, we derive nontrivial objectives and derandomization to meet the targets. Finally, we apply the derivations to various CO problems. Via extensive experiments on synthetic and real-world graphs, we validate the correctness of our derivations and show our empirical superiority w.r.t. both optimization quality and speed.

Multi-objective combinatorial optimization (MOCO) problems are prevalent in various real-world applications. Most existing neural MOCO methods rely on problem decomposition to transform an MOCO problem into a series of singe-objective combinatorial optimization (SOCO) problems. However, these methods often approximate partial regions of the Pareto front and spend excessive time on diversity enhancement because of ambiguous decomposition and time-consuming precise hypervolume calculation. To address these limitations, we design a Geometry-Aware Pareto set Learning algorithm named GAPL, which provides a novel geometric perspective for neural MOCO via a Pareto attention model based on hypervolume expectation maximization. In addition, we propose a hypervolume residual update strategy to enable the Pareto attention model to capture both local and non-local information of the Pareto set/front. We also design a novel inference approach to further improve quality of the solution set and speed up hypervolume calculation. Experimental results on three classic MOCO problems demonstrate that our GAPL outperforms several state-of-the-art baselines via superior decomposition and efficient diversity enhancement.