In this work, we present several tools for efficient sequential hierarchical least-squares programming (S-HLSP) for lexicographical optimization tailored to robot control and planning. As its main step, S-HLSP relies on approximations of the original non-linear hierarchical least-squares programming (NL-HLSP) to a hierarchical least-squares programming (HLSP) by the hierarchical Newton's method or the hierarchical Gauss-Newton algorithm. We present a threshold adaptation strategy for appropriate switches between the two. This ensures optimality of infeasible constraints, promotes numerical stability when solving the HLSP's and enhances optimality of lower priority levels by avoiding regularized local minima. We introduce the solver $\mathcal{N}$ADM$_2$, an alternating direction method of multipliers for HLSP based on nullspace projections of active constraints. The required basis of nullspace of the active constraints is provided by a computationally efficient turnback algorithm for system dynamics discretized by the Euler method. It is based on an upper bound on the bandwidth of linearly independent column subsets within the linearized constraint matrices. Importantly, an expensive initial rank-revealing matrix factorization is unnecessary. We show how the high sparsity of the basis in the fully-actuated case can be preserved in the under-actuated case. $\mathcal{N}$ADM$_2$ consistently shows faster computations times than competing off-the-shelf solvers on NL-HLSP composed of test-functions and whole-body trajectory optimization for fully-actuated and under-actuated robotic systems. We demonstrate how the inherently lower accuracy solutions of the alternating direction method of multipliers can be used to warm-start the non-linear solver for efficient computation of high accuracy solutions to non-linear hierarchical least-squares programs.

Recent advancements in the flexible job-shop scheduling problem (FJSSP) are primarily based on deep reinforcement learning (DRL) due to its ability to generate high-quality, real-time solutions. However, DRL approaches often fail to fully harness the strengths of existing techniques such as exact methods or constraint programming (CP), which can excel at finding optimal or near-optimal solutions for smaller instances. This paper aims to integrate CP within a deep learning (DL) based methodology, leveraging the benefits of both. In this paper, we introduce a method that involves training a DL model using optimal solutions generated by CP, ensuring the model learns from high-quality data, thereby eliminating the need for the extensive exploration typical in DRL and enhancing overall performance. Further, we integrate CP into our DL framework to jointly construct solutions, utilizing DL for the initial complex stages and transitioning to CP for optimal resolution as the problem is simplified. Our hybrid approach has been extensively tested on three public FJSSP benchmarks, demonstrating superior performance over five state-of-the-art DRL approaches and a widely-used CP solver. Additionally, with the objective of exploring the application to other combinatorial optimization problems, promising preliminary results are presented on applying our hybrid approach to the traveling salesman problem, combining an exact method with a well-known DRL method.

The authors propose a new approach for solving constraint satisfaction problems (CSPs) in a neural architecture. While previous attempts in this direction have largely relied on stochastic neuron models to prevent a neural network from getting stuck in local optima (e.g. through the use of Boltzmann machines), the proposed architecture is capable of finding optimal solutions to problems using purely deterministic network dynamics. This is achieved by coupling oscillator modules such that the strength of their mutual interaction depends on their phase difference. Although the paper does not provide a rigorous theoretical explanation for it, these phase differences are observed to vary irregularly in simulations, thus providing sufficient exploratory drive to escape local optima. In small simulations of a CSP with ten binary variables and nine tertiary constraints the network is observed to find either of the two correct solutions (which satisfy all constraints) in each trial.

We present a recurrent neuronal network, modeled as a continuous-time dynamical system, that can solve constraint satisfaction problems. Discrete variables are represented by coupled Winner-Take-All (WTA) networks, and their values are encoded in localized patterns of oscillations that are learned by the recurrent weights in these networks. Constraints over the variables are encoded in the network connectivity. Although there are no sources of noise, the network can escape from local optima in its search for solutions that satisfy all constraints by modifying the effective network connectivity through oscillations. If there is no solution that satisfies all constraints, the network state changes in a seemingly random manner and its trajectory approximates a sampling procedure that selects a variable assignment with a probability that increases with the fraction of constraints satisfied by this assignment. External evidence, or input to the network, can force variables to specific values. When new inputs are applied, the network re-evaluates the entire set of variables in its search for states that satisfy the maximum number of constraints, while being consistent with the external input. Our results demonstrate that the proposed network architecture can perform a deterministic search for the optimal solution to problems with non-convex cost functions. The network is inspired by canonical microcircuit models of the cortex and suggests possible dynamical mechanisms to solve constraint satisfaction problems that can be present in biological networks, or implemented in neuromorphic electronic circuits.

We investigate the problem of learning the structure of a Markov network from data. It is shown that the structure of such networks can be described in terms of constraints which enables the use of existing solver technology with optimization capabilities to compute optimal networks starting from initial scores computed from the data. To achieve efficient encodings, we develop a novel characterization of Markov network structure using a balancing condition on the separators between cliques forming the network. The resulting translations into propositional satisfiability and its extensions such as maximum satisfiability, satisfiability modulo theories, and answer set programming, enable us to prove optimal certain networks which have been previously found by stochastic search.

The cutting plane method is an augmentative constrained optimization procedure that is often used with continuous-domain optimization techniques such as linear and convex programs. We investigate the viability of a similar idea within message passing - for integral solutions in the context of two combinatorial problems: 1) For Traveling Salesman Problem (TSP), we propose a factor-graph based on Held-Karp formulation, with an exponential number of constraint factors, each of which has an exponential but sparse tabular form.

Metric labeling is a special case of energy minimization for pairwise Markov random fields. The energy function consists of arbitrary unary potentials, and pairwise potentials that are proportional to a given metric distance function over the label set.

Symmetry breaking is a technique for speeding up propositional satisfiability testing by adding constraints to the theory that restrict the search space while preserving satisfiability. In this work, we extend symmetry breaking to the problem of model finding in weighted and unweighted relational theories, a class of problems that includes MAP inference in Markov Logic and similar statistical-relational languages. We introduce term symmetries, which are induced by an evidence set and extend to symmetries over a relational theory. We provide the important special case of term equivalent symmetries, showing that such symmetries can be found in low-degree polynomial time. We show how to break an exponential number of these symmetries with added constraints whose number is linear in the size of the domain. We demonstrate the effectiveness of these techniques through experiments in two relational domains. We also discuss the connections between relational symmetry breaking and work on lifted inference in statistical-relational reasoning.

Foundation models pre-trained on web-scale data are shown to encapsulate extensive world knowledge beneficial for robotic manipulation in the form of task planning. However, the actual physical implementation of these plans often relies on task-specific learning methods, which require significant data collection and struggle with generalizability. In this work, we introduce Robotic Manipulation through Spatial Constraints of Parts (CoPa), a novel framework that leverages the common sense knowledge embedded within foundation models to generate a sequence of 6-DoF end-effector poses for open-world robotic manipulation. Specifically, we decompose the manipulation process into two phases: task-oriented grasping and task-aware motion planning. In the task-oriented grasping phase, we employ foundation vision-language models (VLMs) to select the object's grasping part through a novel coarse-to-fine grounding mechanism. During the task-aware motion planning phase, VLMs are utilized again to identify the spatial geometry constraints of task-relevant object parts, which are then used to derive post-grasp poses. We also demonstrate how CoPa can be seamlessly integrated with existing robotic planning algorithms to accomplish complex, long-horizon tasks. Our comprehensive real-world experiments show that CoPa possesses a fine-grained physical understanding of scenes, capable of handling open-set instructions and objects with minimal prompt engineering and without additional training. Project page: https://copa-2024.github.io/

Lifted inference rules exploit symmetries for fast reasoning in statistical relational models. Computational complexity of these rules is highly dependent on the choice of the constraint language they operate on and therefore coming up with the right kind of representation is critical to the success of lifted inference. In this paper, we propose a new constraint language, called setineq, which allows subset, equality and inequality constraints, to represent substitutions over the variables in the theory. Our constraint formulation is strictly more expressive than existing representations, yet easy to operate on. We reformulate the three main lifting rules: decomposer, generalized binomial and the recently proposed single occurrence for MAP inference, to work with our constraint representation. Experiments on benchmark MLNs for exact and sampling based inference demonstrate the effectiveness of our approach over several other existing techniques.