Generalized Conflict Learning for Hybrid Discrete/Linear Optimization

Abstract

Abstract. Conflict-directed search algorithms have formed the core of practical, model-based reasoning systems for the last three decades. At the core of many of these applications is a series of discrete constraint optimization problems and a conflict-directed search algorithm, which uses conflicts in the forward search step to focus search away from known infeasibilities and towards the optimal feasible solution. In the arena of model-based autonomy, deep space probes have given way to more agile vehicles, such as coordinated vehicle control, which must robustly control their continuous dynamics. Controlling these systems requires optimizing over continuous, as well as discrete variables, using linear as well as logical constraints. This paper explores the development of algorithms for solving hybrid discrete/linear optimization problems that use conflicts in the forward search direction, carried from the conflict-directed search algorithm in model-based reasoning. We introduce a novel algorithm called Generalized Conflict-Directed Branch and Bound (GCD-BB). GCD-BB extends traditional Branch and Bound (B&B), by first constructing conflicts from nodes of the search tree that are found to be infeasible or suboptimal, and then by using these conflicts to guide the forward search away from known infeasible and sub-optimal states. Evaluated empirically on a range of test problems of coordinated air vehicle control, GCD-BB demonstrates a substantial improvement in performance compared to a traditional B&B algorithm applied to either disjunctive linear programs or an equivalent binary integer programming encoding.

A probabilistic approach to optimal robust path planning with obstacles

“…The robust probabilistic path planning problem with obstacles can therefore be expressed as a Disjunctive Linear Program, and solved efficiently using existing methods [11]. The additional computational complexity required to handle uncertainty compared to the original, deterministic path planning problem, is small, consisting only of a single lookup table evaluation per constraint of the original problem.…”

“…By including temporally flexible state plans, [10] was able to generate optimal trajectories for UAVs with time-critical mission plans. [11] showed that using Disjunctive Linear Programming [12] rather than Mixed-Integer Linear Programming, and using conflicts [13] to guide the search process leads to more efficiency in solving the optimization problem. As an alternative to these constrained optimization approaches, [14] used a randomized learning-and-query approach known as Probabilistic Roadmaps.…”

“…The key insight is that the probability of colliding with an obstacle can be upper bounded using a disjunction of linear chance constraints. With our method, these chance constraints are then converted to deterministic linear constraints [18] in order to yield a Disjunctive Linear Program that can be solved using techniques such as those developed by [11], with similar computational complexity as the original path planning problem, without uncertainty.…”

Regression-based LP solver for chance-constrained finite horizon optimal control with nonconvex constraints

“…We proposed a bounding approach in [1], whereby the relaxed problems are constructed by removing all constraints below the corresponding disjunction. This approach was used by [16] and [17] for a different problem known as disjunctive linear programming.…”

A cell decomposition approach to cooperative path planning and collision avoidance via disjunctive programming

“…Another disadvantage of existing approaches is that, in the presence of many obstacles or long time horizons, the solutions becomes computationally prohibitive. Moreover, existing DP approaches cannot be used to find paths in environments containing concave obstacles or narrow passages, because the corresponding linear boundary constraints can give rise to an infeasible mixedinteger program [14], [15].…”