[Paper Review] Complex Number Formulation and Convex Relaxations for Aircraft Conflict Resolution

David Rey, Hassan Hijazi|arXiv (Cornell University)|Mar 20, 2017

Spacecraft Dynamics and Control4 citations

TL;DR

This paper introduces a complex number formulation for aircraft conflict resolution that models speed and heading control as quadratic constraints, enabling tight convex relaxations via MIQP and MIQCP. The approach achieves global optimality for previously open instances and reduces computation time by up to two orders of magnitude on standard benchmarks.

ABSTRACT

We present a novel complex number formulation along with tight convex relaxations for the aircraft conflict resolution problem. Our approach combines both speed and heading control and provides global optimality guarantees despite non-convexities in the feasible region. As a side result, we present a new characterization of the conflict separation condition in the form of disjunctive linear constraints. Our formulation features one binary variable per pair of aircraft, is free of trigonometric functions, and captures the non-convexity in a set of quadratic concave constraints. Using our approach, we are able to close a number of open instances and reduce computational time by up to two orders of magnitude on standard instances.

Motivation & Objective

• Address the non-convexity in aircraft conflict resolution due to trigonometric constraints in speed and heading control.
• Develop a global optimization framework that guarantees optimality despite non-convex feasible regions.
• Close open instances in standard benchmark sets that were previously unsolved with existing methods.
• Reduce computational time significantly compared to state-of-the-art exact approaches.
• Provide a scalable solution that handles up to 40 aircraft with high conflict density using a three-step algorithmic strategy.

Proposed method

• Represent aircraft velocity control using complex numbers to unify speed and heading adjustments into a single mathematical framework.
• Formulate the conflict separation condition as a disjunctive linear constraint based on relative position and velocity vectors.
• Model trajectory crossings via a single disjunction that captures the order of aircraft passage at intersection points.
• Derive convex relaxations in the form of Mixed-Integer Quadratic Programs (MIQP) and Mixed-Integer Quadratically-Constrained Programs (MIQCP).
• Implement a three-step algorithm: first solve MIQP relaxation, then MIQCP if needed, and finally use NLP for local solutions when gaps remain.
• Use bound-violating constraints to guide heuristic recovery and improve solution quality in high-density conflict scenarios.

Experimental results

Research questions

• RQ1 Can a complex number formulation effectively capture the non-convex dynamics of speed and heading control in aircraft conflict resolution?
• RQ2 How tight are the convex relaxations (MIQP and MIQCP) in approximating the true non-convex problem, and do they yield global optimality?
• RQ3 Can the proposed method resolve previously open benchmark instances that were intractable with prior exact approaches?
• RQ4 To what extent does the method reduce computational time compared to existing state-of-the-art algorithms?
• RQ5 How does the algorithm perform on high-density conflict scenarios with up to 40 aircraft?

Key findings

• The proposed method solved all 10- and 20-aircraft random circle problem (RCP) instances to global optimality within one second on average.
• For 30-aircraft RCP instances, 71% were solved to global optimality in one step, and an additional 12% in two steps, with an average optimality gap of 4.8% for the remaining 17%.
• Among 40-aircraft RCP instances, only 17% were solved to global optimality, but 75% were solved to local optimality with an average optimality gap of 13.9% and standard deviation of 16.1%.
• The method reduced computational time by up to two orders of magnitude compared to existing approaches on standard benchmark problems.
• The algorithm successfully closed multiple open instances, extending the state of the art beyond the previous limit of 7 aircraft.
• The MIQP relaxation was sufficient for 100% of 10- and 20-aircraft instances, while MIQCP was required for 17% of 30-aircraft and 75% of 40-aircraft instances due to higher bound violations.